https://www.youtube.com/watch?v=lvmzgVtZtUQ

There's a really important step missing in the article which has been aluded to by a couple of comments already. It's the step between:

   1. a series of notes defined by the harmonic series, that is, exact integer multiples (or ratios, if you prefer) of a base frequency, a physical phenomena found in nature

   2. the well-tempered scales used by most western music, which adjusts the ratios to allow certain compositional "tricks" to be used without sounding dissonant.

The "problem" with just intonation, if indeed it is a problem, is that if you define two series of notes (call them a scale, if you like) using these integer ratios but starting from two different notes, you will trivially find cases where "the next higher E" is a different frequency. So starting from (say) A and moving by integer ratios gets you to a different (say) F than if you start from (say) B.

This means that if you are writing in a just intonation scale with (say) A as the root, you have a set of notes that are not actually the same frequencies as if your scale started on (say) B.

By itself, this is not a problem at all - there is all kinds of lovely music written through history that works just fine with notes and scales defined this way. You just stay in the same scale throughout, and there are no issues. There are even some scale changes you can make that still work, you just have to know what they are (and they depend on the root note and the set of integer ratios you're using, so it gets complicated).

But ... somewhere in the period mentioned above, a subset of western musical culture started to want to experiment with "modulation" - changing from one scale to another in the midst of piece. On continuous pitch instruments (e.g. violin, voice), this is entirely possible to do, since they can play any frequency in their range at all. However, it does require significant skill on the part of the performer, since the pitch of a (say) "F" will differ depending on the scale currently in use in the piece.

The breaking point, such as it was, came with the development of fixed pitch instruments (keyboards). These can only play the notes they are currently tuned to, and so if the (say) F in two scales is a different frequency, you cannot play in both scales without retuning - an obvious impossibility in the middle of a piece.

So, "well-tempered" tuning was developed - the ratios described in TFA. These are tweaked by relatively small amounts so that the notes are close to where Just Intonation (integer ratios) would have placed them, but not precisely the same. These small shifts mean that the (say) "F" is the same frequency whether your scale began on (say) A or B. You can now play a fixed-tuning instrument like a harpsichord or a piano, change from one scale to another, and everything remains "in tune".

Of course, to ears used to Just Intonation, "well-tempered" tuning sounds out of tune. But in the west, most people (even within our musical academies) have grown up used to the sound of well-tempered tuning, and it is Just Intonation that sounds "off".

A nice example is this:

On an 7 octave (85 key) piano, if you start on the lowest note and go up in perfect fifths, after going up 12 fifths you will be on the highest note, i.e. up 7 octaves.

In just intonation you multiply the frequency by 3/2 to go up a fifth, and you multiply the frequency by 2 to go up an octave.

But clearly (3/2)^12 is not equal to 2^7 (129.7ish versus 128).

Does it really? Listeners are pretty forgiving, so I think in practice, you will just have drift away from concert pitch, and you maybe just have to be a bit careful because the open strings will jolt you back to concert pitch.

But in a capella vocal singing, there's no such forcing function, so you just need the ensemble to be good at locking in with each other, especially on long chords where you really want the audience to feel the consonance or dissonance. Which isn't simple, but it's also not rocket science.

It would be very hard to perform exact pitches within a couple cents of exact frequencies, if that were necessary. I just think theorists vastly overstate how much listeners care about tuning in most cases. I made this mistake myself. For research, I was interested in creating a sythesizer that allowed composers to switch between tuning schemes mid piece. And I could barely hear the difference, and much to my disappointment, it wasn't very compositionally useful.

Good point about the open strings though. Less of an issue on instruments from other cultures like the kamancheh, where you would rarely play an open string, but certainly true on multi-strings where the open string situation is "oft-used".

https://en.wikipedia.org/wiki/Syntonic_comma#Comma_pump

Also, my understanding is that a big part of the magic of barbershop quartet singing is ringing (https://en.wikipedia.org/wiki/Barbershop_music#Ringing_chord...). Those chords that sound so pure that it gives you goosebumps. That requires singing in just intonation.

The interesting thing about the Barbershop 7th is that it doesn't function like a dominant 7th chord, which feels unstable and pulls towards the relative tonic chord (e.g. G7 -> C). The Barbershop 7th chord has a slightly flattened 7th to line it up with the overtones of the root note, blending it in and making it stable.

You might say it's a microtonal effect in that you hypothetically could notate and sing a C7 differently depending on its function in the piece. I don't know whether this is done in practice, though.

The order of intervals in the harmonic series is:

Octave

Perfect Fifth

Perfect Fourth

Major Third

Minor Third

Subminor Third

Diminished Third

Supermajor Second

Major Second

Neutral Second

Minor Second

And that's just the leading edge of the series. Between distant notes in the series you will find the neutral third, subfourth, superfourth, superaugmented fourth, subfifth, superaugmented fifth, supermajor sixth, and neutral seventh.

If you want to claim that our intervals all come from the harmonic series, then in order to include the Minor Second, which everyone would agree is pretty dang important for our music, you must also include all the intervals listed above that nobody's ever heard of. It's simply wrong.

The major scale has nothing to do with the harmonic series except by coincidence. The Tritone for instance, never appears in the harmonic series, and any music theorist will tell you that's way more important to our music than the Minor Second.

> The Tritone for instance, never appears in the harmonic series

The first of multiple tritones that show up is the one from harmonic 5 to harmonic 7. We're not talking weird abstract, esoteric harmonics here.

The major scale has SOMETHING to do with harmonics of course. It's true that Just Intonation purists can be dogmatic and make claims that don't hold up to scrutiny, but jumping to the conclusion that their claims have NOTHING to them is even more stupid a position.

The major scale became the reference scale in chordal music specifically because of the way it facilitates blended harmonies in closely related chords. It's because people discovered the blend of the major chord which is a thing because of how harmonics work. And you put three overlapping major chords together and you get the major scale.

Before chordal-harmony, other modes and scales were just as or more prominent.

The harmonic series helps to explain the relationships between the twelve tones that we (people composing and performing in "western" musical traditions) use to divide the octave. It's likely not a coincidence that many other cultures, despite their use of additional tones within the octave, also seem to "stumbled" as we did onto the properties we get from 12T-per-octave (regardless of the precise tuning of those 12 tones).

I'm not sure where you are getting your information about tritones from. Everything I've learnt about tritones would say that you're wrong to claim they occur between the 5th and 7th harmonic. Got any references on that?

Also, this thing about communities of experts ... one of the problems with this is that at least in the USA and Europe, the majority of the community of experts appears to have deliberately constrained their expertise to a somewhat narrow view of tuning. There's a reason why Bohlen-Pierce and the tunings of Harry Partch are considered "out there": it's not just that they are quite challenging for 12T(ET) ears to listen to, but also that our musical academies have shrunk the possibilities for tuning down to a sadly constrained range. Consider for example, the bulk of musical set theory, which is incredibly interesting at the scale (mode) level, but still appears from the literature to overwhelming assume only a handful of possible tuning systems (and a big chunk of it considers only one possible tuning system).

The tritone point doesn't need any reference. Go and play the 5th harmonic and then the 7th harmonic on a string. That's an interval of a tritone. Tritone means 3 whole steps, 3 whole tones. It's a general scale-based interval amount.

The 7/5 ratio is 583¢, that's a tritone. It's an interval that takes 3 whole tones to reach.

Put in traditional language, the 5th harmonic is a major third (2 octaves above the fundamental) and the 7th is the minor 7th (2 octaves above the fundamental). And the interval from a major third to a minor seventh is a tritone.

There's no references needed.

> the majority of the community of experts appears to have deliberately constrained their expertise to a somewhat narrow view of tuning

Exactly, that's my point. All those so-called experts are basically ignorant to be blunt. What I mean is that there ARE people out there (on the margins to a degree), all the people who develop https://en.xen.wiki/ for example, and from their (okay, our, I'm one of them) perspective, all the chatter here is a mix of thoughtful people like you trying to talk some sense with a bunch of ignorant folks who are clueless.

So, you have more than some clues, you seem to have some knowledge, but you would be a beginning student in the world of people who actually understand the topic of tuning. So, understanding of this stuff exists in the world, and the reasons that it remains on the margins are themselves also understood. Suffice to say, Partch being a reference known by the already small subset of clued-in people is already pretty tiny. His work is foundational but quirky and old and nowhere near reaching real understanding. A lot of people into tuning are intentionally weird sounding, so it's hard to separate that form the topic. And if you're NOT weird sounding, then nobody is noticing the tuning. As soon as the tuning draws your attention, it's already weird. It's all too much to get into here. It does seem that there's some prospect of breaking out of this situation in the next few years finally, but don't hold your breath.

You're using "harmonic" here in a way this is disconnected from a lot of the rest of the discussion here. We are not talking about scale degrees or anything like that, but the physical harmonic series. The 5th and 7th partial of the harmonic series has nothing to do with the 11th, which the basis for the ratio that normally is identified as the tritone, modulo some culture-specific tweaks to fit in with other musical practice (eg. definining it as 64:45 or 7:5 or sqrt(2) in various tuning systems).

> There's no references needed.

There's plenty of existing references that make it clear that the tritone is based on the 11th harmonic of the harmonic series, folded down into the octave as a ratio of 11:8. The "tritone is 3 whole steps" is a tuning-specific simplification of this.

And yeah, I know the xen site, and have a learned a lot from it. I still see some fairly clear impositions of "worldviews" on some of the material there, maybe not as bad as it would be at a music school, but not quite as context-free as some people would prefer.

I'm the author of a cross-platform DAW, and one of my goals for the next few years it to get support for arbitrary tuning systems deeply embedded into the software.

This is such a confused statement. I'm talking about the physical harmonic series, and OH!! I see the CONFUSION NOW HAHAHA. I do NOT MEAN the 3rd and 4th harmonics that HAPPEN to be on the 5th and 7th frets!!

I meant the actual harmonic series, the 5th and 7th, which is at the 4th fret and in the middle between the 2nd and 3rd frets (and also available at other nodes along the string).

The 5th and 7th HARMONICS in the harmonic series are a 5:7 ratio, which is a tritone!

What this means is:

The tritone is NOT based on the 11th harmonic which is not used in much Western music and is significantly flat. It's more of a 2.75 tone than a tri (3) tone. Yes, it's the closest tritone that is precisely a tritone above some octave of the fundamental, but octaves are a real thing. Saying that 11/8 is a tritone is not more compelling than saying that 7/5 is a tritone. The actual 11th harmonic isn't a tritone, it's three octaves plus an almost-tritone.

The idea of deferring to the harmonic series but sticking strictly to the fundamental as the reference and yet also ignoring octaves is a few layers of common mistake in this topic.

Here's how the 7/5 tritone works: It literally is the major third to the minor 7th of a harmonic 7th chord. In other words, when you play it on two notes of an instrument, you play that tritone, first, it sounds like a tritone because it is one. Second, the two notes are part of a harmonic series that starts a couple octaves and a major third below the notes you are playing. With difference tones on a distorted guitar, you'll actually generate the missing fundamental.

Put another way, when you play 7/5 tritone calling it C and G♭, what you're doing is creating a harmony that is part of an A♭7 chord and you're not playing the A♭. There's no reason that the concept of tritone above C needs to be within a harmonic series starting on C, that's not how this stuff works.

I'm not using letters because it's how I think. I understand the whole system of JI theory and prefer thinking in relations and ratios. I'm using letters so you can follow clearly.

The same issue arises with minor thirds. You could erroneously say that the first minor third in the harmonic series is the 19th harmonic because you are stuck on the idea of a minor third strictly above some octave of the fundamental. But the minor thirds that are prominent and make the harmonic basis for minor third intervals in blended chords are the ratios 6/5 and 7/6, which means the intervals from the fifth to the sixth and from the sixth to the seventh harmonics of the series. Incidentally, the 6/5 minor third might be called the upminor or large-minor, it's wider than 12edo. The 7/6 is septimal minor or small-minor or downminor. It's awesome and bluesy. Neither sounds just like 12edo, and they are different from one another, but there's no doubt that these are minor thirds. Consider the standard premise that the minor third is the difference between the major third and perfect fifth — that's what 6/5 is, 5/4 is the major third, and 5/4 * 6/5 = 3/2. And 7/6 is appropriately described as the minor third that you get from the fifth to the seventh of a harmonically-tuned dominant seventh chord.

> I still see some fairly clear impositions of "worldviews" on some of the material [at xen site]

Yes, I agree with you completely. In fact, I'm fairly critical of aspects of the xen community too. They get too dogmatic or mathy without enough give for psychology and real-world practice. I don't defer blindly to them or agree with the community on everything. But there are some people in the community with real depth of understanding without problematic dogma or cultural bias. Consider that Partch fits in a quirky part of that community effectively, and his works and concepts are valuable but are also a quite incomplete and debatable angle.

> I'm the author of a cross-platform DAW, and one of my goals for the next few years it to get support for arbitrary tuning systems deeply embedded into the software.

YES I KNOW, I'M A PAID MONTHLY SUPPORTER, and THANK YOU THANK YOU!

I am a dedicated Free/Libre/Open advocate who uses and promotes Ardour to students and everyone. And having good support for tuning is a key thing I value in music tools. I'm very excited to hear of your interest! Maybe I should post on the Ardour forum and have a different and more productive engagement than this awkward chat here.

Oh, and P.S. : on discourse.ardour.org we ban the use of western classical terminology when discussing tuning :)))

I will quote to you from wikipedia:

> The harmonic series is an arithmetic progression (f, 2f, 3f, 4f, 5f, ...). In terms of frequency (measured in cycles per second, or hertz, where f is the fundamental frequency),

> The second harmonic, whose frequency is twice the fundamental, sounds an octave higher; the third harmonic, three times the frequency of the fundamental, sounds a perfect fifth above the second harmonic. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth above the third harmonic (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher).

> The frequencies of the harmonic series, being integer multiples of the fundamental frequency, are naturally related to each other by whole-numbered ratios and small whole-numbered ratios are likely the basis of the consonance of musical intervals

https://en.wikipedia.org/wiki/Harmonic_series_(music)

What you are describing is a musically significant set of intervals that are merely a subset of the harmonic series, created by what the wikipedia page describes as:

> If the harmonics are octave displaced and compressed into the span of one octave, some of them are approximated by the notes of what the West has adopted as the chromatic scale based on the fundamental tone

is not the same thing as

> the chromatic scale is derived from the harmonic series

which is what the OP article claims.

You can find the chromatic scale in the harmonic series, yeah, if you ignore the majority of the notes in the harmonic series.

To find the chromatic scale in the harmonic series, you need to take the 2nd, 3rd, 4th, 5th, 9th, 15th, and 44th harmonics, and ignore of the rest. That's not a mathematically justified derivation, that's a post-hoc rationalization built on coincidence alone.

(in pitch/frequency order) 1st, 17th, 9th, 19th, 5th, 21st, 11th, 3rd, 13th, 27th, 7th, 15th

In harmonic order: 1,3,7,9,11,13,15,17,19,21,27

So, sure, fair question why these harmonics and not any of the others?

Well, powers of 2 are out because they are just higher octaves. Then we have a whole series of harmonics that are equivalent ratios to the fundamental when folded down into the octave range (3 (3:2),6 (6:4), 12 (12:8)), (5 (5:4), 10 (10:8)), (7,14,28), (13,26) and so on.

You'll notice the pattern: the harmonics the define the intervals in a 12T system are those that introduce new ratios into the list of intervals, so they lean toward being prime or only having factors not already introduced.

By the time you go through the list, it's easy to see that going up to the 31st harmonic really only leaves out a couple of possibilities from a 12T system: 25, 29, 31 and as far as I am aware this is because introducing them into the pitch class produces results extremely close to already existing members.

And sure, you could go higher, but the pattern will repeat: harmonics whose ratio folded into the octave range are identical, or which give rise to pitches extremely close to pitches defined already.

It's not a post-hoc rationalization at all.

Let's consider the major sixth.

It is the 27th harmonic of the fundamental. Expressing that in the conventional octave range (1:1 .. 2:1) requires us to write it as 27:16. This is Pythagorean major sixth.

JI can use the same ratio, but it is common there to use 5-limit tuning, which builds all intervals from ratios built from powers of 2, 3 and 5. This is conceptually equivalent to the sort of "ratio distortion" that occurs with ET, though for an entirely different purpose.

The Pythagorean major sixth (27:16) is expressed in decimal form as 1.6875. The closest 5-limit tuning ratio to that is 5:3, or 1.6666.... By contrast, the ET major sixth is 2^(9⁄12) or 1.681793.

The description of the 5-limit tuning major sixth as 5:3 is no different in its deviation from the Pythagorean version (27:16) than the reason why the ET version also does not match the ratio given by the harmonic series: musical/compositional preference. Neither version precisely matches the harmonic series, but has its own musically-rooted reasons to use a nearby ratio (that will sometimes be expressed with denominators that are not powers of two).

I would also note that in some other musical cultures, they also express the same musical ambivalence. In some Indian scales for example the pitch denoted "Dha" can be either 27:16 or 5:3 with respect to "Sa", the fundamental.

So sure, in one sense, "the harmonic series cannot explain frequency ratios which does not have a power of 2 as a denominator" is true. But the full explanation is that "the real ratio is fully explained by the harmonic series, but for performance/instrument/compositional reasons, many music cultures use a ratio that deviates from the real value, just as is the case for much of ET tuning and other ratios derived from the harmonic series".

>It is the 27th harmonic of the fundamental.

This is the problem I have with discussions concerning the harmonic series, it is presented as equivalent to the notes of the scale, even though no musician before the 19th century was even aware of it. Even in the heyday of just intonation (in theory not in practice), no theorist used the Pythagorean ratio for it, the very simple reason being that the intervals were obtained via dividing the octave and subsequent ratios. This is how it was done in the most important treatises such as Zarlino, Galilei etc, and subsequent theorist followed in their footsteps. The 5:3 major sixth differs from the ET and Pythagorean one for one very good reason: it was the one derivation that theorists actually used. Though how the major sixth was used in practice is a much more difficult question, and that was the reason for Galilei's break with Zarlino showed.

I just want to re-iterate that all these musical concepts have existed long before the discovery of the harmonic series, and were justified differently. The harmonic series definitely does not "explain fully" any ratio as this comment thread has amply shown. It is an invention of the 19th century that people have tried to bolt on to existing concepts which nevertheless requires significant contortions and even so come up deficient.

For the audience, I highly recommend reading through the first chapter of Galilei's Dialogo which presents a historical derivation of the intervals of the scale and the many difficulties attending to just intonation.

https://digital.library.unt.edu/ark:/67531/metadc500437/

1. a claim about the history of how common western pitches were selected over time.

2. a claim about acoustics and physics shaping the overall process of how humans (and some other animals) select pitches.

I am not suggesting for one second that the actual history of the 12T scale has been explicitly rooted in an understanding of the actual physical harmonic series.

I am suggesting that many of the choices made throughout time, both in western europe and other cultures, and also by some other species that sing (e.g. some birds) have been shaped by the physics of the harmonic series.

From: https://musicscience.net/2021/09/12/separating-the-cultural-...

> "The preference for consonance varies across cultures but the aversion to harsh dissonance is universal"

Just browsing the paper and I come up with some shocking inaccuracies, consider

>Notably, low preference for the major triad among the Northwest Pakistani tribes is not corroborating the theory according to which the attractiveness of consonance is due to harmonic similarity to human vocalizations and is in line with the historical observation according to which the major third became consonant only over time in the framework of Western music as well.

The preference of the Pakistani tribes for the minor triad over the major triad (which is shown in the paper) is most certainly not in line with the gradual acceptance of the major third in Western music for the very simple reason that in the west the two types of thirds were accepted together. And if we consider the acceptance of thirds in the final chord of the piece, the western experience is exactly the mirror of the Pakistani tribes: the minor triad was deemed less 'stable' than the major triad and it took much longer for the minor triad to be regarded as acceptable to the ending of pieces.

Shoddy research like this which betray a lack of historical awareness is why I have very low opinion of most these types of social research. They are not good writers on music, and some, I assume, are good people.

[1] https://nyaspubs.onlinelibrary.wiley.com/doi/full/10.1111/ny...

Again, Plomp & Levelt (1965!). It's not about "simple ratios".

You can try this on a guitar. If you pluck a string right in the middle to kill all the even harmonics, major 7ths and minor 9ths don't sound so dissonant as they did when the original string had its 2nd harmonic octave present.

The tritone is the eleventh harmonic of the series, folded back down into the octave to give a ratio of ... "well, it depends".

Again, from wikipedia:

> The ratio of the eleventh harmonic, 11:8 (551.318 cents; approximated as Fhalf sharp4 above C1), known as the lesser undecimal tritone or undecimal semi-augmented fourth, is found in some just tunings and on many instruments. For example, very long alphorns may reach the twelfth harmonic and transcriptions of their music usually show the eleventh harmonic sharp (F♯ above C, for example), as in Brahms's First Symphony

The actual 11th harmonic is the neutral second against the 10th harmonic, which corresponds to the super-fourth compared to the root.

It appears fairly conventional to say at least that the lesser undecimal tritone is the 11th harmonic, which corresponds to a ratio (folded into the octave) of 11/8, but other defintions (64/45 or 45/32 for example) also exist.

Like, if you want to say that our musical system revolves aroundt the frequency ratios of the octave (1:2) and fifth (2:3), then that's one thing and it's basically correct.

But that's very different from claiming that western music is derived from the universal mathematical properties of the harmonic series, and is entirely irrelevant for making progress towards actual important concepts in music theory, like octave equivalence, let alone scales or dominant resolutions.

Western (classical) music is marked (one might even say "distinguished") by its attention to harmonicity, which in turns is rooted in ideas about consonance and dissonace. These ideas are absolutely rooted in the mathematical properties of the harmonic series. As noted elsewhere in the comments, the work of Plomp & Levelt and then Sethares has shown how human perception of consonance (with dissonance being its reciprocal) is rooted in the averaged amplitude-weighted sum of the pair-wise disonnances between all the partials.

There's nothing inherently "good" or "bad" about dissonances and consonances, the Western ear has been trained to recognise instability in dissonances and stability in consonances, hence the ancient prohibition of dissonances in the final chord of the piece (a rule that has since been discarded). The trajectory of classical music in the 20th century shows that people can indeed get used to music which do not take that as granted. In fact, the almost unspeakable secret about dissonances is that they sound as good, even better than consonances. How else can the suspension play such an important part in western music up until then?

Sethares is of course free to define consonance and dissonance in his idiosyncratic way unconnected to common usage, but of course his definition and discoveries would then be of no use to those who use the ordinary definition.

If you define a term in such a way that it is completely culturally determined, you can then prove this concept is subjective and culturally determined?

The 4th against the bass changes the perception of the root. This is why it was avoided. Using the word dissonance to refer to this is just confusing.

The definition I use and is used in the work you are rebutting is consistent, has a physics explanation and is largely cross cultural.

At best all that is going on here is a disagreement about the definition of a word.

At worst this discussion is some sort of proxy for a metaphysical disagreement about postmodernism and the subjectivity and cultural framing of all reality.

If it was true that western musical ideas of consonance is rooted in simple harmonic ratios, and that our musical scales were based on the harmonic series, then we should be seeing subminor thirds (6:7) and supermajor seconds (7:8) everywhere. They are way more "consonant" than major seconds (9:10), or minor seconds (15:16). But in reality, they are seen as exotic and wrong.

This is a good overview (linked elsewhere in the comments): https://sethares.engr.wisc.edu/consemi.html

Sethares extended their conclusions a bit by noting that since the partial spectrum is the definition of timbre, the most consonant/dissonant intervals would vary by timbre, which they claimed is observed in the real world.

(1) Indeed, the first step of their work, defining the "human dissonance response curve" relies on being able to use a signal generator to create pure sine tones.

Even as you pointed out some of the harmonic series tracks closely to the scale tones, and the relationship between the harmonic series and the scale tones is a useful unit of analysis towards understanding consonance and dissonance and such things.

I think of the 12 tone system as a technology.

Somehow the sound of 12 tones took root, like a meme, we don't know when or why, but it's like the Omicron of tuning systems, and 2500+ years later, it's pandemic.

We know that as a technology, it has co-evolved with culture. Technologies do that. We know something about its history based on things that people wrote. For instance the harmonic series was known to the ancient Greeks (attributed to Pythagoras). Music that used the harmonic series probably existed before Pythagoras, for him to have discovered its underlying secret. We know that they used "diatonic" scales, but don't precisely know which notes they chose for which scales, for instance whether their Dorian scale is the same as ours.

I suspect that the harmonic series made it easy for musicians to make and tune their own instruments, when that became important to them. Musical culture did not just consist of music, but also knowledge about making and playing instruments, and singing. That's what I mean by a technology. A harpsichord needed to be tuned before every performance, by the musician. The need for a tune-able music technology persisted for centuries, swept along by the 12 tone pandemic.

Jacob Collier has a good example of this[1] (I'm a huge fan).

Separately, I've previously heard this referred to as "equal temperament" tuning which makes me wonder what led to this difference in wording.

[1]: https://www.youtube.com/watch?v=XwRSS7jeo5s

Using the harmonic series to derive the notes of the scale is a distinctly modern phenomenon, and it does not represent how musicians thought about the issue in history. There are already presented plenty of criticisms to this perspective, so I will simply give the historical view of deriving the diatonic notes(or "Just Intonation") via ratios alla Galilei. Here the larger interval is always divide into two ratios which when multiplied together gives back the original ratio. The numbers in the smaller ratios are selected to be as small as possible. This is repeated until you have all the intervals. We start with the 8ve (2:1) since that's the simplest possible ratio which is not unity.

The 8ve (2:1) is divided into two unequal portions: the 4th (4:3) and the 5th (3:2).

The 5th (3:2) is divided into two unequal portions: the major 3rd (5:4) and the minor 3rd (6:5).

The major 3rd (5:4) is divided into two unequal portions: the major tone (9:8) and the minor third (10:9). Note that there are actually two different tones!

The major tone (9:8) is divided into two unequal portions: the major semitone (16:15) and the minor semitone (25:24). There are also two different semitones!

Now the rest of the intervals in the octave can be obtained by adding up the intervals:

The major 6th (5:3) is the fourth (4:3) plus the major third (5:4).

The minor 6th (8:5) is the fifth (3:2) plus the major semitone (16:15).

The minor 7th (15:8) is the fifth (3:2) plus the major third (5:4).

The minor 7th (9:5) is the fifth (3:2) plus the minor third (6:5).

Note that Galilei takes all the ratios for the intervals as granted, so these "derivations" are no less arbitrary than those from the harmonic series. But it at least has historical standing and is in my opinion more aesthetically pleasing. If you keep on going and dividing the ratios derived with each other you'll sometimes end up with ratios already derived, and sometimes ratios which are slightly different, this is the root of the issue with Just Intonation, and Galilei spends the second half of his treatise explaining precisely why nobody uses it in real life.

Music theory is, in my experience, typically taught as a list of facts to remember. Deriving it from first principles, insofar as it can be, is not common. This article is attempting to start that process.

Some commentators are saying that the article doesn't address why we have 12 notes but I wonder if they even know why (one answer is because it's a good compromise between number of notes and that have simple/reduce ratios to each other [0]). I'm also skeptical of music theorists that can't even attempt an answer to basic questions about note length frequency, why some chords are "sad" or "happy" and other basic questions. It's difficult because music theory is a hodge podge of theory that's attempting to describe what's effectively an evolved language (with different music evolution for different regions), but there are some basic tenets that probably apply.

I don't claim to have deep knowledge but there are a few key facts about music and music theory that are really obscure unless you know where to look. Any attempt at coming to music theory from a more rigorous foundation should be encouraged.

[0] "Measures of Consonances in a Goodness-of-fit Model for Equal-tempered Scales" by Aline Honigh (https://github.com/abetusk/papers/blob/release/Music/measure...)

EDIT: corrected spelling (thanks for the comments pointing it out)

I've never met a jazz pianist that for example will work under the assumption that a 2-5-1 sounds sad-transition-happy, it sounds like a 2-5-1 because the pianist has a sophisticated enough musical understanding to have a concept on their brain that is a 2-5-1 and has a specific sound that is associated to it. It's the whole deal with ear training intervals, scales and chords. The whole major happy minor sad thing is like a pedagogy trick to get people started into getting those kind of sounds into their language. Same thing with intervals, a minor third is maybe sad but then a whole or a half step on the major scale are what? super sad? they sound like a train wreck harmonically but happy when you play them all together on the scale? What's happening is that people just associate musical phrases, sounds, harmony, to stuff and that's kinda outside of musical theoretic research and more like getting into the fields of cognitive science or psychology.

Have you actually asked any music theorist about this? I would be surprised if you hadn't gotten this kind of answer because in my experience talking to working theorists over lunch or on social events they usually agree on this.

I tend to think of things in consonance vs dissonance, and different colors for chord qualities and voicings. Where the goal is managing contrast/tension and release.

I am no expert in music theory but I've been playing for most of my life and write music for fun. I know that A4 is 440 and that's about it. I don't know how knowing any other frequency, or even that one for that matter, would ever be helpful in me reading a chord sheet and walking a bass line to it nor would it be very helpful in me writing a melody or chord progression for a song I might be working on.

As far as I know, the happy/sadness qualities of chords isn't a universal thing. It's likely not a question for music theory to explain.

I read through the article a bit and I think it's fine but the way most people learn music theory is by first learning an instrument and learning the relevant ideas associated with it and if still interested will likely move onto more advanced concepts. It's like any other field really it all sort of builds on each other. Some of it has to be rigorous I think. If you can't read music it's probably not going to be very useful to know G->B is a major 3rd. The author says they want to write music which you don't really need to know music theory to do but you need to learn to walk before you run and it's not clear to me whether the author even knows how to play an instrument.

This learning style is not universal and, in my view, why this article is specifically titled "Music theory for nerds". Programmers especially can make highly complex music without ever learning a classic instrument. Algoraves, chiptune, generative music, DAWs, real-time music programming environments etc. are all a thing and participants often don't need any concept of a "real" instrument to create quality music therein.

I think you're right about the happy/sad being a cultural quality but there might be portions of it that might be explained by other means. Here's my attempt, which is almost surely at least partly wrong: "Major" chords have more constructively interfering waves than do "Minor" chords, especially in the lower frequencies, where more of the power lays.

I would say this more as programming and math centered folks prefer subjects that are amenable to first principle analysis, which is what drew them to math and programming in the first place. And folks with that mental inclination will tend to use those tools to bear on all problems and subjects. When all you have is a hammer, etc.

But it's important to remember that not every subject is built from first principles. Not everything in the world is reducible to a simple system with emergent properties. There is no Grand Unified Theory of history, no five axioms of mammalian biology. And, while I like the mother sauces, there is no culinary theory that can logically prove which recipes will taste good.

We saw attempt to formalize in the 50s and 60s around spoken language. Chomsky and other linguists hoped to fully systematize spoken languages and analyze them according to formal grammars. Computer scientistics hoped to figure out all the rules of human languages so precisely that computers could parse them with perfect accuracy. The efforts failed. It turns out that while there are certainly patterns and grammars to how humans speak, the rules are fuzzy, ever-changing and deliberately broken.

Music theory is also particularly tempting for people who like thinking mathematically, because it does have some numbers and stuff around frequency, intervals, etc. But music is about 10% systematic and 90% arbitrary cultural history. One can certainly focus on the subset that fits in that 10% and have an enjoyable experience making music while doing so. But a whole lot of music won't "make sense" in any real way if you discount "because people did it that way and others got used to hearing it" as a valid justification for some musical practice.

There is no mathematically sound reason why we prefer the sound of a stretched out metal spring rattling against a taut piece of plastic on the second and fourth beat of every measure. Any explanation for why thousands of songs do that today has to involve discussions of goat skin, the Atlantic slave trade, the American fad for Hawaiian music in the 1930s, etc.

Also, I guess my comment does sort of imply that I meant a classical instrument but arguably some of the things you listed could also be described as instruments, in which case, my point still stands. If you want to learn western music theory with the intentions of writing music, which is what this article is describing more or less, you need to have a way to make western music, whether that be a cello, a tracker, or a gameboy it doesn't matter.

You're absolutely correct that we all learn differently but I would bet the people who learned and understood music theory well prior to learning and/or experimenting with some sort of sound generation first are in the minority. Most people will learn both at the same time. Playing music goes along way with reinforcing why things are done the way they are done and reading about the concepts after hearing them, playing them, experimenting with them goes a long way.

If you're getting a practical education in this stuff you learn bits of theory right at the point they make perfect sense and you "Grok" them quickly and internalize them without having to think of them as memorizable facts.

E.x. you are directly associating a particular part of the music theory with a direct link to muscle memory on your chosen instrument. It's burned into your muscle memory already and now you have just associated the theory with the practice. Every time you perform that technique or type of passage it just reinforces the theory.

Another example.. understanding the difference between a minor chord and major chord or 7th/diminished/augmented chord, or the difference in how two different intervals relate means a totally different thing to a person who has practiced those things till they can differentiate them by ear than a person who cannot differentiate them by ear.

Perhaps all this is a weakness in an age where a lot of people are making music on computers in ways that break this type of link.

It's all very analogous IMO to someone who has read a book about a sport but has never actually played the sport.

There IS a notation that they use among themselves about microtones above and below the standard pitch, and they have to agree what pitch, exactly, they're playing at any given note.

Furthermore, the pitch they would use depends on whether the musical line is going up or down; in other words, it could be different in different parts of the same piece.

There are a lot of things that serious musicians know well enough to explain, but almost no one ever asks them.

The reason they're the "same" is because of a limitation of musical instruments in their ability to play in multiple keys that led most Western instruments to be tuned in the 12-tone equal temperament system. If you select a different temperament, or use an instrument that can't be constrained by the 12-tone temperament, such as a computer, they can be different.

Some violin players will also tell you that they're not the same notes.

They're the same on an instrument that can only play a defined set of notes, e.g. a piano or fretted instrument. It has nothing to do with the piano being in equal temperament.

If a violinist is playing an F#, his finger isn't necessarily on the exact same spot as when he's playing a Gb. As he explained answering my question, it isn't even in the exact same spot for the same F# every time in the same piece. The string players have very slight variations that they can describe precisely to each other so they sound good together.

And yes, fretless instruments such as violins can be precise enough to differentiate between the two.

"nothing" other than the fact that there's only one black key which has to serve both purposes?

Or a guitar... I think multistring fretted instruments can't be in equal temperament.

First, no. I went hard into music in my early life, I both learned basic piano and intermediate saxophone, and I went through the full suite of band/orchestra/jazz/soloing, to the point where I was a fairly decent-ish musician. It's not until later that people get taught this stuff in my experience, I struggled with any practical connection to music theory probably until late middle-school.

But second, what you're describing is exactly why the article is useful; because the people who are trying to get into music haven't played an instrument for so long that they can improvise on it and start to put together intuitively that certain ratios or key signatures or whatever sound good.

What happens with people who are taught to play instruments is that they spend a lot of time honing those instincts by playing the instrument, but at the same time, what they're playing is constantly attached to this notation so that when they intuitively connect that minor/major/augmented chords sound good or bad in different situations, they simultaneously connect them to the notation as a way of expressing those concepts.

People who don't have formal instruction don't go through that process the same way. They might be composing music without ever thinking about the notation. And when they go to seek out instruction or learn more about techniques, they don't have a bridge to connect their intuitions about what does and doesn't sound good with the notation, everyone just expects them to find the notation intuitive.

I had a lot of formal instruction on playing jazz. A lot of it was about honing intuition, listening to jazz, replaying other famous improve tracks, improving over those tracks, memorizing licks. But that intuition and practical experience was also coupled with being able to break down and describe what those other musicians were doing. That required a shared notation that I could understand, it required being able to understand what an instructor meant when they talked about why certain riffs worked or common ways to resolve a run of notes if I got lost in the middle of an improv solo and didn't what to do next. But I could follow that instruction because I knew the notation and I had a connection between my practical experience of how music sounds and the formal notation about key signatures and chords and crud.

A lot of people don't have that because they haven't spent X years learning to read sheet music. Articles like this are helpful for them.

It just doesn't work like that. Music could have gone in various directions, and Western music happened to pick the various directions it developed in, more or less arbitrarily.

They happen to emphasise a combination of parallel blocks and horizontal lines, and various quite complex kinds of motivic and structural elaboration and development.

Post-classical music - especially electronica - is much more focused on rhythm and timbre. It has more complex sounds and sound combinations and less complex structures.

There isn't much in the way of rhythm-and-timbre theory yet, but it's on its way and will be a thing within a few decades.

All of it is a mess because music has always been about styles, experiments, and conventions, and those have changed over time.

And it's also written from the musician's POV. Not a mathematician's POV. Or a dabbler's POV. Or a nerdy POV.

For example - sooner or later people get to transposing instruments and the obvious question is "Why would anyone sane do that?" And the answer is because it's convenient for the players.

That's it. That's the rationale. For all of it.

So while it's awkward and not very elegant, it has a kind of consistency that's somewhat useful across multiple styles.

It's impossibly hard to invent a system that is so much better for this that it's worth throwing the old system away. People keep trying and failing, because you can't just rationalise part of it, you have to improve all of it. And it's such a huge thing that's not a practical project.

So it persists - partly as a relic, partly as a developing system.

Electronica might be rhythm and timbre based but even looking at rhythm you can start to ask some basic questions (what note length frequency is more pleasing? Why? What frequency of two underlying beats sounds good? Why?).

Sometimes theory can be nothing more than a collection of discovered tricks. Sometimes it can be something more fundamental. To waive your hands and say that all of it is arbitrary is doing a disservice to anyone trying to learn and understand.

There are no fundamental rules to grammar, but we still teach writers what a simile is. Even in spelling, English has no consistency about a lot of this stuff. Are spelling rules useful though for non-native learners? Heck yes. Do we still have grammar books? Yes. It turns out that learning the exceptions is sometimes faster than memorizing everything. Music is a communicative medium based in large part around social norms of what combinations of sounds tend to feel good to people who grew up in a specific environment, and music theory is an attempt to break apart that social consensus into basic rules that can be imitated and built up into more complicated genres.

The rules are somewhat arbitrary and made up, but that doesn't mean they're not real, and that doesn't mean you can't sort-of derive some of them from more basic rules or from basic principles.

Music could have gone in multiple directions, but for whatever reason Western music went in the direction it did, and many people who are trying to learn music aren't trying to learn the full spectrum of every direction music could have gone. They're trying to learn how to imitate Western sounds, and for that purpose specifically music has rules -- many of them derivable from other rules, flexible and breakable as they may occasionally be.

Music theory is a language used to be able to have conversations about harmonic and melodic aspects of a piece of music. Rythmic aspects are less well supported (that's putting it mildly) by western musical theory, but the same is true there for whatever formalism or terminology/nomenclature you might pick: it's not a set of rules, it's a language to facilitate non-musical discusison.

For example, the vast majority of people who have taken Music Theory 101 have completed coursework that is the equivalent of "Hello, World!". That is barely scratching the surface and certainly give the aspiring programmer the skills necessary to create something non-trivial. The same is true of music theory. You don't instill in someone the musical understanding of Brahms with one music theory class.

I think that there are rules, but they're less about a teacher being an asshole. Musical styles evolve over time. We wouldn't have classical without baroque, and we wouldn't have romantic without classical. The style itself imposes a "rule" that is really a best practice. How do we in technology treat best practices? Well, they're basically rules that we shouldn't violate without careful thought. The same is true of music. People who study basic music theory learn about the cadences used most commonly in church music ... V-I ... IV-I ... and the deceptive I-IV-V-vi cadence! You feel where music is going because you've learned the rules by listening to other music and embedding yourself within its best practices. V-vi is deceptive because it violates a best practice, but used effectively, it works.

As music evolved, and as the world became smaller in terms of ease of travel and exposure to other cultures' music, we started to see new influences. Debussy and others were influenced by indonesian gamelan music. Without that influence, we might not have the jazz that we have today. Why? The impressionist style that Debussy practiced created floating pillars of sound, chromaticism, etc. that became important elements of early Jazz.

It's all connected and has evolved over time, just like technology. Personally, I do think they are rules, but like most rules, they're meant to be broken.

As another example, I've spent a ton of time learning game design rules, and all of them are optional, but they're still useful. "Rules" in this context means, yes, communicative terminology, but communicative terminology about particularly effective ways to build commonly understood musical motifs/phrases that affect Western listeners in somewhat predictable ways.

Music theory is a language for talking about a language, music itself. It is partially social convention that leads us to have a 12 tone scale, and it is definitely social convention that leads us to call that a "scale". However, the average Western listener will respond to the notes of that scale in predictable ways, and there are "rules" that you can learn that will allow you to more easily and predictably manipulate that listener's emotions and communicate broader ideas through your music -- many of those underlying rules about tuning, ratios between notes, and so on are derivable from mathematical principles or at least describable in mathematical terms, even if ultimately the reason why listeners respond to some of those ratios is social and arbitrary.

Of course, the rules are not concrete or immutable, they can be broken and often are. But the majority of rules we learn in most subjects are not concrete.

Similarly, there are no concrete rules in writing, and the terminology we use to describe story structure is arbitrary and made up. However, learning the "rules" of writing will make you a better writer for typical audiences that live near you, and those rules are expressed through common set of terms and concepts that many professional/hobby writers have decided to use -- many of which can be partially derived or explained by talking about psychology or history or whatever.

Sure, these are not immutable, scientific principles baked into the heart of the universe, but:

A) the author goes out of her way to say that she isn't claiming that, and

B) breaking down rules and building them back up from different starting points or looking at them mathematically is still a reasonable thing to do with rules that have a social origin, and

C) even though a lot of why certain chords sound good is baked into culture rather than biology, it's also a kind of strong claim to say that all of it is purely social. But I don't think it would matter much even if it was purely social, people who do music theory for a living still talk about math sometimes.

"Western musical theory" is full of ideas about harmony (OMFG, so many rules), often with non-musical semantics overlaid on top of the actual musical elements. These are the "rules" that you get play with as a musician, and the very best of them do in fact break them frequently (but expertly). Why do so many people remember "Take 5" ... because it's in 5 not 4! Why do people consider Coltrane to be a genius ... because of the games he played with harmonic relationships mostly connected to the circle of fifths but deeply subverted. Why do some of us still celebrate the "genius" of Schoenberg, Stravinsky or Bartok ... they upended traditional rules about harmonic resolution, even the very notion of tonal harmony in some cases. Even within this thread, we see people noting a striking detail of a recent Adele song that consists of (almost certainly deliberately) singing somewhat off-key to strong effect.

There is a huge amount of recorded music that plays entirely by "the rules" (the ones that go way beyond TFA), but a lot of what people think of as musical genius is precisely the stuff that flouts the rules (with enough knowledge of the rules to make this work).

I've commented to the same effect elsewhere, but people really underestimate how much the "barely anything" notation concepts are a real barrier to people who are unfamiliar with the ___domain. https://xkcd.com/2501/ comes to mind here; TFA is 5000 words and ends with multiple links to further tutorials and reading. That's a completely fine place to start. If you want to learn the rules of western music theory and get into the meat of what you're talking about, it is going to be a lot more work if you don't know what a scale or a chord is, and I don't see anything wrong with teaching those basic concepts from a mathematical perspective.

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> with enough knowledge of the rules to make this work

That's the part that the author's target demographic wants. They want to be able to either:

A) imitate the rules well enough to write something passable (say for game soundtracks), but aren't looking to innovate, or to

B) learn the rules well enough to innovate.

In both of those cases, even the most basic notation concepts like "what scales and chords are" is a serious barrier to entry for many people.

One thing that often goes "wrong" with this is that the mathematics of frequency ratios is a huge barrier that's most often irrelevant to actual music making. If you literally know nothing about music, there's a good case for just letting the tuners deal with it for the time being, and starting from the old Do, Re, Mi etc. that teaches you both solfège (sight reading/aural skills) and the musical syntax of scale degrees. Then sing a whole lot of music in (movable do; fixed do is pointless except for specialists) solfège and try to make up simple embellishments and variations on what you sing. There's a "programmer's favorite" way of learning these too (these are called diminutions) based on the scale-degree leap that you're traversing, hypothetical elementary operations of musical syntax and whatnot; but good musical intuition will always be helpful. Guess what, now you're well on your way to improvising simple music at the keyboard, and later on you can even get started on learning counterpoint without being lost in all the details. Because, unlike actual college students who are forced to take a counterpoint class as part of studying bookish "music theory", you'll have the fundamentals down pat.

The problem with teaching them from the particular mathematical perspective taken in this work is that...it doesn't actually teach them, and it throws up it's hands and says I don't really know about fairly basic stuff. This isn't an alternate pedagogical route chosen by someone who has a different view of how to get people up to speed for the ___domain, it's a smattering of trivia that isn't directed at learning the rest because the author doesn't understand the basics, much less have a particular pedagogical approach to them.

I think that's a separate conversation from whether this article specifically should be the entire basis for someone learning how to compose music. I agree that I would not point someone at this article and say, "this will be enough to get you on the road to learning how to compose music", I would want something more involved by someone who has more experience.

But that's different from saying it's wrong for the author to talk about sound frequencies.

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To your point in specific, I don't see where the author ever claims that this is an alternate pedagogical route to learning music, the author actively discourages people who already know music from reading the post. The post explains a few basic concepts like what a scale is -- and fairly accurately (or at least about as accurate as most of the other explanations that you'll find online). It openly works through the stuff that the author understands, and openly admits what the author doesn't understand, while sympathizing with the target reader that it's hard to pick up a new subject when it sounds like everyone else is speaking a different language from you.

So basically, it is every single technical blog post written on any subject by anyone who is openly learning about a thing online, and that is something that should be encouraged, not derided.

I also still kind of disagree with people who are saying that this is just trivia or too basic to even talk about, commenters are still underestimating how little normal people know about music.

If people come out of this article understanding stuff like:

- There are 12 "steps" in an octave

- An octave is doubling the frequency of a pitch

- Written down on a staff, we compress those 12 "steps" into roughly 7 spaces.

- A scale is 7 different notes.

- Because of weird ratio stuff and social consensus about which "steps" in an octave are most commonly used, some parts of the scale move 2 "steps" and some move 1

- Transposition exists as a concept, you can play the same song starting at different pitches

- Notes like B# and C can overlap, except some really professional players might treat them differently because it turns out the math we use for different pitches doesn't completely work out correctly in all scenarios.

That's all stuff that people who are unfamiliar with music don't know. Okay, the author doesn't really understand what's going on with minor keys, but this isn't a textbook, and there is value even in something as simple as "an octave is doubling the frequency of a pitch."

I'm weirded out by how upset HN is being about an amateur blog post with reasonably correct information by someone who is actively trying to teach themselves how to write music. This is exactly the content that we should want people to write about online, and it's written in exactly the style that we regularly encourage bloggers to write in when they're exploring new concepts/domains.

Starting off an explanation of music theory by talking about whole number frequency ratios is like starting off English class with a diagram of the glottis.

> Western music has twelve distinct pitches. This is somewhat arbitrary — twelve has a few nice mathematical properties, but it’s not absolutely necessary. You could create your own set of notes with eleven pitches, or seventeen, or a hundred, or five. There are forms of music elsewhere in the world that do just that.

[...]

> I get the feeling that treating the whole chord/key ecosystem as a set of rules is like studying Renaissance paintings and deciding that’s how art is. It’s not. Do what you want, if it sounds good. I’m gonna go try that. Consensus seems to be that the real heart of music is managing contrast — like every other form of art.

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> Starting off an explanation of music theory by talking about whole number frequency ratios is like starting off English class with a diagram of the glottis.

I would say it's more like starting out an explanation of color theory by talking about wavelengths, even though you don't really need to know any of that to understand composition, and even though most of the rules of image composition are contextual to specific cultures and aren't hard requirements that artists have to obey.

The math behind color is definitely not a requirement that artists should be forced to learn, but some people find it helpful to know what mathematical formulas were used to create a given color wheel.

Do you play the piano? I think the piano is the best vehicle for grokking music—it allows for melody and harmony (and rhythm) all in one instrument, in sort of a visual way, that is hard to get in monophonic instruments, or even with guitar (for more complicated reasons). I think I developed my musical sense in two main ways: 1. listening to songs on records and on the radio, and figuring out how to replicate them at the piano, and 2. playing lots of music on the piano from sheet music, where I was constantly exposed to chord progressions and the like in other people's music; the "tropes" of Western classical and popular music, if you like. Neither of these are trivial, but both are within reach of most people, I think. I will posit that there are few shortcuts to musical mastery. Grokking music is definitely not like (as easy as) learning a new programming language.

6 years ago, I accidentally abandoned my preferences and disengaged disgust in an attempt to stop automatically judging. And then I returned to music for the first time in 18 years with the idea that there is no wrong note. Here are some of my observations:

- I wasn't deeply/actively/mindfully listening to music when I was younger. This was part of a cycle of an attention-degrading cycle where I'd hear music and then distract myself with judgements of it, emotions that followed the judgments, and judgments about those emotions.

- Songs that "feel sad" don't. How a person hears/attends to music, how their body has been conditioned to receive different notes/chords/progressions, the meaning given to the sounds/words, the judgments around those things, and how they learned to feel in response to those judgments is what's being described when someone says a song feels a certain way to them. It's not universal and those feelings can be disengaged.

- How I feel in response to music has gotten way deeper and more expansive as a result of releasing all those conditioned feelings. I did this, in part, by pressing a bunch of random keys down on an organ and sitting with the feelings that came up and let them pass. Part of the practice involves letting up on one key at a time and then pressing it again, back and forth, until I can distinguish the note in the cacophony and have reached a feeling of contentment with the sounds. Rinse and repeat for each note.

- Music theory is absolutely unnecessary and in many ways hinders the learning process. I play improvisationally and not necessarily to play a song I've heard. Doing this for a year led to the emergence of a skill where I can start picking out melodies I remember.

- Styles and genres are largely based around preferences. I have a friend whose connective tissue disease (Ehrler-Danlos Syndrome) keeps them from feeling music. As a result, they've developed a strong liking of noise music, which would likely feel very uncomfortable in the body to people whose bodies have been trained on more traditional forms of music.

- The discomfort with dissonance is largely a learned thing, due to cultures of being unwilling to sit with dissonance. It can be turned off.

- If what you're looking to do is make music people can easily like based on commonly cultivated body feels around different sounds and preferences, focus on rhythms that don't change much, avoiding dissonance by default, repetition and also on novelty.

- If you want to make music people can easily ENJOY, repetition, improvisation, and novelty are keys. If you don't know any music theory, novelty will be easy. If you can make some sounds and then vaguely make the same sounds again, repetition will come (this can take a bit of practice, especially if not very connected with the body). Improvisation is such a broad idea because there's improvising around a melody or theme, around rhythm, around volume, around whatever the inspiration is (whether it be sounds, rhythms, feels, words, stories, colors, whatever), and probably more. Allowing and accepting whatever happens is a key to improvisation.

- Music can be enjoyable, even with abrupt transitions from slow to fast, soft to loud, harmonious to dissonant, rhythmic to droning.

- If I want to play what's in my mind, I can sing it and play along with how I'm singing.

- "Wrong notes" played rhythmically can fit into anything.

- If there's no wrong sound, I can play any instrument.

- Everything is a drum.

- Everything is music. Including silence. They cannot compel the silence to cease. So you are music trying to music in some ways and making music without trying. Trying to do that in a way that gains approval or caters to preferences is one way to live. Experimenting in the ways I describe here is another.

I happened to go to an experimental concert just last week, involving organs (since you mentioned them). You might enjoy it. It's available for streaming here: https://roulette.org/event/cleek-schrey-and-weston-olencki-o...

Here's the instructions for doing the nonjudgment practice that led to all these realizations. I highly recommend them. The effects are quite literally lifechanging, unlocking/unblocking joys I didn't even know were possible.

https://news.ycombinator.com/item?id=29764591

I've had to really staunchly stand up for myself and my music to other musicians and to my past notions of music, and have definitely gotten defensive around playing music like this.

No more. I'll just send people to this. And I'm definitely going to focus on playing more like this again.

Thanks again for bringing this into our lives! <3

I sometimes feel like one could take this sentence and substitute just about anything for "music."

"Science is what we understand well enough to explain to a computer. Art is everything else we do."

There is nothing wrong with trying to turn an art into a science. Music just has a bunch of implicit rules that people want to make explicit. Musicians find this frustrating because they aren't interested in understanding music this way. Both perspectives are fine.

And yet I get the feeling that if someone does investigates that (and adjacent topics) in depth with success there would be a bunch of interesting consequences to it.

Maybe, I have got no knowledge in the field - it just sound like something interesting, why the heck wouldn't we want to know it?

Thanks, I misunderstood you.

It's a craft, not a rigid system that you have to follow. What's called 'Rules' in music is more like patterns that have been found to sound good. For example, the famous 'Rule of the Octave' is a pattern based originally on playing thirds and sixths ("first inversion chords" if you want to phrase it that way) over a "walking" baseline. But the rule is far from fixed, and almost immediately the basic pattern using thirds and sixths was altered to use thirds and fifths ("root position chords") on the tonic and dominant degrees. Then the other chords were altered in turn, giving each bass scale degree a sonic identity of its own, with ascending and descending, major and minor varieties etc. Thus what literally started as a simple and basic rule has become incredibly complex. This just shows how a craft can evolve to become more interesting; rigid, foolproof rules cannot. And the "rule of the octave" is a tiny part of music as a whole (though it's pretty foundational, all things considered); but guess what, other "rules" work the exact same way!

'Music theory' is much more of a working practice musicians use to communicate with each other and document their work and discoveries. How programmers feel about their notation systems isn't something working musicians should be concerned with, I don't think.

There is no fundamental thing that makes some notes or combinations pleasing and others not. It is all culturally mediated. I know that possibly seems unlikely and uncomfortable, but nonetheless it is true. If you drill down far enough into music you find the harmonic ratios, and human culture. That's it, there is nothing else down there at the foundations.

IMO asking why a melody sounds pleasing would be like asking why a plot of the Mandelbrot set is interesting; math isn't going to be able to answer that.

I would be fine with a music theory that is a set of tools to deal with sounds, just as math is a set of tools to deal with numbers. Whether certain properties are desirable are up to the theorist.

Music theory can tell you why a specific combination of notes and a chord progression in a musical piece are pleasant to the ear, but does not guarantee that using similar combinations and progressions will result in pleasant sounds.

Part of the reason is subjectivity, in the sense of music is intended to be heard by subjects. And these subjects have their own experiences of music, which in turn conditions their expectations and their reactions to sounds. What sounds good is different from person to person, let alone from culture to culture. Coming from a classical background, when I heard jazz at a young age, I considered it chaotic and awful. Today, it's the most pleasant style of music for my ears. I've seen people who had never, ever heard opera music listen to an aria and laugh to tears because for them it was comically awful.

Another is reason, I think, is that music is embodied by who plays it. I can play Blackbird in the exact same way Paul McCartney does, but when I do it the best I can expect is people saying "that's nice!". When McCartney plays it, the entire stadium hold their breath and get chills, some are in tears. I don't see how that level of emotion can be explained in a scientific way.

GP didn't say there are no rules.

“The rules are culturally, and path, dependent and are not derived or derivable from a simple set of universal first principles” is very far from “there are no rules”.

It does mean that certain approaches to learning the rules that might fit well with people's abstract preferences for how they’d like to approach the field don't work well in practice, though.

What "there are no rules!" is kinda the excuse for music theorists to keep on working on a field that has had so far little applications on how music is created. Maybe yeah on how some people learn it, but that's it. It's basically them admitting that actually successful musicians usually work under a set of rules and techniques so simple from the theoretical standpoint that the whole field gets invalidated.

If anything, it's just the opposite. The 'rules' (again, this should be understood as patterns, or perhaps rules of thumb) practitioners implicitly rely on are far more complex than most theorists are implicitly comfortable with. This is why bookish music theory was mired for centuries in pointless discussions about acoustics and ratios, or some incredibly weird formalisms for representing meter, whilst practitioners got on simply focusing on what sounded good.

(The 'Rule of the Octave', which is the kind of rule I'm pointing to, was quite exceptional in being explicitly taught in actual published treatises about music - and even that was practically forgotten later on; it's not usually taught in "introductory music theory" classes even though it arguably should be!)

I hit my head... and other people's heads... and tables and walls... until I realized that:

1. Yes, there are "rules"

2. They are far far far more arbitrary than I realized

3. They are far from universally applicable.

Personally, I expected music to be heavily math based, with universal rules.

But my current understanding is that while there IS a lot of math, what sounds good / what people in any given culture like, is all arbitrary. You can use math to describe some of it, but not to derive it.

As well, unfortunately, when people say "Music Theory", they largely mean a variation on one or both of:

1. Note reading - Western music notation, which I don't consider "Music Theory" but rather notation, but many will disagree

2. Western Music Theory, largely though not entirely put down by 18th century old cranky europeans based on what they happened to like at the time

There is FAR less "universalness" in Western Music theory than I thought.

So there is this combination of:

* Huge englightnment and pattern recognition and increased understanding as I learned more music theory

and

* tremendous frustration at inconsistency and arbitrariness, until I realized it wasn't my music teacher who hated me, it's that what they are teaching me is arbitrary

So personally, I think it's important for programmer to accept "there are no rules (in the sense that we think of "rules", boolean logic and math etc), though there are guidelines and recommendations (which are completely arbitrary and cultural-based)"

It turns out "what rhythm / note / combination / frequency is more pleasing" is 99% how you were brought up / what you are used to, rather than some math/physics correlation. Move to another part of the world and throw out your 4/4 and 12-note equal temperament and I-V-vi-IV

You absolutely can, because that's essentially exactly what western classical music (and more or less all other forms of music) already did, just spread out over a very long period of time.

Take Western music. Early music is mostly vocal, and early principles of music theory evolved around what would be possible (and practical) to sing. Different people have different vocal ranges, and the constraints that puts on the music lead to certain constraints in counterpoint. Eventually instrumental music becomes more socially important, and that changes what kinds of music can be made. Mathematics progresses in such a way that new tuning systems (12 tone equal temperament, and its precursors) make it easier to modulate between keys, and more modulation (and chromaticism) becomes common.

Even things like the way the music is structured depend on social practices, they're not spontaneous. The sonata form depends on an audience that listens attentively to music so that they can perceive the way the themes are gradually transformed.

I could go on, but nearly everything in music goes this way--there are principles, but they only have a limited explanatory power, you need to get into historical contingency to really understand why things evolve the way they do.

I certainly agree that historical context has always been central to the way music has developed. It's not for nothing that most of Europe refers to "the church modes" rather than using a more abstract term for a set of interval rotations. But that historical context is only absolutely necessary if you want to try to understand why music evolved in the way that it did. It's not necessary if your goal is to understand the way we understand, compose and perform music today.

Of course, I'm all for more understanding of music, so I'd favor historical context every time. It's just that it's not a necessary feature for understanding where we currently are.

There is no one true "music theory". Music theory as it is typically taught is no more than an elaborate system of nomenclature of the stylistic preferences of European music in the last three centuries. It is a cultural description of a cultural phenomenon.

To get into more specifics, when explaining music theory at an elementary level, you might say that a frequency ratio of 1:2 is called an octave and all the notes with an octave relationship to each other are considered equivalent. That is true, if you are making European-style music. Most other cultures around the world have a name for the interval called the octave, but most of them don't consider all octaves to be the same note. "octave equivalency" is fundamental to Western music, but it's not a universal law, it's a stylistic choice. To imply otherwise by claiming that your explanation of this European convention is essential to music writ large is to do a disservice to the many musical cultures around the world that don't follow that convention.

> The harmonic series is relevant, but you can't start from the harmonic series and find your way to Western classical music.

As big as fan as I am of being aware of non-western musical culture, I was commenting on the specific idea of moving from the harmonic series to a specific musical culture (the western classical one). This is why the chemistry analogy is (roughly) appropriate, because there are in fact a substantial number of (western) music theorists who consider there to be only a single western classic music theory.

I try to almost never use the words "music theory" without prefixing them with a temporal and/or geographic cultural qualifier (though I likely often fail here).

It's not just a "convenient for the players" moment. It's also because changing over to non-transposing instruments would create a generation-long transition period of broken pedagogy. It's the same reason oboe fingerings are so nonsensical. It's backwards compatibility to preserve previous fingerings that makes the new fingerings awkward and arbitrary.

My experience with being taught music theory lines up with your experience, and I spent a lot of time learning both piano and instrumental music leading up to college, to the point where I considered myself to be a fairly decent-ish musician. I was not taught why anything exists I was taught rules: rules that I constantly saw being broken around me, and that I was constantly told, "well, it's OK for them to break the rules once they understand them." It was not until I started to think of music-theory as a grammar based on conventions and social norms that any of it clicked for me, before that point I hated music theory. Certainly it was rare for people to try and break things down to first principles or to describe why things were the way they were.

What happens as an early musician (in my experience) is that over time you just kind of get used to the social conventions and it becomes intuitive because you've spent a lot of time on it. I learned about stuff like ratios once I started getting into intermediate jazz on the saxophone; it was not an early concept taught to me in piano lessons.

And then people look at articles like this and say, "well, that's all super-basic stuff, and it's not completely right, and she's doing a bad job of explaining..." Nonsense, she's doing fine. Even if there are a few inaccuracies, this is a good article: it's written in a way that's sympathetic to people who know nothing about music theory, it gets across the immediate points that musicians might not even realize are problems (why is a C a C, why is B# C), it is a (mostly successful) attempt to tie music theory to something that readers might already understand. It's helping bridge this gap of "repeat this rule until it becomes intuitive", and that's a good thing to do.

Even with really simple concepts, people forget how big the barrier of entry is around notation. It's extremely common for me to run into people who say they would read more scientific papers or statistical research if the math notation didn't throw them off. Music is the same, there is a common language that is very efficient and nice for people who know it, and is indecipherable for everyone else. Anything that helps lower that gap is good, and looking at this article as someone who knows the notation already my immediate reaction is that it seems pretty helpful and a lot of "intro" music courses won't explain these concepts, they'll expect you to tough it out and just memorize the rules.

An electronic musician working in a DAW needs to understand notes as ratios and frequencies lest they produce a mix that sounds like goop on club speakers, but a pianist can get away with not having any idea. Put them in a room and ask them to explain the difference between a mode and a scale, and you'll need a hazmat crew.

Realizing all this led to a handy heuristic: the best musicians to know and work with are those who can navigate those different conceptions and traditions without fear or judgement. There's no reason an orchestral composer can't learn from riddim without tripping over triplets.

There is definitely some subset of musicians that cannot conceive the notion that someone somewhere on the planet might be learning music just for fun and don't WANT to take it uber-seriously.

Just in case it’s helpful - “tenets”

Is there an actual reason or are all of these post-hoc explanations? Depending on the culture you'll have different scales and ways of organizing the sound. From what I understand it always comes down to "some people thought it sounded good and from there they made music and people got used to it and it snowballed".

But there is an explanation that does try and get at it a bit more analytically and that's in the paper I linked to. With the above caveats about the cultural momentum, the 12 not equal tempered scale provides a happy compromise between the number of notes to provide a basic building block for music (characters or digits would be an analogy to notes in an octave) vs the number of "good" note pair combinations (where "good" is if the frequencies have a small/simple fraction approximation).

Some of it is hand-waivey, to be sure, but at least it provides a potential reason and a starting point.

https://en.wikipedia.org/wiki/Pentatonic_scale#Use_of_pentat...

As far as I can tell, 12 ET is the smallest equal temperament that acceptably encodes the common variants of pentatonic scale. Meanwhile, the pentatonic scale probably appears because (3/2)^5 ≈ 8.

I assume it's written partly tongue-in-cheek, to show how arbitrary a lot of music tradition is, and how it creates a barrier to entry.

And the article didn't even mention different clefs or transposing instruments!

Also (tangent / tiny nit): "tenants" (inhabitants) -> "tenets" (rules)

Frequency is a physical measurement of a periodic waveform.

Generally, a tone is composed of harmonics, which have frequencies that are integer multiples of a fundamental frequency. The pitch physically corresponds to the fundamental frequency of harmonic sound (or a sound that is mostly harmonic). But the relationship is complicated, as we can perceive two tones as having the same pitch (and fundamental frequency) even if the actual spectrum of one of them does not actually contain a component at the fundamental frequency (see https://en.wikipedia.org/wiki/Missing_fundamental).

This is because our brain will fill in the fundamental frequency if a tone has most of its harmonics. This is why you can hear bass notes of a song even if the speaker you're listening to doesn't have the frequency response to actually reproduce the fundamental frequency.

I hope some of this nuance is making this make sense.

There's also a concept of "pitch class", which is the idea of what you might call "C-ness" of every C note on the piano. In other words, octave equivalance, or the fact that you can substitute nearby C for each other without ruining harmony. Pitch, in some ways, is the intersection of pitch class and a specific fundamental frequency.

Pitch is the perceptual (i.e., human auditory perception) correlate of frequency. Pitch is what your brain/auditory system interprets that it hears from the sound wave at that frequency.

For a piano chord, there is likely the fundamental frequency (or f0, the lowest frequency), and its upper harmonics (integer multiples of the fundamental). That's why musical instruments (like the piano) sound richer than simple sine waves: their physical bodies create richer, more complex waveforms from the harmonics, giving it a unique timbre.

There are some subtle differences. For example, (very oversimplified), if you were made to listen to a waveform containing the frequencies of 60 Hz and 90 Hz, your auditory system would "hear" a pitch corresponding to a fundamental frequency of 30 Hz, since 60 Hz and 90 Hz are integer multiples of 30 Hz.

Eh, so if you add 60hz and 90hz you get a wave that repeats every 33ish milliseconds. You don't "hear a 30hz sine wave" that your brain is inventing, that's a very common misconception. You just hear a 30hz "thing" that is not a sine wave because there's a 30hz thing that is not a sine wave playing. Just try adding an actual 30hz sine wave to the 60+90 thing and you'll see it sounds very different to what you hear when you add those two sine waves. If you keep on adding waves spaced 30hz adding the 30hz or 15hz one wouldn't make much of a difference because what you are listening to is something very similar to a low pass filtered 30hz pulse wave.

> your auditory system would "hear" a pitch corresponding to a fundamental frequency of 30 Hz

If one were to make a pitch classifier machine, would it always agree with the human ear?

Regarding a pitch classifier machine, typically algorithms for pitch classification/tracking/detection focus only on the fundamental frequency (or f0) of the input sound wave. So, _technically_ pitch and f0 are used interchangeably where they shouldn't be.

I have yet to see a pitch classifier machine that tries to implement the special workings of the human auditory system.

Periodic sounds above about 20 Hz are perceived is being `pitched'. Our perception of pitch is logarithmic. I.e. if you keep multiplying the frequency of a sound by the same number we perceive the pitch as going up in equal steps.

For example, to go up (an equally-tempered) semitone (aka half step or half tone) — which is usually the smallest pitch distance used in music — you multiply the frequency by 2^(1/12) (roughly 1.059463).

To go up an octave in pitch, you multiply the frequency by 2. But going from the pitch A3 (220 Hz) to the note an octave higher, A4 (440 Hz), sounds like the same `size' of increase in pitch as going up an octave from A4 (440 Hz) to A5 (880 Hz), even though you are now going up 440 Hz instead of 220 Hz for the previous octave.

Yes, but accompanying those facts is a set of phenomena that you're supposed to experience physically.

In short: no you're factually wrong. Human ear can discern smaller pitch changes and there actually music composed on smaller subdivisions than 12

For instance https://en.wikipedia.org/wiki/19_equal_temperament

You can check done pieces on YouTube.

Yes it sounds unusual. Yes it sounds strange. But thats just because you've been conditioned to listen 12 tet all your life.

But saying 12 notes because we can't perceive smaller changes is, again, wrong and there's actual counterexamples of music done with smaller changes

“Ragas are precise melody forms. A raga is not a mere scale nor is it a mode. Each Raga has it’s own ascending and descending movement. And those subtle touches and uses of micro tones and stresses on particular notes like this…”

Microtones are less than the difference between two adjacent pitches in western scales and are discernable, even to people like me who have a hard time telling adjacent pitches apart.

For example, lots of the notes in the vocal of the Beatles `Come Together' are clearly (often very) flat — wonderfully so, subversively so, even, in our pitch-corrected age!

A very small gap gives a subtle chorus effect, a little wider takes one into honky-tonk piano territory, and wider still perhaps more like a bell, but still sounding like one note.

But, as you say this doesn't negate the audibility of microtonal inflections in a melodic line.

However, pitch steps smaller that those of 12tet can still provide new chords. For example, in 24tet the triad with a third half-way between minor and major is a distinct and interesting sound. But yes, in 24tet two consecutive pitches sounded together sounds more like a single note with an interesting timbre.

There are systems that have more than 12 notes.

Plenty of instruments hit these "microtones". E.g. string instruments with frets can bend strings, and string instruments without frets can just play those notes.

Genres of music like the blues frequently utilize these microtones.

12TET is just the custom in western music, but we can absolutely discern smaller intervals.

http://terpstrakeyboard.com/web-app/keys.htm

I'm sure at some point those tones will converge into a single note to human ears, but the fact that a 53-note keyboard exists shows that people can absolutely hear smaller divisions than just the 12 standard notes.

https://www.sciencedirect.com/topics/medicine-and-dentistry/...

Sometimes the math is just there in the background: "This chord progression is based on certain physical facts related to the harmonic series. That makes it sound good, and I like it". Sometimes it is brought to the forefront: "All of this music appears to be generated from a short pattern of notes which is recognizable even as it is subjected to rotations, elongations, truncations, etc. That is beautiful."

Certain types of music play up the lovely-pattern-ness of music, and some don't. Neither is better than the other (denigrate the art of Justin Bieber who dare), but you can probably guess which I prefer :)

Consciousness, pattern-recognition, spatial rotation of abstract objects, natural mathematical beauty, poetry, drama, acting/performance, nonverbal collaboration, shared rhythm: music is a nexus for all of these things. Music is nothing short of awesome, in the truest sense of that word.

Dissonance (a "broken mathematical rule") has as much a right to exist in music as consonance does. There is a constant tension between consonance and dissonance, and it is that dynamic tension which is exploited for affect or structural definition.

So what we have in the case of Schubert, Beethoven, and Chopin is not so much the breaking of mathematical rules. In my opinion, they are just examples of expert judgment applied to the tension and release within an artistic work.

Computer programs must follow the rules... the rules get made up first and the programs get built to conform to the rules.

Music theory isn't like that. The music comes first. The theory gets made up to try and describe & understand what is going on with the music. It's not at all like a program where the rules & theory come first and you build something that fits into that.

If you write a piece of music that doesn't conform to some kind of arbitrary rule in music theory that's fine. If your music is successful and important enough someone will modify music theory to explain why your music was successful and important.

There is a cycle where you learn some music theory to help learn music but the theory just helps you understand what's going on, it doesn't force you to do anything a certain way. You can't come at music and think about it like you're building a computer program.

I think the absolutionist take here is unproductive.

I would argue a writing a fugue is analogous to "building a computer program"; to say little of 12 tone chromatic atonal music compositions that usually sacrifice 7 tone based harmonies for interesting structure, like palindromic reflection of ascending and descending musical ideas, and optimising for a high number of pitches in a melody.

Additionally, 7 note diatonic music theory can be completely summed up with two statements:

1) tick-tick-tock-tick-tick-tick-tock

2) permuate all the things!

Most common western instruments are literally built to only play 7 tone diatonic music theory.

Look at a piano. It can literally ONLY play "semi-tones". It's impossible to explore music beyond the 12 note chromatic scale without first modifiying the physical construction of the piano.

What makes good music good is ultimately human ears. Fugues, for example, might seem to be programmatical at first glance, but get your hands dirty and compose one yourself, based on nothing but the mechanical rules. Doesn't sound good? It doesn't.

It's like writing English based on syntactical rules. You'll end up with a music equivalent of "Colorless green ideas sleep furiously".

To me, and most digital art, I build it very much like I build my computer programs. An evolving exploration of a ___domain, leveraged by macros that encapsulate former techniques that work well to produce a desired sound/effect. I'm essentially building a framework for churning out songs through this process. Then you can experiment within those frameworks to build out related songs(in mixing and tone and techniques) producing a collection of related works.

Even traditional rock artists were experimenting and building up frameworks to help them produce cohesive works.

... or you can go the other way and look at Atonal music such as Schoenberg :)

https://www.youtube.com/watch?v=oJg4XbzSV9Q

[but] The author is missing or skipping key pieces of information about why we use the notes we use, how they were derived, and why they sound the way they do to us.

Edit. There was zero sarcasm in my comment. I really love this style of writing.

The missing concept here is the harmonic series. A pitched note is not generally a sine wave, it is properly modeled as a series of sines waves combined. These sine waves go up in mathematical multiples.

So say we have a note at 100hz. Simplistically it is actually 100, 200, 300, 400 etc all at once. So if 100 were A, 200 is A, 400 is A, 300 is … E, so 150 is E. Etc.

None of this is arbitrary because the When the waves combine the result is more ordered or more chaotic, depending on the ratios of the waves, which our ears hear as consonant or dissonant.

Best I can do from a phone.

PS I love OP's writing style, too. Fearless learning in public is to everyone's benefit -- given proper disclosure/transparency.

When two patterns are overlayed there is a third pattern which is the interference of the two patterns, or a pattern of the sums and differences of the two patterns.

If the first two patterns have a simple fractional relationship you can visualize the resulting pattern in your head, and the brain when hearing perceives this as predictable and regular, and predictability is boring and sounds consonant.

If the fractions are more complicated we end up with a more complex combined pattern which is harder for the brain to predict and is perceived as richer or more dissonant.

If the patterns get very far from the fractional relationship (for example If an instrument is out of tune) these patterns start to feel non deterministic or non repeating to the brain and we tend to hear this as unpleasant.

If the patterns have no particular mathematical relationship we end up with a completely non repeating pattern that is indistinguishable to the brain from randomness, and we call this unpitched or noise.

    0    1.000          = 1:1   (unison)
    1    1.059                  (semitone; minor second)
    2    1.122  ≈ 1.125 = 9:8   (whole tone; major second)
    3    1.189                  (minor third)
    4    1.260  ≈ 1.250 = 5:4   (major third)
    5    1.335  ≈ 1.333 = 4:3   (perfect fourth)
    6    1.414
    7    1.498  ≈ 1.500 = 3:2   (perfect fifth)
    8    1.587                  (minor sixth)
    9    1.682  ≈ 1.667 = 5:3   (major sixth)
    10   1.782                  (minor seventh)
    11   1.888  ≈ 1.889 = 17:9  (major seventh)
    12   2              = 2:1   (octave)

In my own experimentation, the just intonation intervals are jarringly perfect, and I'd say that the 5:4 major third is too narrow. Has anyone checked that barbershop quartets are actually singing such narrow major thirds? My understand is that many studies have shown that string instrumentalists and singers, even when they claim to be using just intonation, play their major thirds somewhere between 5:4 and equal temperament.

It's also extraordinarly difficult to really tune basic harmony justly. The ii V I progression is a fun one... if you're not careful you shift the tonic by a syntonic comma.

I don’t know if anyone has checked, but why wouldn’t they want to? Surely the low integer ratios are best going to get the intended sound of barbershop harmony, unless there are other acoustic reasons why slightly changing the intervals would work better (something akin to octave stretching in pianos, although I’m not sure that sort of thing would be as relevant to barbershop harmony).

There's more to music than acoustics. Four-part harmony has a lot of moving parts (literally and figuratively), and you have to make concessions to acoustically pure intervals to make it sound like it's in tune through time.

Is the D a fourth below G or a minor (or harmonic) seventh below C? Once you've chosen your D, is A a perfect fifth above it? If so, what kind of major second relation does it now have to G?

Is the F a perfect fourth above C, or is it a major third above your D? Or is it supposed to be the minor (or harmonic) seventh above G?

I think barbershop tends to skirt these questions by instead using a II V I progression (a secondary dominant), since the F# doesn't have to stand in careful relation to other notes of the plain C major scale.

(That's interesting Schenkerian analysis can help understand these issues. I know only about it at a very thin surface level.)

My impression of barbershop harmony is that they're not particularly concerned with how the melodies are tuned. I don't think there's much deliberate intention there, other than that you obviously don't want frequencies to drift too much throughout a song because you want to be in the right ranges for each vocalist. Barbershop harmony seems much more concerned about the "vertical" tuning and my hypothesis is that they do tend to hit very close to the low integer ratios. Can you think of a reason they wouldn't intend to do that? The harmonies tend to be pretty close (unlike a piano) and I would guess that the human voice is less inharmonic than piano strings (although bowed string instruments are very harmonic, right?).

I pulled up a random barbershop recording and looked at a spectrogram for the final chord, since the final chord is supposed to ring as much as possible, and there are only vertical considerations, so you'd presume they'd tune it as justly as possible, right? Here's what I found:

They tuned their perfect fifth justly as 3 : 2 almost perfectly. That's not unexpected, even equal temperament gets perfect fifths close to right (though a little flat).

They tuned their major third between 1.253 : 1 and 1.258 : 1. A just major third is 1.25 : 1, an equal tempered major third is about 1.26 : 1, and the 55-EDO system mentioned in the book I referenced in my first comment would give a major third of about 1.255 : 1. Like I expected, they tuned their major third sharp with respect to just intonation, but still flat with respect to equal temperament.

What's the theory behind why string quartets aim slightly sharper than 5:4? Is it something to do with inharmonicity or acoustics? Could it also have something to do with avoiding going flat? Is it possible that listeners are so accustomed to equal tempered thirds that a 5:4 third actually sounds flat?

One thing I like is that the slight dissonance takes the edge off the ringing. I find that to be a rather strong flavor.

The book I mentioned offered suggestions why sharper major thirds are used, but I don't remember there being anything convincing other than observations about how harmony works. Maybe your thought that going too flat sours the third makes sense: if it's sharp, you have a little more freedom to adjust your intonation without accidentally going flat. Plus, thirds are fairly forgiving anyway since they're corresponding with the fifth harmonic, which over two octaves higher.

It's worth mentioning that the Pythagorean major third is about 1.266 : 1, which is quite sharp, yet it's still just intonation since it's from going up by 3 : 2 fifths.

Most of what he covered is stuff you learn the first day playing and never really need to worry about again.

The interactions of how different notes & chords in a scale draw our ears, how 7ths and other intervals work/sound. Modes, chord voicings, inversions, modulations.. all that stuff gets incredibly fascinating.

In defense of the article, I'd note that these are all choices you make within the framework set up by 12TET, not the framework itself.

Something that's very important to note is that music theory and playing music are very different. Music theory allows us to understand what someone played, and lets us communicate to other musicians in a compact manner. Theory by no means imposes rules, unless you are specifically wanting to write music exactly like Bach or something.

As a logical-minded person myself, it's tempting to really dive into theory and the root of things, but I've found that it's a trap most of the time. For getting better at songwriting or composing, I've found the best thing to do is transcribe the songs you like (or portions of them) and see what they do, then incorporate those ideas into your own playing.

I also get the frustration around enharmonics (e.g. C# vs Db), but honestly it's not an issue in practice. I had no opinion until I started typesetting songs myself, and it just looks cleaner to sometimes have Cb or B# depending on the context. The circle of 5ths also neatly puts scales in the order of how many sharps or flats they have. Also, C# is not a commonly used key anyways.

From a theoretical perspective C major and A minor share the same notes, but when I improvise I do find the distinction meaningful. Having multiple ways to think about things is helpful. There are other things too: C minor 7 is the same as Eb major 6. Or if you play a rootless voicing, the harmony could be ambiguous. Etc.

Yes, our music notation system has some issues. But this article is overanalyzing it from an armchair. It's mostly fine in practice.

It starts making more sense when you're working with harmony and chords. The wikipedia example section on enharmonic equivalents has a nice example of a Shubert sonata that makes use of this [1].

[1]https://en.wikipedia.org/wiki/Enharmonic

Something that people with casual exposure to musical notation probably don't know is that before the standard 5-staff notation became ubiquitus there were quite a few alternatives, maybe the most interesting from a mathematical perspective being the Byzantine notation

Broadly speaking, while the modern system focuses on pitch values and an elaborate (modulated) map from frequency space to physical space (paper), older systems used in the Byzantium used "deltas" or the first differences of pitch values.

So you start with a base note (lets say C) and then you go +1, +1, -2 to indicate pitch changes (in semitones). This is quite well adapted to monophonic chant. This notation was never developed to cope with the complexity of modern music but its not immediately obvious that it can't be done

There is no easily accessible exposition of this musical notatin style, this cheatsheet gives a flavor http://www.byzantinechant.org/notation/Table%20of%20Byzantin...

Sine waves work to visualize the concepts but real instrument notes almost always are better modeled as a large number of sine waves with a mathematical relationship that sound simultaneously and in varying amplitudes that combine to create the timbre of the note.

Then when those notes sound together we get combinations of all those sine waves together to create a very complex wave that sounds a certain way to our ear based on the interactions between all of these waves at once.

So to visualize it or reason about it, my comment here makes sense:

https://news.ycombinator.com/item?id=30363653

But to hear it you would be much better off with triangle or sawtooth waves or samples of actual musical instruments.

Here's a 31-equal temperament song. Microtonal music has a reputation for sounding bad, but it doesn't have to.

https://www.youtube.com/watch?v=u-PEhbSOh74&list=PLC6ZSKWKnV...

> ... but it’s probably why Western music settled on twelve.

The curious part is that Western music settled on twelve a long time before equal temperament. They used to have 12 notes but use for example quarter-comma meantone. Great mozart meantone piano piece (mozart never used equal temperament):

https://youtu.be/lzsEdK48CDY?t=700

To my ear it sounds like there are way more major thirds than there normally would be in a Bach piece like this... anything else I ought to be hearing?

https://www.dailymotion.com/video/x2j1gy2

Terry Riley's Harp of New Albion is awesome as well.

https://www.youtube.com/watch?v=k6tptdvQbos

Harry Partch - Two Studies on Ancient Greek Scales is so haunting https://www.youtube.com/watch?v=GglpEGSXpXo

This one is fascinating too - it's just intonation vs 12edo comparison. You can really hear all the beating in the harmonics in the EDO one.

https://www.youtube.com/watch?v=AgB4TrV57AU

https://en.wikipedia.org/wiki/Comma_(music)

[0] https://youtu.be/XnGqVIW51JA, https://youtu.be/mPZn4x3uOac [1] https://youtu.be/PqR36Drhn_k [2] https://youtu.be/iRsSjh5TTqI [3] https://youtu.be/P14JcRyJCEI [4] https://youtu.be/7GhAuZH6phs [5] https://youtu.be/xVZy9GUeMqY

> Einstein's violin playing took a gypsy turn when he improvised, often while thinking about mathematics. He was a very good musician. He could navigate the piano well enough. But his real love was for the violin. Even so, Einstein's violin playing was not of the first rank. Yet such was his love for the violin that he played and practiced anyway. Such is the spell that musical instruments cast on people's minds.

> In addition to studying the classic violin pieces of Bach and Mozart that he adored, Einstein was also given to improvisations. Some have told of the haunting, gypsy-like quality of these rhapsodies. Einstein himself stated that many of his greatest mathematical ideas came when he was improvising on the violin.

[0] https://pianobynumber.com/blogs/readingroom/einsteins-violin...

Improvisation has little to do with music theory; most Folk and Jazz music is improvised; even what we know today as "classical music" was often improvised in the past; many great composers were also great improvisers.

Pro folk musicians are some of the most theory adept people I know, my uncle is one and he knows 1000s of songs, and can basically play any song in any key through his theory chops.

Jazz musicians... I mean, cmon, Jazz is all about doing wild things with theory that surprise and amaze you. Good luck being a jazz musician without understanding seventh chords, modal/metric modulation, etc.

Improvisation doesn't mean "making up random stuff", it means knowing your instrument and the ways in which chords, melodies, harmonies interact, knowing changes and progressions, inside out so well that you naturally can use that knowledge to come up with something unique.

If what you mean is "some musicians can implicitly understand this stuff without knowing the academic language", well yeah, music theory isn't prescriptive, it's a language used to communicate with other musicians. You can get away without knowing the terms, but knowing what it's describing is important.

Edit: I should add, knowing what it's describing is important up until the point that you want to readily communicate with other musicians and have them communicate with you with full understanding, as quickly as possible, sometimes during performance - then the language of music theory becomes very useful.

A skilled/talented musician just hears that. It's like cooking: a good cook knows from experience which spices go with which dishes, without having to know the basic physics.

> making up random stuff

It always has random components, that's how humans operate; but good improvisers can produce music in real-time which sounds like composed/arranged music; the purpose of music is to entertain and please people; this goal can be fully achieved without any theory or prior arrangement.

Last night I was improvising a melody based on a chord progression I had just come up with. I heard the chord progression in my head without thinking about what the chords were. But then when I thought about it, I knew what they were, tested on my instrument, and wrote them down.

Beginning to melodically improvise on the progression (with no backing) I was initially looking at my chart the whole time and thinking in theoretical terms. But, before long, I was having moments of losing track of where I was in the chart, but my mind's ear and my fingers still knew what to do to keep going within the harmony.

I.e., you reach a point where you have internalised the sound of the chord changes, and you can just hear the lines you want to play, and your fingers just know how to play the notes that you hear.

But to get to the point of being able to do that in a reasonably sophisticated way, one is mightily helped by thinking about things and practising things in a theory-informed way.

> And knowing lots of recipes allows you to much better understand how to tweak and improvise on your existing knowledge

Sure, practice is helpful. And it is to be hoped that the conservatories are also helpful for something, if society invests so much money in them and so many professional musicians come out of them.

You do not need theory to perform existing pieces all that much. Which is why you can learn to play classical music while ignoring theory.

> To improvise you first of all require musical talent on both harmonic and rhythmic dimensions

I mean, you need ability to keep rhythm and play notes. But beyond that, you absolutely can get away with theory only with basically zero special talent. You wont become world famous, but you will be able to match what the rest of group is playing.

> even what we know today as "classical music" was often improvised in the past; many great composers were also great improvisers.

Also going back to original comment, you seem to put theory and improvisation into some kind of dichotomy. But, both being great composer and great improviser are heavily facilitated by understanding theory.

When jazz music appeared, there were no conservatories or theory for it. The first jazz conservatories emerged fifty years later. Today they find many graduates, for example, from Berklee. But this is a development that has only begun in the last forty years. It didn't make the music better, it just made it different.

> You wont become world famous

Very very few world famous exponents of popular music have a formal musical education.

This is just false. If you start to dig into this a little bit, you will easily find connections between almost all famous musicians, and universities. If they didn't have formal education themselves, they often had private lessons from someone with a degree. Or someone in the band had a degree etc. Or in the case of the beatles, the "producer" just happened to be a trained composer who was giving a lot of "suggestions".

Other examples: punk band green day songwriter billie joe had 15 years of lessons with a jazz guitarist with a university degree. Bluegrass guitarist tony rice had private theory lessons with a berklee graduate. etc etc.

It's part of the marketing to downplay the theory and knowledge involved, people want to relate to musicians, and not feel like their being outsmarted, or manipulated by "some formula".

Are you from the industry? I am, with forty years of experience under my belt. Believe me, there is no correlation between success and formal musical education.

Obviously, there are also plenty of artists who only have this influence indirectly, through other artists that inspire them. Or in the case of hip hop and electronic, through sampling. So I think their point was overstated.

Yes, to study the tropes and patterns of theory applied by the great jazz musicians, who certainly knew what they were doing with theory, even if "the theory of jazz" didn't exist. You don't have to know "jazz theory" in order to use theory to create a new genre of music that became known as "jazz". There were a community of performing musicians who had at least some formal theory knowledge (if not to a rigorous Berklee standard), who shared ideas and patterns and tropes, which eventually became a recognised genre, and later on an academic field of study.

It's funny, because out of all the niche subgenres of music out there today, jazz is almost certainly one of the most "theory influenced" genres of the last century.

Also, I'm just plain not as good a musician as, say, Duke Ellington was, and there's a benefit to being able to understand what he was doing so I can try to imitate that style and incorporate his techniques into my own playing -- I don't want to rediscover all of his lessons completely on my own because I'm not talented enough to do that; very few people are.

One way to 'improvise' is to have an understanding of the relationships between keys and while moving around at pleasing intervals and making interesting chords, regularly jump keys at points where they overlap -- that's .. pretty music-theory driven, and can be very mathematical! There's a lot of fun math in jazz.

It's not the only way to improvise, certainly. But someone who knows the rules can make them run a long way.

Some of the grammar rules are sometimes useful to general public.

I don't think theory is strictly necessary, there are great musicians who don't know anything about theory. Theory should never be used as a way of gatekeeping or deriding people's efforts to build things. However, unless you are also a musical genius who has spent decades honing your craft, there will probably be some benefit from learning from the people who came before you.

It's not necessary essential, but very little theory is essential. You can technically be a great game designer without ever looking at other games, you can technically be a great UX designer without learning UX rules. The rules just make it much, much easier to get to that point -- because the rules allow you to more quickly learn from a large body of extremely talented individuals who shaped the entire genre.

Trying to separate theory from its basis is folly. Like the scientific method and the body of scientific knowledge, it's important to remember one follows the other, but I'm still employing centuries of science every day even if I can't name it all.

Jazz musicians who can't cite theory don't use 7ths by accident.

> Jazz musicians who can't cite theory don't use 7ths by accident.

I don't understand this argument. It's unavoidable to use 7ths, regardless whether the musician can name the interval or not.

Music and music theory are two different things. A person who is an expert in music theory is not necessarily a good musician and vice versa. There are approaches to music theory based on mathematics (see book list below); here at the latest, your assumption is likely to be correct, but hardly in general.

Book list:

Mathematics and Music - Composition, Perception, and Performance

Musimathics Volume 1 and 2

Music and Mathematics - From Pythagoras to Fractals

Mathematical and Computational Modeling of Tonality - Theory and Applications