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> 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.

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).

> 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.

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.

I used the term "significant skill" to refer to something that I don't think an untrained person could do, but that a reasonably well trained performer could (whether on violin or with their voice). Certainly not rocket science. As a reference, it took my wife about a month of twice-weekly singing Bulgarian music to really be able to "hear" the correct tones, which of course were even further from her own musical practice (she had been a professional singer earlier in life) due to not just JI/WT but quarter tones etc.

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".

I have heard the a capella singers will naturally sing intervals and chords using just intonation and that because of that you can craft melodies that will lead them farther and farther from the original tuning:

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.

Yep yep.

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 first point is just really wrong. The harmonic series is not integer multiples, and you don't have to get very far in the harmonic series before you encounter radical microtones that no European musician would have ever considered in-tune for the last thousand years.

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.

It does somewhat though. You build it by taking the 3rd harmonic (perfect fifth) cut it in half to get it back into the octave, rinse, repeat.

That's not the harmonic series though, that's the first 3 members a bunch of different harmonic serieses smooshed together. It's extremely arbitrary.

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.

Sure. Just saying the harmonic series is not completely and utterly irrelevant.

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.

> But that's very different from claiming that western music is derived from the universal mathematical properties of the harmonic series,

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.

They are related to the mathematical properties of the harmonic series, but they are definitely not rooted in it.

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.

Unfortunately, nobody said "western musical ideas of consonance is rooted in simple harmonic ratios". The whole point of Plomb & Levelt's work (recommended if you have no read it) is that provides an explanation for consonance that has simple harmonic ratios as an existential but insufficient component. From their work, what matters is not the ratios of the two fundamentals, because there are essentially no natural tones that consist of only the fundamental (1). Instead, consonance/dissonance is a result of the sum of the consonsance relationships between each pair of partials in the two tones.

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.

Sethares showed shocking ignorance towards the history of the actual usage of dissonance and consonance by practicing musicians and theorists in the history of Western music. The most obvious being that the category of dissonance and consonance has shifted over time. The most famous example being the perfect 4th, which has a simple ratio of 4:3 and should be a consonance but was treated as a dissonance against the bass for most of music history in the west (even though Sethares' graph categorise it clearly as a consonance, contrary to common practice). At around around the same time, the thirds and sixths began to be considered as consonances either, whereas previously they were regarded as dissonant and were used sparingly (composers generally avoided thirds in the final chord until around the end of the 15th century).

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.

I don't think this is a fair interpretation of Plompt & Levelt's work, which is by far the more important of what's under discussion (IMO), and I don't think it's a particularly good interpretation of the work by Sethares that interests me. Why not? Because the interesting work is not about musical practice but listener perception, about audio psychology, not compositional style.

I find this line of argument to be self proving. It seems like a variation of begging the question.

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.

This is just wrong. You are using western musical culture terms for what are actually physical phenomena.

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

> some of the pitches in the harmonic series are approximated by the notes of the chromatic scale

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.

day-after update: 12T in the octave, based on the following harmonics:

(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.

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

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 eleventh harmonic is a neutral second. Even according to the wikipedia section you quoted, you have to round up from F half-sharp to F sharp to get it, which doesn't make sense if you're trying to derive a musical scale from the harmonic series, as you would have already passed over the neutral third, diminished third, supermajor second, and all the rest of the intervals listed above which require no rounding at all.

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

You are confusing the physical harmonic series with the series of "harmonics" typically named in western music.

You're the one saying that F half-sharp (the actual 11th harmonic of C) is the same thing as F-sharp.

The problem here is that "tritone" has varying definitions, including at least: half an octave, three whole tones, two minor thirds, and that particular in JI, that leaves things a bit wooly to say the least.

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.

Thank you. The insistence on using the harmonic series to explain the source of the notes diatonic scale is anachronistic, post hoc and never really made any sense even on its own terms. There is nothing inherently "natural" about the seven basic notes that make up the diatonic scale, and there are even more problems than the major omissions that have been made in order to fit the members of the harmonic series into a diatonic mold that you have mentioned. The harmonic series cannot explain frequency ratios which does not have a power of 2 as a denominator, unless you invoke reciprocals. Furthermore the association of dissonance and consonance to the simplicity of the ratios is not consistent since the idea of dissonance is a social category, not a scientific one.

> Furthermore the association of dissonance and consonance to the simplicity of the ratios is not consistent since the idea of dissonance is a social category, not a scientific one.

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

No, but it is partly about beats and matching partials. See https://sethares.engr.wisc.edu/ttss.html which describes how the spectra of timbres affects which harmonies blend. You can synthesize inharmonic timbres to get arbitrary scales and tunings to blend more.

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 harmonic series cannot explain frequency ratios which does not have a power of 2 as a denominator

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".

>Let's consider the major sixth.

>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/

It seems to me that you are confusing two claims.

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.

> Furthermore the association of dissonance and consonance to the simplicity of the ratios is not consistent since the idea of dissonance is a social category, not a scientific one.

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"

The cited paper uses chord clusters containing four notes, and say nothing about dissonance as a dyadic concept which is what the discussion of ratios have been focusing on. The results are not what the PR piece states since Figure 1 of the cited paper[1] which sums up the results show that for the non-Westerners (Just Pakistani tribes of Kalash and Khow. If that is enough to claim "universality" in the social sciences then God help us.) low valence ratings for chromatic clusters are similar to their responses to major chords, so there is uncertainty about interpreting this result as their aversion to harsh dissonance or simply something they are unfamiliar with.

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...

Meta: this whole conversation amounts to a bunch of people who don't really understand tuning talking over-confidently about it. There are communities of experts on the subject, and if you bother to engage with any of them, you can learn. But you don't learn by starting out sure that you already know everything.

> 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 does not define the major scale, which is a heptatonic (7 tone) scale, at least not any more (or less) than it defines any other scale.

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).

I wasn't saying that the major scale is defined by harmonics, I'm saying that the harmonics which define what tones blend into chords are the reason for the major scale becoming our primary reference scale, whether or not it is tempered in the end.

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.

> Go and play the 5th harmonic and then the 7th harmonic on a string.

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.

> 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.

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.

Well, THANK YOU, and by all means start up a topic on the forums. Because you're wrong about the tritone :)

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

I'm a double bassist, so I have more than 12 pitches, unfortunately. ;-)

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.

I wonder how hard it would be to make tool that can take an existing audio file and digitally alter the frequencies in each section to give us a perfect just-intonation all throughout (by subtly lowering or raising the frequencies to form perfect ratios with other frequencies being played at the same time). You could even get this to run in real time on digital pianos, giving you perfect just-intonation depending on the chord or keys you play. The change in frequency would be unnoticeable alone but might be noticeable when played together with other keys, making the song more melodious.

> On continuous pitch instruments (e.g. violin, voice), this is entirely possible to do

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

>The set of notes defined by (1) are typically referred to as "Just Intonation", and was the basis for most western music (and some non-western music) until somewhere between about 1400 and 1600 (lots of room for discussion/debate there).

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.