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 :)))