What I Learned Writing an Album in Just Intonation

I'd love to get some feedback on this, because I'm not sure I've hit the original goal of this article.

For a long time, the topics of just intonation, harmonic lattices, and of their relationship with temperaments were difficult for me to understand, despite reading the theory on them. It's not until I applied the concepts that could say I understood them.

I tried writing the "missing article" that would have given me this understanding right away, banking on the added value of interactive figures. Right now I suspect that I failed in the same way as my early readings did: this article might be hard to understand unless you're already familiar with the topic, which is a bit of a chicken-and-egg problem. I think I glossed over some concepts, and it sometimes looks like I'm pulling numbers out of thin air in some examples. Nonetheless I hope it will help some people understand these topics. Maybe what's needed is a slower-paced version of this as a longer series.

For practical performance of alternate intonations, you can tune a guitar so that a particular chord shape presents perfect roots and 5ths, while arranging for a single string to be a perfect third (e.g, a few cents "flat" compared to equal temperament.)

When playing this same chord shape (or an appropriate derivative) up the neck, the relative intervals is maintained for other chords, allowing perfect intonation not just of the root chord but of other chords in the key family.

For chord shapes where a flatted string cannot be used as a 2nd or 3rd, it will sound objectionably flat, but for many chord shapes it's possible to fret the flat string with additional pressure to bring it back into tolerance.

Mastering this technique makes the guitar sound much more in tune with itself which is a wonderful sensation as a player.

Here are three additional references on applied Just Intonation.

The first book is abstracted away from composition and provides a more general description of tuning systems.

The next two are composition focused, explaining how tuning system is an option for musical works.

1) The book "How Equal Temperament Ruined Harmony" provides excellent historical context on why ET was a great solution and how we kind of got stuck with it.

While it could be read by anybody, having a bit of musical knowledge OR physics of acoustics knowledge may benefit you.

https://www.amazon.com/Equal-Temperament-Ruined-Harmony-Shou...

There are also two composers who recognize ET as a constraint (as opposed to the "the only option").

2) Harry Partch built many of his own instruments to circumvent issues from 12ET and composed music just for them. He describes his conceptualizations of JI in "Genesis Of A Music":

https://www.amazon.com/Genesis-Music-Account-Creative-Fulfil...

3) Paul Hindemtith also has some opinions on the matter, though both Partch and I disagree with his points of view. Nonetheless he has some at-the-time innovating ideas, some of which are valid and others of which may not have aged well. You can read about them in "The Craft of Musical Composition"

https://www.amazon.com/Craft-Musical-Composition-Theoretical...

Sometimes I like writing stuff in odd scales and intonations. The "Lucy" tuning is also very pleasant [0]. I once met a producer who had guitars fretted to this tuning. It doesn't lend itself well to DAWs that have an underlying MIDI note logic. Practically it's better to work in Pd/Max with raw oscillators and arithmetic than to try shifting MIDI note values around with bend. The ZynAddSub engine (also used in Yoshimi synth) has a nice scales filter.

Probably worth mentioning Jacob Joaquin [1] who did this stuff in Csound back in the 90s. On one of his patches I based a generative music system for a game, it was the "Harmonic tree" - very similar to the system in the article - repeated multiplication by integer fractions at each branch.

Using simple timbres (like flute/organ/sine) you can make big but crystal clear sounding chords, and then slowly changing one "note" at a time you get an ever evolving soundscape that blows anything "Eno" off the map. Its in a sweet creative area between composition and additive synthesis.

[0] http://www.harmonics.com/lucy/lsd/colors.html

[1] https://github.com/jacobjoaquin

Haven't read it fully (to be honest this topic is new to me), but it's funny that as I read this article, I also happen to listen to a microtonal music: Flying Microtonal Banana by King Gizzard & The Lizard Wizard.

Both the name of the album and the name of the band sounds funny, trust me when I say this: It's one of the best microtonal album out there. They really mixed the sound element of turkish(?) music into the album. Highly recommended!

Should be noted that any ensemble that consists only of continuous-pitch instruments (eg. strings, trombone, human voice, fretless guitar) is free from the shackles of 12TET and its harmonies naturally tune to as close to "pure" intervals as the performers’ skills allow.

Overall a good intro on the subject. I feel like at the start it might benefit from being a bit more explicit about pitches being frequencies and ratios being intervals; probably just worth a reminder for people who aren’t as familiar

Might be interesting to talk about how the usual ratios come from the harmonic series. For sounds that don’t produce a harmonic series, other potentially non-integer ratios can actually sound more consonant. The youtube channel New Tonality[0] has a bunch of great videos about this

Also wanted to mention that I’ve been working on a piece of commercial software[1] for working with freestyle/adaptive just intonation, if anyone’s interested

[0]: https://www.youtube.com/@new_tonality [1]: https://www.dmitrivolkov.com/projects/pivotuner/

This reminds me of when I stumpled into a music theory flame war by proposing the "scale": 1, 15/14, 15/13, ..., 15/8, 2. https://music.stackexchange.com/questions/66620/what-is-a-te...

This set of 8 nodes (can be transposed to whichever base frequency) has the lowest common denominator between any pair of nodes among all 8-node scales.

I thought that might make it a good choice for a just intonation scale (which can really be any subset of frequencies you build you music from), but it turned out a sensitive subject.

What a marvelous article about intonation and temperament. It is one of those expositions that approach perfection in conveying the subject matter. I am no music theorist but rather an engineer with an appreciation for all things wave- or signal-oriented. I will want to read it multiple times to savor the explanations. The author's album based on just intonation is linked in the article ("A Walk Through The Ambient Garden") and I highly recommend it: it is a mesmerizing composition.

Readers who enjoyed this essay might also enjoy this one, which goes into further depth on the mathematics of musical intervals, the connection between the "blue note" in Equal Tempered jazz music, and more:

https://iopscience.iop.org/article/10.1088/0034-4885/72/7/07...

As an added bonus, the author draws an extended analogy between musical intervals and energy levels in quantum mechanics toward the end of the article, and indeed the study of the patterns in intervals between and among energy levels in the simple harmonic osciallator remains an active area of reasearch!

For more pieces in just intonation, check out https://en.wikipedia.org/wiki/List_of_compositions_in_just_i...

When I compose, I usually begin by sitting quietly and humming or singing out loud until I have a melody that I like. Because I was born and raised in the west, acoustically I'm naturally accustomed to a 12 tone equal temperament.

Atonal music is more difficult for me to grok, and I kind of imagine that for you to be fluent and compose in these sort of microtonal scales you would need to be introduced to it at a young age - in much the same way that it's significantly easier to learn a second language if you grow up in a bilingual environment.

This is interesting, but the harpsichord based samples really made me not wanna listen. But that’s probably just me disliking harpsichord

If the author is here, the audio clips don't play for me. Firefox console has: Uncaught (in promise) TypeError: document.querySelector(...).contentDocument is null

I feel like I can hear beat frequencies in the 12TET chords that I don't hear in the JI chord examples.

Anybody else hearing that? Or is that my imagination?

One of the linked music examples, “occultation” by Pure Code, is beautiful ambient music.

“ This also has the reputation of being terribly impractical, which is always a selling point for me.”

That was truly LOL. Thank you.

Avez-vous la version française, monsieur ?

When I started with violin, since my academic major was math, looked into the 12th root of 2 and ratios of small whole numbers, dissonance from beats, etc. stuff.

Then I learned (partly from a really good violinist):

(1) On violin, can use the ratios of small whole numbers to check intonation. So, get to check the intervals of octave, fifth, fourth, major third. That is, play two notes, each on its own string, together, listen for the beats, and adjust pitch (move a finger) until the beats go away.

That approach starts with the tuning of the violin, notes G, D, A, E, each on its own string, starting with the first G below middle C and going up by fifths (frequency ratio 3/2). The beats are easy to hear, and making the beats go away is what essentially all violin players do, whatever slightly different frequencies might come from other approaches to tuning.

(2) For the rest, there is a single, simple overriding principle: Make it sound good. Uh, there is no practical way to play a violin and get all the frequencies anywhere except close, varying with vibrato, and sounding good!

(3) The math of the 12th root of 2, etc. is too precise for violin -- can't expect to play frequencies to such precision and with vibrato, on that wooden box, with flesh fingers, real violin strings, bow rosin, etc. and really don't need to.

E.g., for the math, there is no good guarantee that a note played on violin is really accurately what the math and physics (a solution of a certain differential equation) define as a periodic function -- thus, lots of considerations of overtones start to become imprecise and, thus, moot.

Again, make it sound good! So, sure, for a major third, ratios of small whole numbers can be relevant, but for a minor third, mostly just guess at the semi-tone and make the interval sound good!

(4) For semi-tone guessing: The famous Bach E-major Partita starts with E (one octave above the open E string, half way along the string), D# (down one semi-tone), and then the E again. So, it's the three notes, played quickly, with the D# inserted and left quickly, with the musical effect an introductory bright heralding -- the relevance of the semi-tone as the 12th root of 2 is lost and instead just play the E with the little finger and for the D# squeeze in the next finger until it passes for a semi-tone and gets the heralding musical effect.

(5) For much more, can play the Bach Chaconne with its many chords, sometimes on all 4 strings -- no way can a violinist try to honor some precise tuning arithmetic for all the intervals played. E.g., the piece starts in D minor and with the "D minor" chord, that is, D, F, A. On a piano that would be the first D above middle C, the first F, that is, 3 semi-tones, above the D, and the A, a fifth above the D. But on violin that would have both the D and the F on the same string, so Bach put the F an octave higher, on the E string, a semi-tone above the E string. Sooooo, to play that opening chord, play the D and the A together, and, continuing right away all with one down bow, with the A and the F together. Then the F is from the index finger at a guess of a semi-tone above the E string. So, the bow starts on the D and A strings, both open (no fingers on the strings) and, thus, brilliant, and then both the open A string and the fingered F. So, the A and F 'would' be just an okay major third but the F is up an octave so that the A and F are an interval of 8 semi-tones and, actually, dissonant. Since the F is on the E string which is still brilliant even though fingered and not open, all three strings are brilliant, and the opening is dramatic and in a minor key. Some of the piano and orchestra arrangements make this first chord dramatic!

In all that for that opening chord, on violin, the 12th root of 2 and ratios of small whole numbers don't play a central role, and the same for the rest of the piece, nearly all chords.

Sure Bach knew very well what he was doing, indeed, in that music, SHOWING that could make music in all the keys, and had all those pages and pages, for violin, in another book, for cello, in another book, piano with the intervals and chords "sounding good".