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Tiny Musical Intervals

Music theorists have studied many fractions of the form

that are close to 1. They’re called 5-limit commas. Especially cherished are those that have fairly small exponents—given how close they are to 1. I discussed a bunch here:

• Well temperaments (part 2).

and I explained the tiniest named one, the utterly astounding ‘atom of Kirnberger’, here:

• Well temperaments (part 3).

The atom of Kirnberger equals

Two pitches differing by this ratio sound the same to everyone except certain cleverly designed machines. But remarkably, the atom of Kirnberger shows up rather naturally in music—and it was discovered by a student of Bach! Read my article for details.

All this made me want to systematically explore such tiny intervals. Below is a table of them, where I list the best ones: the ones that are closest to 1 for a given complexity.
The first eleven have names, and many of them play important roles in music! But beyond that point, all but one remain unnamed—or at least I don’t know their names. That’s because they’re too small to be audible, and—except for one—not even considered to be of great theoretical importance.

I’ll list these numbers in decimal form and also in cents, where we take the logarithm of the number in base 2 and multiply by 100. (I dislike this blend of base 2 and base 10, but it’s traditional in music theory.)

Most importantly, I list the monzo of each numbers. This is the vector of exponents: for example, the monzo of

is

In case you’re wondering, this term was named after the music theorist Joseph Monzo.

Finally, I list the Tenney height of each number. This is a measure of the number’s complexity: the Tenney height of

is

The table below purports to list only 5-limit commas that are close to 1 as possible for a given Tenney height. More precisely, it should list numbers of the form 2i 3j 5k that are > 1 and closer to 1 than any number with smaller Tenney height—except of course for 1 itself.

You’ll see there’s a big increase in Tenney height after the schisma. This is very interesting: it suggests that the schisma is the last ‘useful’ interval. It’s useful only in that it’s the ratio of two musically important commas, the syntonic comma and the Pythagorean comma. Life in music would be simpler if these were equal, and in well-tempered tuning systems it’s common to pretend that they are.

All the intervals in this table up to the schisma were discovered by musicians a long time ago, and they all have standard names! After the schisma, interest drops off dramatically.

The atom of Kirnberger has such amazing properties that it was worth naming. The rest, maybe not. But as you can see, I’ve taken the liberty of naming the smallest interval in the table the ‘quark of Baez’. This is much smaller than all that come before. It’s in bad taste to name things after oneself—indeed this is item 25 on the crackpot index—but I hope it’s allowed as a joke.

I also hope that in the future this is considered my smallest mathematical discovery.

Here is the Python code that should generate the above information. If you’re good at programming, please review it and check it! Someone gave me a gift subscription to Claude, and it (more precisely Opus 4.5) created this code. It seems to make sense, and I’ve checked a bunch of the results, but I don’t know Python.

Gene Ward Smith

In studying this subject I discovered that tiny 5-limit intervals were studied by Gene Ward Smith, a mathematician I used to see around on sci.math and the like. I never knew he worked on microtonal music! I am sad to hear that he died from COVID-19 in January 2021.

I may just be redoing a tiny part of his work: if anyone can find details, please let me know. In his memory, I’ll conclude with this article from the Xenharmonic Wiki:

Gene Ward Smith (1947–2021) was an American mathematician, music theorist, and composer.

In mathematics, he worked in the areas of Galois theory and Moonshine theory.

In music theory, he introduced wedge products as a way of classifying regular temperaments. In this system, a temperament is specified by means of a wedgie, which may technically be identified as a point on a Grassmannian. He had long drawn attention to the relationship between equal divisions of the octave and the Riemann zeta function.[1][2][3] He early on identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the regular mapping paradigm.

In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a crystal oscillator. The device in turn was controlled by HP 9800 series desktop computers, initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and pajara.

Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note highschool scales, led to interesting (omnitetrachordal) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.

Gene was among the first to consider extending the Tonnetz of Hugo Riemann beyond the 5-limit and hence into higher dimensional lattices. In three dimensions, the hexagonal lattice of 5-limit harmony extends to a lattice of type A3 ~ D3. He is also the first to write music in a number of exotic intonation systems.

Historical interest

• Usenet post from 1990 by Gene Smith on homomorphisms and kernels
• Usenet post from 1995 by Gene Smith on homomorphisms and kernels

See also

• Microtonal music by Gene Ward Smith
• Hypergenesis58 (a scale described by Gene Ward Smith)

References

[1] Rusin, Dave. “Why 12 tones per octave?”

[2] OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: OEIS: A117536.

[3] OEIS. Increasingly large integrals of the Z function between zeros: OEIS: A117538.

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