[HN Gopher] The Mathematics of Tuning Systems
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The Mathematics of Tuning Systems
Author : u1hcw9nx
Score : 79 points
Date : 2026-01-31 09:04 UTC (4 days ago)
(HTM) web link (math.ucr.edu)
(TXT) w3m dump (math.ucr.edu)
| kensai wrote:
| When you think the article is ending, you get more links for
| further indulging! Thanks.
| aa-jv wrote:
| A very nicely written and detailed article, with many details I
| have not personally learned of, in my own music technology
| hacking so far .. if anyone else is interested in writing
| software for musical tuning systems, both Xenharmlib and PyTuning
| are very extensive libraries for the job .. Xenharmlib for C++
| and PyTuning for python.
|
| Xenharmlib has deep support for intervals, chords, scales, non-
| standard notations, and advanced topics like non-Western
| harmonics, diatonic set theory, and non-octave-repeating systems
| and also allows for the mathematical manipulation of ratios and
| structures (harmonic exploration).
|
| PyTuning allows for generating scales from ratios/cents, EDO,
| just intonation, and custom temperaments and facilitates
| calculations such as frequency ratios, comma approximations, and
| temperament comparisons, aligning with the article's derivations
| and trade-off discussions at a similar depth.
|
| I hope to see more math of music articles in the future .. its a
| fascinating subject indeed!
| uwagar wrote:
| As we'll see, seeking the Pythagorean ideal causes trouble. It
| will unleash the devil in music.
|
| ^^ from the article. this kind of thing bothers me. what is the
| devil? for whom is it the devil? music doesnt havent to be
| "christian" to "sound good".
| giraffe_lady wrote:
| It's not a literal allusion to satan and never was, it's akin
| to the idiom "the devil is in the details."
|
| It was nearly or actually impossible to harmonize or progress
| tritones while following the medieval "rules" for church music
| composition. So it is someone complaining about their work, not
| a statement about musical morality or whatever.
| uwagar wrote:
| multiply by 3^m/2^n is a human construction. and it breaks
| after a bunch of m,n. to expect it "not to break" is well an
| expectation. nature always has a comma, its not clockwork.
| giraffe_lady wrote:
| Very cool explanation. Something I've come across a few times on
| here was wanting to explain how 12tet "includes" or handles
| approximations of intervals from other scales, and how that
| affects the musical choices of musicians or especially the
| notation choices for transcription of improvised music.
|
| But it's impossible to explain without getting into like, what is
| even the problem solved by tuning systems. Without the intuition
| that comes from making music, programmers and engineers see the
| fractions & obvious series and get too fixated on finding the
| "perfect" system. When these are much more physical tools,
| created over time to make certain processes easier. Tuning
| systems are more like a woodworker's knives than like the unit
| circle: being perfect does not make them better tools for
| creation if they are already fit enough.
| TheOtherHobbes wrote:
| 12tet is a _practical_ compromise between overtone
| approximation and playability.
|
| Any fretted or keyed acoustic instrument with more than 12
| notes/octave is extremely difficult and expensive to build and
| hard to tune and play.
|
| The music is also hard to notate, because there are so many
| more possible pitch positions.
|
| So 12 notes became a practical default for instrument builders.
| And from there, ET became a practical default for tuning to
| allow smooth-ish modulation through different keys. The errors
| in the tuning are the same for every key, so the tuning
| relationships stay the same while the key root changes.
|
| That doesn't quite happen because overtone perception and
| beating effects aren't completely linear. But it avoids the
| obvious out-of-tune notes you get when playing in distant keys
| with non-ET tunings.
| tzs wrote:
| An interesting option for fretted instruments is to build it
| with 12 regular frets per octave, and then add temporary
| extra frets if you want to make specific notes more in tune
| for the key you are in.
|
| This was common for lute players in the Renaissance and
| Baroque periods. Lute frets were usually made of the same
| material as strings and were tied on around the neck, so
| adding an extra full width fret was not hard.
|
| Of course being tied on lute frets were also movable, so they
| could also reposition the regular 12 frets instead of adding
| extras.
|
| Instead of tying on an extra full width frets another common
| option was partial frets. They would glue a piece of string
| or wood under a specific string, to just give an option to
| play one note with either the regular fret or the more in
| tone little fret (called a "tastino" (plural is "tastini")).
|
| In this video [1] Brandon Acker uses a guitar that his
| luthier friend was building that had not yet had the frets
| installed to demonstrate with lute style tied on frets some
| different 12 tone tunings, then demonstrates using tastini to
| improve specific notes, such as little fret on the G string a
| little before the first regular fret so that G# and A can be
| different notes.
|
| The tastini he uses in the video are simply pieces of string
| held on with a piece of tape, so quick and easy to add and
| remove.
|
| [1] https://www.youtube.com/watch?v=tiKCORN-6m8
| HPsquared wrote:
| With real instruments, you must also manage inharmonicity.
|
| A real piano string, for instance, is made of metal and resists
| bending slightly unlike an idealised string. This affects higher
| harmonics more than lower ones (think of all the bends in the
| string on the 7th harmonic, for example). This increases the
| harmonic frequencies slightly above exact integer multiples of
| the fundamental.
|
| As a result, pianos require "stretched tuning" so the harmonics
| better match the higher notes. It's always a bit of a compromise.
| The higher harmonics will be more "off" than the lower ones.
|
| So even if you were to tune the fundamental frequency of all the
| keys on a piano "perfectly" in a given key (so-called Just
| Intonation), the harmonics would not perfectly match up.
| jankhg wrote:
| There is actually an interesting mathematical method for piano
| tuning that takes harmonics into account [1]. The core idea is
| to minimize the integral over the logarithm of the sum of all
| spectra. This basically favors spectra that are smeared out
| less. Instead of the logarithm, one could also use another
| sublinear function, such as the square root, but I guess it
| just makes for a better story to call it ,,piano entropy
| tuner". The paper also shows a nice plot about the ,,stretched
| tuning" that you mentioned.
|
| [1] https://arxiv.org/abs/1203.5101
| heikkilevanto wrote:
| A bit too mathematical for my taste. I learned my tuning theory
| by owning a harpsichord, and learning to tune it. A harpsichord
| is more sensitive to the "rounding errors" in equal tuning, owing
| to the richer overtones, so equal temperament does not sound
| quite as good a compromise as it does on a piano. And those
| historical temperaments are so much easier to tune by ear.
| Besides that is what the they used at the time of Bach, so
| historically correct for playing Baroque music.
| nayuki wrote:
| > Some people who never play the piano claim it would be easier
| if had all white keys, or simply white alternating with black.
|
| Actually, I'm one of those people. For over 20 years, I struggled
| with playing piano because I would have to memorize a different
| fingering pattern for the major scale in 12 different keys. I
| knew the mechanical process of it, but it was hard to develop the
| muscle memory and play songs by ear based on intuition alone. So
| I was most comfortable playing in C major (white keys only) and
| using mechanical/electronic transposition.
|
| In the year 2024, I stumbled upon the Janko piano layout (
| https://en.wikipedia.org/wiki/Jank%C3%B3_keyboard ), which turns
| out to be the smallest modification to standard piano that
| results in an isomorphic keyboard. I kid you not, I was up and
| running in less than 5 minutes - I just treated the layout as if
| it was a pattern to learn on standard piano, except that it was
| the only pattern I ever had to learn. On Janko, I found it much
| easier to play songs by ear, in any key. I wish I discovered
| Janko earlier, as standard piano was never a good fit for my
| brain.
|
| For anyone who is curious to try, here's a software Janko piano
| keyboard that you can play right in the web browser:
| https://novayashkola.org/janko/
| simojo wrote:
| I suppose that's why the harpejji [0] has recently gained
| popularity? I too have wished for an isomorphic keyboard. All
| of the non-stacked ones become either too wide or the keys
| become too skinny. Example: Dodeka Keyboard [1]. I know that
| the Lumatone [2] exists too, but it is too progressive for my
| taste :)
|
| As a side note, the traditional keyboard size is not
| representative of the average pianist's hand size. David
| Steinbuhler [3] has been making modified traditional keyboard
| layouts by varying the width of the keys slightly, and people
| rave about it. I've had the chance to visit his shop in
| Titusville, Pennsylvania, where he designs them. It's a totally
| enhanced playing experience, even for someone like me who can
| play a 10th without difficulty.
|
| [0] https://en.wikipedia.org/wiki/Harpejji [1]
| https://dodekamusic.com/ [2] https://www.lumatone.io/ [3]
| http://dsstandardfoundation.org/the-standards/
| mbo wrote:
| If anyone wants to hear the practical effects of a 1/4 comma
| meantone temperament compared to an equal temperament, Brandon
| Acker gives a wonderful overview on the classical guitar:
| https://www.youtube.com/watch?v=tiKCORN-6m8
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