The Mathematics of Tuning Systems(math.ucr.edu)
math.ucr.edu
The Mathematics of Tuning Systems
https://math.ucr.edu/home/baez/tuning_talk/
18 comments
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/
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/
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.
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.
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
[1] https://arxiv.org/abs/1203.5101
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.
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!
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!
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.
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.
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.
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.
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
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
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
Really enjoyed this writeup!
Having watched a piano tuner tune an old piano, they measured the frequencies of a few notes and then tuned pairs of notes by ear.
I wonder where that ends up compared to the formal analysis here.
(For me, the math is much more approachable, I am not musically inclined!)
Having watched a piano tuner tune an old piano, they measured the frequencies of a few notes and then tuned pairs of notes by ear.
I wonder where that ends up compared to the formal analysis here.
(For me, the math is much more approachable, I am not musically inclined!)
See my comment about inharmonicity in real instruments. Unfortunately it's not as simple as 440 Hz * 2^(n/12)
When you think the article is ending, you get more links for further indulging! Thanks.
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".
^^ 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".
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.
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.
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.
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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/