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!
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".
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.
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 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.
> 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/
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.
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
11 comments
[ 5.1 ms ] story [ 31.7 ms ] threadXenharmlib 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!
^^ 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".
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.
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.
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/
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/
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!)