The Mathematics of Tuning Systems
math.ucr.edu
math.ucr.edu
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/
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.
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!
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.
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.
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.
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!)
^^ 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 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.