Musical tuning is essentially a branch of mathematics, especially the way I approach it. Often in mathematics and science the focus is on novelty, on fresh discoveries, where fresh means not previously encountered by the human mind. This focus is unnecessary in math and science, is somewhat distracting or misleading, and will likely serve us much less well in the future. Since the time of Kepler and Galileo, math and science have expanded in a stunning fashion. Predicting the future is a fools game, but it seems unlikely that environmental constraints will permit continuous growth in extracting resources and dumping wastes. The modern trend of constant growth seems destined to end sooner rather than later. Math and science will be of great value in any post-growth society. To keep them alive, though, the focus will need to shift away from novelty.

So here is a historical preface to the schismatic tuning I have (re)discovered: Emilio de’ Cavalieri’s mysterious enharmonic passage - a modern rendition of a renaissance recovery of an ancient Greek tuning! Paul Erlich has written a thorough discussion of tuning A Middle Path Between Just Intonation and the Equal Temperaments - I have barely scratched the surface of this paper! I imagine that schismatic tuning is described in there somewhere! I would just like to share my (re)discovery here of this one small facet of the vast universe of tuning. I offer it as an invitation to explore further!

A quick review of fundamentals. A musical interval is the relationship between two pitches, which can be analyzed as the ratio between their frequencies. If pitches are an octave apart, their frequencies are in a 2:1 ratio; a fifth apart, a 3:2 ratio; a major third apart, a 5:4 ratio. These ratios are ideal. Just Intonation is a tuning that uses these ideal ratios. But for a variety of practical reasons, it is often useful to adjust, or temper, these ratios. There is no perfect solution to the puzzle of temperament. Modern keyboard tuning adjusts the fifth to 2^(7/12) ~= 1.4983 and the major third to 2^(4/12) ~= 1.26. The human ear can detect reasonably well the difference between this tempered major third and the ideal of 1.25.

Schismatic tuning is actually a family of tunings. I will present one version, based on dividing an octave into 53 equal steps, rather than the conventional 12. With 53 steps available, a fifth is tempered to ~1.499941 (31 microsteps) and a major third to ~1.248984 (17 microsteps). The fifth is improved, but the conventional tuning was already very good; the main improvement is in the major third.

How can such an improved tuning be adapted to a conventional keyboard? Here is my proposed schismatic tuning:

The top row names the keys on the keyboard. The second row gives the number of microsteps from the low C to the particular key of that column. The bottom row re-expresses that pitch in terms of cents. The conventional tuning would result in pitches of 0, 100, 200, 300, etc. cents. So this last row makes clear the difference in pitch between the schismatic tuning and conventional tuning, e.g. D is 3.774 cents sharper.

Some points to observe:

- Almost all of the fifths are 31 microsteps, i.e. very accurate. From D to A is only 30 microsteps, though.
- Four of the major thirds are the ideal 17 steps: C to E, F to A, G to B, and D to F#. The others are sharp by a microstep, i.e. closer to a pythagorean major third, 81:64.
- The sizes of the chromatic intervals in this tuning are not all the same: 4, 5, 4, 4, 5, 4, 5, 4, 5, 5, 4, 5.
- The syntonic comma is not tempered. E.g. moving by fifths up from C to E, one must cross the "wolf" fifth from D to A. This is a distinctly unconventional tuning.

One can certainly play in any key signature with this tuning - none of the intervals is too far off. But certainly a piece of music will sound different when the key signature is changed. This tuning does allow though a simple dynamic shift as outlined in my post Dynamically Tuned Piano. With perhaps a push of a foot pedal, A can be sharpened by a syntonic comma:

or a different pedal could instead flatten the D by a syntonic comma:

These shifts will move the wolf fifth up or down a fifth, and also rotate which major thirds are pythagorean, etc.

lots of history here: https://en.wikipedia.org/wiki/Schismatic_temperament

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