Würschmidt comma
| Interval information |
The Würschmidt comma ([17 1 -8⟩ = 393216/390625) is a small 5-limit comma of 11.4 cents.
It is the amount by which an octave-reduced stack of eight classical major thirds falls short of a perfect fifth: (5/4)8(393216/390625)/4 = 3/2, which comes from 5/4 being a convergent in the continued fraction of [math]\displaystyle{ \sqrt[8]{6} }[/math]. (Therefore, it is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves), that is, (5/4)7(393216/390625)/4 = 6/5.)
In terms of commas, it is the difference between:
- the difference between the diesis and the magic comma, (128/125)/(3125/3072); tempering both leads to 3edo
- between two diaschismas and the tetracot comma, (2048/2025)2/(20000/19683), corresponding to 34edo
- or equivalently, between one diaschisma and the kleisma, (2048/2025)/(15625/15552), thus also corresponding to 34edo, and finally,
- between two dieses and the just chromatic semitone, (128/125)2/(25/24), corresponding to 3edo
The last expression means that if you temper it out in any nontrivial tuning (that is, not 3edo), there is an exact neutral third between 5/4 and 6/5, which usually represents ~11/9 (or more accurately 49/40, tempering S49 instead of (or in addition to) S9/11).
Notice that magic is a lower-accuracy analogue of würschmidt, reaching 3/1 with (5/4)5, and a trivial analogue of wurschmidt is dicot, where 3/2 is reached by (5/4)2. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of 5/4 go 2 (dicot), 5 (magic), 8 (wurschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of 25/16 by another 128/125; finally, at (5/4)11 / (12/1), we get magus, a sharp-major-third analogue of würschmidt, which is in some sense the logical dual of magic, which tunes 5/4 flat. There is no real reason to use magus unless you want a sharp 5/4 and/or want to use a temperament that happens to support it, a notable tuning of which is 46edo.
Temperaments
Tempering it out leads to the würschmidt family of temperaments. Similar to meantone, it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat. Unlike meantone, it is far more accurate; an ideal tuning of wurschmidt sharpens the 5/4 by up to 1.43 ¢ (corresponding to 1/8-comma wurschmidt, where 3/2's are pure). Combining it with meantone gives 31edo as the first real tuning but increasingly good 5-limit edo tunings after 31 (all of which distinguish the syntonic comma) are 34edo and especially 65edo, although 34+65 = 99edo certainly makes sense if you prefer its tuning properties. 65edo has the distinguishing property of being the smallest würschmidt edo with a 5/4 in the aforementioned ideal tuning range, and corresponds to combining it with schismic (especially the extension to include prime 19 called nestoria) and gravity, so is a very accurate 5-limit tuning that extends naturally to prime 11 (through the aforementioned 243/242 or equivalently through S9/S10 or S10/S11) and prime 19 (through nestoria), among others. In an ideal tuning of wurschmidt, 5/4 is sharpened by ¢