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 syntonic comma and the semicomma, (81/80)/(2109375/2097152); tempering both leads to 31edo
- the difference between the diesis and the magic comma, (128/125)/(3125/3072); tempering both leads to the trivial tuning 3edo
- the difference between two diaschismas and the tetracot comma, (2048/2025)2/(20000/19683); tempering both leads to 34edo
- equivalently, between one diaschisma and the kleisma, (2048/2025)/(15625/15552); tempering both thus also corresponds to 34edo
- finally, between two dieses and the just chromatic semitone, (128/125)2/(25/24); tempering both leads to the trivial tuning 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 (which exceeds 3/1 by the magic comma), 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 (würschmidt), 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 ¢