4000/3993
4000/3993, the wizardharry comma or pine comma is an unnoticeable 11-limit (specifically 2.3.5.11-subgroup) comma with a size of roughly 3.03 cents. It is the amount by which a stack of three 11/10 submajor seconds falls short of the 4/3 perfect fourth; in other words it is equal to (4/3)/(11/10)3.
| Interval information |
pine comma
Triluyo comma
In terms of commas, it is the difference between 100/99 (S10) and 121/120 (S11), itself having an S-expression of S10/S11. It is also the difference between 540/539 (S12/S14) and 9801/9800 (S99). It factors into 13-limit commas as (1575/1573)⋅(2080/2079) or (625/624)⋅(6656/6655).
Temperaments
Tempering out this comma means the fourth is divided into an equal stepped tetrachord, the step of which is a trienfourth (from 1/3 of a fourth) or submajor second and because it is an ultraparticular it makes 12/11 and 10/9 equidistant from 11/10.
On the low-accuracy end, this may be reminiscent of how the porcupine comma splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup porcupine equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making porcupine as efficient and elegant as it can reasonably be. (As a rank-3 detemperament of porcupine, one way to extend it to include prime 7 is by tempering out 385/384, which septimal porcupine also tempers out.)
Tempering it out along with the schisma results in the rank-2 tertiaschis temperament. Tempering it out with the trimitone comma 8019/8000 (S9/S10, so that three 10/9's are also an 11/8) implies also tempering out the semiparticular 243/242 (S9/S11) = (3/2)/(11/9)2 leading to larry in the gravity family. Tempering it out with both the schisma and trimitone comma gives a description of 65edo in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.
Another strategy is to take advantage of the size of 121/120 (S11) so as to equate it with 144/143 (S12) = (16/13)/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, 325/324 (S10/S12), a comma with various advantages.
Observing that 4000/3993 = (540/539)/(9801/9800) also shows a natural path, if one is willing to split the octave in half, leading to hades.
There is a massive wealth of possible high-accuracy temperaments that temper out this comma, including 311edo.
Etymology
Wizardharry comma was presumably given by Gene Ward Smith in 2010[1]. It combines wizard and harry, two of the temperaments where the comma is tempered out, the latter being named after Harry Partch. Harry gave rise to a number of names related to the Harry Potter books, which might have been an inspiration for this combination.
Pine comma was given by Godtone in 2023, for its relevance and importance to the pine (7L 1s) scale.