4000/3993

Interval information

Ratio	4000/3993
Factorization	25 × 3-1 × 53 × 11-3
Monzo	[5 -1 3 0 -3⟩
Size in cents	3.032314¢
Names	wizardharry comma, pine comma
Color name	1u3y31, triluyo unison, Triluyo comma
FJS name	[math]\displaystyle{ \text{AA1}^{5,5,5}_{11,11,11} }[/math]
Special properties	reduced
Tenney height (log2 nd)	23.929
Weil height (log2 max(n, d))	23.9316
Wilson height (sopfr(nd))	61
Harmonic entropy (Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])	~1.28366 bits
Comma size	unnoticeable
S-expression	S10 / S11
Open this interval in xen-calc

4000/3993, the wizardharry comma or pine comma is an unnoticeable 11-limit 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, therefore it is equal to (4/3)/(11/10)³ = S10/S11.

In terms of commas, it is the difference between 100/99 (S10) and 121/120 (S11) or 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 and it is for these reasons that 2.3.5.11 porcupine equates 11/10 with the superparticulars adjacent to it of 12/11 and 10/9, tempering out S10 and S11 as the porcupine comma is S10² × S11 so through the extension to the 11-limit we make porcupine as efficient and elegant as it can reasonably be.

Tempering it out along with the schisma results in the rank-2 tertiaschis temperament. Tempering it out with the trimitone comma S9/S10 = 8019/8000 (so that three 10/9's are also an 11/8) implies also tempering out the semiparticular 243/242 = (3/2)/(11/9)² = S9/S11 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 S11 so as to equate it with 144/143 = (16/13)/(11/9) = S12, 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, which extends naturally to the 17-limit by tempering out 289/288 = (17/12)/(24/17).

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. 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.