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]\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]\sqrt{nd}[/math])	~1.28366 bits
Comma size	unnoticeable
S-expression	S10 / S11
open this interval in xen-calc

4000/3993, the wizardharry comma (from wizard and harry (named after Harry Partch)) or pine comma (for its relevance and importance to 7L 1s) 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 (12/9)/(11/10)³ = S10/S11.

In terms of commas, it is trivially the difference between S10 = 100/99 and S11 = 121/120 or less trivially between S12/S14 = 540/539 and S99 = 9801/9800. It factors into 13-limit commas as (1575/1573)(2080/2079) or (625/624)(6656/6655).

Temperaments

Tempering it out 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 porkypine equates 11/10 with the superparticulars adjacent to it of 12/11 and 10/9, tempering S10 and S11 as the porcupine comma is 250/243 = S10² * S11 so through porkypine 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) allows us to also temper the semiparticular 243/242 = (12/8)/(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 S12 = 144/143 = (16/13)/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering S10/S12 = S25*S26 = 325/324, a comma with various advantages. Observing that S10/S11 = (S12/S14)/(S33/S35 = S99) then 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 S17 = (17/12)/(24/17).

These are just a few examples, but there is a massive wealth of possible high-accuracy temperaments that temper out this comma, including the 41-limit temperament 311edo.