User:2^67-1/Sandbox
Testing
The blackwood-dicot-semaphore equivalence continuum is a continuum of 7-limit rank-3 temperaments describing the set of all 7-limit rank-3 temperaments supported by 10edo. Any rank-2 temperament supported by 10edo can thus be represented by a line between two points in this continuum.
All temperaments in the continuum satisfy (25/24)p(49/48)q ~ 256/243, equating a stack of dicot commas (25/24) and semaphore commas (49/48) with the blackwood comma (256/243).
The blackwood comma is the characteristic 3-limit comma tempered out in 10edo.
User:Godtone notes the following JIP's, each one corresponding to a 1D continua contained therein, for which increasingly efficient approximations generally represents increasingly efficient 7-limit temperaments:
- log2(256/243) / log2(25/24 * 49/48) = 0.8482245109 ; this is the JIP of p=1, q=1 (equiv. to p=-1, q=-1)
- log2(256/243) / log2(25/24) = 1.2766647429 ; this is the JIP of p=1, q=0 (equiv. to p=-1, q=0)
- log2(256/243) / log2(50/49) = 2.5796543166 ; this is the JIP of p=1, q=-1 (equiv. to p=-1, q=1)
- log2(256/243) / log2(2401/2400) = 125.1... ; this is the JIP of p=-1, q=2 (equiv. to p=1, q=-2)
- log2(256/243) / log2(49/48) = 2.5275365063 ; this is the JIP of p=0, q=1 (equiv. to p=0, q=-1)
- log2(256/243) / log2(25/24 * 50/49) = 0.8540148427 ; this is the JIP of p=2, q=-1 (equiv. to p=-2, q=1)
Importantly, each JIP corresponds to a rational, so that, for example, (p, q) = (1, -2) is equivalent to (p, q) = (2, -4) and to (p, q) = (-1, 2) but not to (1, 2).
Also note that continua separated by 2401/2400 are meaningfully different, but due to the efficiency of 2401/2400, one may want to examine the continuum of all 7-limit temperaments supported by 10edo for which 2401/2400 is tempered.
| p | q | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| 0 | 0 | Blackwood | 256/243 | [8 -5 0 0⟩ |
| 0 | 1 | Archytas (squared) | 4096/3969 | [12 -4 0 -2⟩ |
| 0 | 2 | Buzzard | 65536/64827 | [16 -3 0 -4⟩ |
| 0 | 3 | Slendric (squared) | 1058841/1048576 | [-20 2 0 6⟩ |
| 0 | ∞ | Semaphore | 49/48 | [-4 -1 0 2⟩ |
| 1 | 0 | Srutal | 2048/2025 | [11 -4 -2 0⟩ |
| 1 | 1 | Mirwomo | 33075/32768 | [15 -3 -2 -2⟩ |
| 1 | ∞ | Jubilic | 50/49 | [1 0 2 -2⟩ |
| 2 | 0 | Negri | 16875/16384 | [-14 3 4 0⟩ |
| ∞ | 0 | Dicot | 25/24 | [-3 -1 2 0⟩ |
| ∞ | 1 | Jubilic | 50/49 | [1 0 2 -2⟩ |
| ∞ | 2 | Breedsmic | 2401/2400 | [-5 -1 -2 4⟩ |