Sensipent
If we take a look at the 5-limit version of sensi called sensipent, we find a high-accuracy extension that specifically only requires prime 31, interpreting the generator accurately as ~40/31~31/24 (by splitting 16/15 into ~32/31~31/30). This can be left as is, or one can extend to other slightly less accurate primes; the main two strategies for doing so are called sendai, focusing on accuracy and adding primes 23 and 29, and sensible, which adds primes 11, 17 and 23 and focuses on adding more primes, in both cases doing so while avoiding the less accurate ~9/7 and ~13/10 interpretations of the sensi generator. They merge meaningfully (though not uniquely) in 65edo, which can be seen by that 65edo is an amazing no-7's no-13's 31-limit temperament, where we've gained prime 19 through a possible extension of either sendai or sensible. Furthermore, 65edo can also be used as a tuning of 7-limit sensi through the 65d val (which corresponds to 65edo roughly supporting garibaldi), though note that if one tries to use its patent but very sharp ~13 (which makes the most sense if one accepts the 65d val) then 13/10 is mapped distinctly and sharp of the ~9/7 sensi generator.
For technical data, see:
- Sensipent family#Sensipent for the 2.3.5.31-subgroup version,
- Sensipent family#Sendai for the 2.3.5.23.29.31-subgroup version, and
- Sensipent family#Sensible for the 2.3.5.11.17.23.31-subgroup version.
Sensipent interval table
Amazingly, in the 2.3.5.31-subgroup-limited 155-odd-limit, every interval of every number of generators up to 23 is given at least one interpretation, so that the 27-note MOS (19L 8s) is surprisingly well-supplied with harmony. The main "holes" are at 24 and 26 gens (as 25 is ~75/64) and that these interpretations tend to be rather complex, requiring a good tuning and a context to justify them. For these reasons, the extensions sensible and sendai are likely to be preferred in practice, whose interval tables are thus also documented here, being alternative but higher-accuracy extensions to 2.3.5.31 sensipent.
| Gens | Cents | Ratios |
|---|---|---|
| 1 | 443.05 | 40/31, 31/24, 162/125 |
| 2 | 886.1 | 5/3 |
| 3 | 129.15 | 100/93, 155/144, 27/25 |
| 4 | 572.2 | 25/18, 216/155 |
| 5 | 1015.25 | 9/5 |
| 6 | 258.3 | 125/108, 36/31, 93/80 |
| 7 | 701.35 | 3/2 |
| 8 | 1144.4 | 60/31, 31/16 |
| 9 | 387.45 | 5/4 |
| 10 | 830.5 | 50/31, 155/96, 81/50 |
| 11 | 73.55 | 25/24, 162/155 |
| 12 | 516.6 | 125/93, 27/20 |
| 13 | 959.65 | 125/72, 54/31 |
| 14 | 202.7 | 9/8 |
| 15 | 645.75 | 45/31, 93/64 |
| 16 | 1088.8 | 15/8 |
| 17 | 331.85 | 75/62, 155/128 |
| 18 | 774.9 | 25/16 |
| 19 | 17.95 | 125/124, 81/80 |
| 20 | 461.0 | 125/96, 81/62 |
| 21 | 904.05 | 27/16 |
| 22 | 147.1 | 135/124 |
| 23 | 590.15 | 45/32 |
Sendai interval table
The following is the table of the 31-odd-limited 2.3.5.23.29.31 equivalents of the intervals of the 19-note MOS (8L 11s) in the CTE tuning of sendai, the sensipent extension 19 & 65 = {465/464 576/575 621/620 900/899}, made by VIxen.
| Gens | Cents | Ratios |
|---|---|---|
| 1 | 442.989338 | 31/24, 40/31 |
| 2 | 885.978677 | 5/3 |
| 3 | 128.968015 | 27/25, 29/27 |
| 4 | 571.957354 | 32/23, 25/18 |
| 5 | 1014.946692 | 9/5 |
| 6 | 257.936031 | 36/31, 29/25 |
| 7 | 700.925369 | 3/2 |
| 8 | 1143.914707 | 31/16, 29/15, 60/31 |
| 9 | 386.904046 | 5/4 |
| 10 | 829.893384 | 29/18, 50/31 |
| 11 | 72.882723 | 24/23, 25/24 |
| 12 | 515.872061 | 27/20, 31/23 |
| 13 | 958.8614 | 40/23, 54/31 |
| 14 | 201.850738 | 9/8 |
| 15 | 644.840076 | 29/20 |
| 16 | 1087.829415 | 15/8, 58/31 |
| 17 | 330.818753 | 29/24 |
| 18 | 773.808092 | 25/16, 36/23 |
Sensible interval table
The following is the table of the 115-odd-limited 2.3.5.11.17.23.31 equivalents of the intervals of the 27-note MOS (19L 8s) in the CTE tuning of sensible, the sensipent extension 46 & 65 = {(S16, S9/S10,) S23, S24, S31, S32, S33}, made by Godtone.
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Todo: expand |