Ennealimmal: Difference between revisions
mNo edit summary |
|||
| Line 159: | Line 159: | ||
The most natural extension to prime 11 is a weak extension Hemiennealimmal (72 & 198) which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can, of course, be extended to prime 17 via S50. | The most natural extension to prime 11 is a weak extension Hemiennealimmal (72 & 198) which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can, of course, be extended to prime 17 via S50. | ||
There is no canonical 13 extension (you can temper out 4096/4095 = S64, but you could instead temper out a different 13-limit comma to extend from an 11-limit strong extension); however, [[270edo]] is an excellent edo tuning in the 13-limit. | There is no canonical 13 extension (you can temper out 4096/4095 = S64 to extend from the 7-limit, but you could instead temper out a different 13-limit comma to extend from an 11-limit strong extension); however, [[270edo]] is an excellent edo tuning in the 13-limit. | ||
== Praxis == | == Praxis == | ||
Latest revision as of 10:51, 25 April 2026
Ennealimmal (from "ennea-" = 9 and "large limma" = 27/25), 72 & 99, is a 7-limit rank-2 microtemperament that tempers out the two smallest 7-limit superparticular ratios:
- 2401/2400 = S49, the difference between the 49/40 neutral third and its 3/2-complement 60/49, or the difference between 49/48 and 50/49
- 4375/4374 = S25/S27, the difference between (27/25)2 and 7/6 or equivalently the difference between (28/27) * (25/24) and 27/25
This implies a period of 1\9 (representing 27/25) and a generator of 49/40. The generator can also be taken to be 5/3.
Theory
Intervals
Tuning shown is pure-2/1, pure-3/2 tuning.
| # gens | 0 | +1 | +2 | +3 | +4 | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| # periods | Cents* | JI | Cents* | JI | Cents* | JI | Cents* | JI | Cents* | JI |
| -4 mod 9 | 666.667 | 72/49 | 1017.644 | 9/5 | 168.622 | 519.599 | 27/20 | 870.577 | ||
| -3 mod 9 | 800.000 | 100/63 | 1150.978 | 35/18 | 301.955 | 25/21 | 652.933 | 35/24 | 1003.910 | 25/14 |
| -2 mod 9 | 933.333 | 12/7 | 84.311 | 21/20 | 435.288 | 9/7 | 786.266 | 63/40 | 1137.243 | 27/14 |
| -1 mod 9 | 1066.667 | 50/27 | 217.644 | (17/15) | 568.622 | 25/18 | 919.599 | 70.577 | 25/24 | |
| 0 mod 9 | 0 | 1/1 | 350.978 | 49/40 | 701.955 | 3/2 | 1052.933 | 90/49 | 203.910 | 9/8 |
| 1 mod 9 | 133.333 | 27/25 | 484.311 | 835.288 | 81/50 | 1186.266 | 125/63 | 337.243 | (17/14) | |
| 2 mod 9 | 266.667 | 7/6 | 617.644 | 10/7 | 968.622 | 7/4 | 119.599 | 15/14 | 470.577 | 21/16 |
| 3 mod 9 | 400.000 | 63/50 | 750.978 | 54/35 | 1101.955 | 189/100 | 252.933 | 603.910 | (17/12) | |
| 4 mod 9 | 533.333 | 49/36 | 884.311 | 5/3 | 35.288 | 49/48, 50/49 | 386.266 | 5/4 | 737.243 | 64/49 |
Derivation of 1\9 period
This can be derived by showing that (1) (27/25)^3 is equated to 63/50 and (2) (63/50)^3 is equated to 2/1.
(1) is easy (~= means "is equated to"):
(27/25)^3 ~= 27/25 * 7/6 = 9/25 * 7/2 = 63/50.
For (2):
(63/50)^2 = (49/40 * 36/35)^2 ~= 3/2 * 81/(49*25) * 16/1 = 3/2 * 27/25 * 3/7 * 1/7 * 16/1 = 3/2 * 27/25 * 6/7 * 1/2 * 1/7 * 16/1 ~= 3/2 * 25/27 * 1/7 * 8/1 = 25/9 * 4/7 = 100/63, the 2/1 complement of 63/50.
Extensions
Ennealimmal readily extends to prime 17 by tempering out 2500/2499 = S50, equating 49/48~50/49 to 51/50. 171edo is an excellent tuning for this extension.
The most natural extension to prime 11 is a weak extension Hemiennealimmal (72 & 198) which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can, of course, be extended to prime 17 via S50.
There is no canonical 13 extension (you can temper out 4096/4095 = S64 to extend from the 7-limit, but you could instead temper out a different 13-limit comma to extend from an 11-limit strong extension); however, 270edo is an excellent edo tuning in the 13-limit.
Praxis
On isomorphic keyboards
To set up Ennealimmal, set one axis to be 1\9 and a different axis to be the ~21/20.
Patent vals
- Main article: Ennealimmal/Patent vals
