A-team
A-Team is a 2.9.21 temperament generated by a tempered 21/16 with a size ranging from 5\13 (461.54¢) to 7\18 (466.67¢), or to about 470¢ if you don't care about tempering out 81/80. Three 21/16's are equated to one 9/8, which means that the latrizo comma (1029/1024) is tempered out. It's natural to consider A-Team a 2.9.21.5 temperament by also equating two 9/8's with one 5/4, tempering out 81/80. The generator generates 5-, 8-, and 13-note MOSes, most notably the 8-note "oneirotonic" MOS; see also 13edo#Modes_and_Harmony_in_the_Oneirotonic_Scale.
13edo, 18edo, 31edo, and 44edo (with generators 5\13, 7\18, 12\31, and 17\44 respectively) all support 2.9.21 A-Team with their closest approximations to 9/8 and 21/16. 13edo, while representing the 2.9.21.5 subgroup less accurately, gives some chords extra interpretations; for example, the chord H-C-D-F (in Cryptic Ruse's notation) represents both 8:10:11:13 and 13:16:18:21 in 13edo. 31edo gives the optimal patent val for the 2.9.21.5 subgroup interpretation. 44edo is similar to 31edo but better approximates primes 11, 13, 17, 19 and 23 with the generator chain.
Its name is a pun on the 18 notes in its proper scale, which is a 13L 5s MOS.
Notation
There are several ways to notate A-Team in a JI-agnostic way:
- The octatonic notation described by Cryptic Ruse (Dylathian = CDEFGHABC) and Kentaku (Dylathian = JKLMNOPQJ).
- Using the pergen (P8, M9/3). Though the tuning lacks perfect fifths, three of the 21/16 generator are equal to twice a perfect fifth (i.e. a conventional major ninth).
- As every other note of the third-fifths pergen (P8, P5/3). This is backwards compatible with having fifths.
A-team tuning spectrum
The 13edo-to-31edo range
Occupies the flat end of the spectrum, from 461.54 to 464.52 cents.
Surprisingly, A-Team tunings in this range can approximate some high-limit JI chords. The most plentiful consonant triad is 4:9:21 or 8:18:21, followed by 4:5:9. The following tuning (and the 44edo tuning) of A-Team[13] has five copies of 4:5:9:13:17:21, three copies of 4:5:9:11, two copies of 4:5:9:(11):13, two copies of 4:5:9:13:19:21, and one copy of 4:9:(15):21:23. The 13-note MODMOS given by flattening just the seventh and eighth degrees of the LsLssLsLssLss mode of A-Team[13] by the 13-generator interval (~25.5 cents) gives the entire 4:5:9:11:13:17:19:21:23 otonal chord over a single root.
| Generators | Cents (*) | Ratios (**) | Octatonic notation | Generators | 2/1 inverse (*) | Ratios (**) | Octatonic notation |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 1/1 | P1 | 0 | 1200 | 2/1 | P9 |
| 1 | 463.5 | 21/16, 13/10 | P4 | -1 | 736.5 | 32/21, 20/13 | P6 |
| 2 | 927.0 | 12/7 | M7 | -2 | 273.0 | 7/6 | m3 |
| 3 | 190.5 | 9/8, 10/9, 19/17 | M2 | -3 | 1009.5 | 16/9, 9/5 | m8 |
| 4 | 654.0 | 16/11, 13/9 | M5 | -4 | 546.0 | 11/8, 18/13 | m5 |
| 5 | 1117.5 | 40/21, 21/11 | M8 | -5 | 82.5 | 21/20, 22/21, 23/22 | m2 |
| 6 | 381.0 | 5/4 | M3 | -6 | 819.0 | 8/5 | m7 |
| 7 | 844.5 | 18/11, 13/8 | A6 | -7 | 355.5 | 11/9, 16/13 | d4 |
| 8 | 108.0 | 17/16 | A1 (the chroma for oneirotonic) | -8 | 1092.0 | (close to 15/8) | d9 |
| 9 | 571.5 | 32/23 | A4 | -9 | 628.5 | 23/16 | d6 |
| 10 | 1035.0 | 20/11 | A7 | -10 | 165.0 | 11/10 | d3 |
| 11 | 298.5 | 13/11, 19/16 | A2 | -11 | 901.5 | 22/13 | d8 |
| 12 | 762.0 | close to 14/9 | A5 | -12 | 438.0 | close to 9/7 | d5 |
| 13 | 25.5 | (***) | AA8 - octave | -13 | 1174.5 | dd2 + octave |
(*) using the 2.9.21.5.11.13 POTE generator; cf. the 463.64¢ generator in 44edo
(**) 2.9.21.5.11.13.17.19.23 interpretations; harmonics are in bold
(***) 65/64 and other commas only tempered out by 13edo
"Superpythagorean" tunings for A-Team
In general sharper subfourths are better if you don't care about approximating 5/4 and only care about optimizing the 4:9:21 triad. Apart from that, there's no common JI interpretation shared by these sharper tunings. One tradeoff is that small steps in the oneirotonic scale get smaller than 1/3-tones (as in 18edo) and become quarter-tones (as in 23edo) and thus become less melodically recognizable.
23edo is a special beast with its own JI identifications, for example two tones represent 14/11 and 33/26, rather than 5/4, and J-O# becomes 5/3 rather than a 13/8. Best to treat it as its own thing?
18edo (466.67 cents) is an edge case, as it tempers out 81/80 but fails to approximate more diverse intervals with the same identifications used by 13edo, 44edo or 23edo.