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. Hence, any EDO that equates three 8/7's with one 3/2 will support A-Team with its 21/16.
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
"Meantone" tunings: 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.
Extending the chain beyond 13 notes can give good (although inconsistent) mappings of 3/2 and 7/4.
| 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
tl;dr: 44edo good
"Superpythagorean" tunings
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 little 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 distinct.
23edo (469.57 cents) and 41edo (468.29 cents): two tones represent 14/11 rather than 5/4, and J-O# (4 large steps + 1 small step) becomes a 5/3 rather than a 13/8.
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. 18edo's oneirotonic is analogous to 17edo's diatonic scale in that L/s = 3, while 13edo is analogous to 12edo.