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 470.78¢ (the pure value for 21/16) if you don't care about tempering out 81/80 or avoiding quartertone-sized steps. 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 3L 2s, 5L 3s, and 5L 8s 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 2.9.21.5 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 with C = 261.62 Hz) and Kentaku (Ilarnekian = JKLMNOPQJ with J ≈ 180 Hz).
- 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
tl;dr: 44edo good
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, owing to the generator being close to the 17/13 ratio and three of them approximating both 9/8 and 10/9, thus approximating their mediant 19/17. The 11th harmonic makes an appearance too, because it is approximated by both 13edo and 31edo. Thus A-Team can be viewed as representing the no-3, no-7 19-odd limit. If you optimize for this 19 limit harmony you also get the 23rd harmonic for free.
The most plentiful consonant triad in A-Team scales is 4:9:21 or 8:18:21 (the voicing of the 21th harmonic is important to make it sound smooth), followed by 13:17:19 and 4:5:9. A-Team[13] in the following tuning has five copies of 4:5:9:13:17:21, five copies of 5:9:13:17:19:21, three copies of 4:5:9:11, two copies of 4:5:9:11:13, two copies of 4:5:9:13:17:19:21, 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 gives good, though irregular, mappings of 3/2 (with -17 generators) and 7/4 (with -15 generators) in the "better" tunings.
| 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, 17/13 | 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, 19/13 | 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
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.
"Superpythagorean" tunings
In general using a sharper 21/16 is 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 possible 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, much like 22edo superpyth[7].
Using a pure 21/16 of 470.78¢ results in an extremely lopsided oneirotonic scale with L/s = 4.60. Harmonically this results in a 9/8 of 212.342 cents which is very much in the superpyth range (for comparison, 17edo's 9/8 is 211.765 cents). Instead of approximating 16/11, the "major tritone" in oneirotonic (J-N in Kentaku notation) will be a very flat fifth of 683.123 cents, constrasting with the very sharp 40/21 fifth (729.2 cents). The flat fifths give shimmery detuned versions of zo (subminor) triads 6:7:9 and sus2 triads 8:9:12. All these intervals contribute to the scale's overall gently shimmery quality which the 23edo version shares too.