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 O#-J-K-M (in Kentaku notation) represents both 8:10:11:13 and 13:16:18:21 in 13edo. 31edo gives the optimal patent val for 2.9.21.5 A-Team and tunes the 13:17:19 chord very accurately. 44edo is similar to 31edo but better approximates 11, 13, 17, 19 and 23 as harmonics 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 for making 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.