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 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:

1. The octatonic notation described by Cryptic Ruse (Dylathian = CDEFGHABC with C = 261.62 Hz) and Kentaku (Ilarnekian = JKLMNOPQJ with J ≈ 180 Hz).
2. 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).
3. 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.