A-team

A-Team is a 2.9.21 temperament generated by a tempered zo fourth (21/16) with a size ranging from 461.54¢ (5\13) to 470.78¢ (the pure value for 21/16). A-Team is one of several oneirotonic temperaments. It can be viewed as every other note of 2.3.7 latrizo or "slendric" temperament. Any EDO that has an interval within the range 461.54¢ to 470.78¢ will support A-Team. Three 21/16's are equated to one 9/8, which means that the latrizo comma (1029/1024) is tempered out. Its name is a pun on the 18 notes in its proper scale, which is a 13L 5s MOS. It is named thus because 18edo supports the temperament.

It's natural to consider A-Team a 2.9.21.5 latrizo & gu temperament by equating two 9/8's with one 5/4, tempering out 81/80. Generators optimized for tempering out 81/80 also tend to generate the melodically best scales. This temperament generates 3L 2s, 5L 3s, and 5L 8s MOSes, most notably the 8-note "oneirotonic" MOS. Any EDO with an interval between 461.54¢ and 466.67¢ can be reasonably said to support 2.9.21.5 A-Team. 13edo, 18edo, 31edo, and 44edo are all valid tunings.

This article is mostly theoretical and based on regular twmperament theory. For other approaches to using A-Team, see 5L 3s.

Tunings for A-Team

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 P1-d4-m5-m7 (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 to within 1.1 cents. 44edo is similar to 31edo but better approximates 11, 13, 17, and 19 as harmonics with the generator chain, and additionally provides the 23th harmonic.

It is also possible to tune A-Team by ear, by tuning a chain of pure harmonic sevenths and taking every other note. This corresponds to using a generator of 64/49 = 462.34819 cents. A chain of fourteen 7/4's are needed to tune the 8-note oneirotonic 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), based on the oneirotonic 5L 3s scale. This article uses the Kentaku notation.
2. As every other generator of the third-fifths pergen (P8, P5/3), which is the pergen for slendric. This is backwards compatible with a notation that has fifths.

A-Team tuning spectrum

"Meantone" A-Team tunings

If you optimize the tuning for tempering out 81/80, the generator size ranges from 13edo's 461.54 cents to 18edo's 466.67 cents. The ratio of large to small melodic steps ranges from 13edo's L:s = 2:1, analogous to 12edo's diatonic scale, to 18edo's L:s = 3:1, analogous to 17edo's diatonic scale.

"19-limit A-Team": The 13edo-to-31edo range

tl;dr: 44edo good

This extension occupies the flat end of the A-Team spectrum, from 13edo's 461.54 cents to 31edo's 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 pretty much 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 4:9:11:(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.

Interval chain

Generators	Cents	Ratios	Octatonic notation	Generators	2/1 inverse	Ratios	Octatonic notation
The "diatonic" 8-note scale has the following intervals:
0	0.00	1/1	P1	0	1200.00	2/1	P9
1	463.17	21/16, 13/10, 17/13	P4	-1	736.83	32/21, 20/13	P6
2	926.35	12/7, 17/10	M7	-2	273.66	7/6, 20/17	m3
3	189.52	9/8, 10/9, 19/17	M2	-3	1010.48	16/9, 9/5	m8
4	652.69	16/11, 13/9, 19/13	M5	-4	547.31	11/8, 18/13	m5
5	1115.86	40/21, 21/11, 19/10	M8	-5	84.14	20/19, 21/20, 22/21, 23/22	m2
6	379.04	5/4	M3	-6	820.97	8/5, 21/13	m7
7	842.21	18/11, 13/8	A6	-7	357.79	11/9, 16/13	d4
The "chromatic" 13-note scale also has the following intervals:
8	105.38	17/16	A1 (the chroma for oneirotonic)	-8	1094.62	close to 15/8	d9
9	568.55	close to 32/23	A4	-9	631.45	close to 23/16	d6
10	1031.73	20/11	A7	-10	168.27	11/10	d3
11	294.90	13/11, 19/16	A2	-11	905.10	22/13	d8
12	758.07	close to 14/9	A5	-12	441.93	close to 9/7	d5

"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 tend towards being quarter-tones (as in 23edo) and thus become less melodically distinct, much like 22edo's 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 larger 5-step interval 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.