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). 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'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.

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.

Oneirotonic MOS (5L 3s)

The term oneirotonic is used for the 8-note MOS 5L 3s, whose brightest mode is LLsLLsLs. The name "oneirotonic" was coined by Cryptic Ruse after the Dreamlands in H.P. Lovecraft's Dream Cycle mythos. It can be viewed as the analogue of diatonic in A-Team; oneirotonic is a distorted diatonic, because it has one extra small step compared to diatonic (5L 2s).

The sizes of the generator, large step and small step of oneirotonic are as follows in various edos:

	13edo	18edo	31edo	64/49 generator	JI intervals represented (2.9.5.21 subgroup)
generator	5\13, 461.54	7\18, 466.67	12\31, 464.52	462.35	21/16
L	2\13, 184.62	3\18, 200.00	5\31, 193.55	187.04	9/8, 10/9
s	1\13, 92.31	1\18, 66.66	2\31, 77.42	88.26	21/20

The names I use for the oneirotonic interval classes are borrowed from diatonic interval categories: "second", "third", "fourth", "tritone" (4-step intervals), "fifth" (5-step intervals), "sixth" (6-step intervals), "seventh" (7-step intervals) and octave. You just have to remember that there's an extra category between fourths and fifths and that fourths and fifths are dissonant. Like in archeotonic you can change the perception of an interval by approaching it from different directions, but in oneirotonic it will change what diatonic interval class you hear it as: say, as both a third and a fourth, rather than both a major and a minor third.

Intervals

Sortable table of Dylathian, the brightest mode:

Degree	Size in 13edo	Size in 18edo	Size in 31edo	Note name on J	Approximate ratios	#Gens up
1	0\13, 0.00	0\18, 0.00	0\31, 0.00	J	1/1	0
2	2\13, 184.62	3\18, 200.00	5\31, 193.55	K	9/8, 10/9	+3
3	4\13, 369.23	6\18, 400.00	10\31, 387.10	L	5/4	+6
4	5\13, 461.54	7\18, 466.67	12\31, 464.52	M	21/16, 13/10	+1
5	7\13, 646.15	10\18, 666.66	17\31, 658.06	N	13/9, 16/11	+4
6	9\13, 830.77	13\18, 866.66	22\31, 851.61	O	13/8, 18/11	+7
7	10\13, 923.08	14\18, 933.33	24\31, 929.03	P	12/7	+2
8	12\13, 1107.69	17\18, 1133.33	29\31, 1122.58	Q		+5

Chords

Despite being melodically familiar, oneirotonic is harmonically complex in that it doesn't represent many lower-complexity JI triads other than 4:9:21, 9:10:13, and the occasional 4:5:9. Hence oneirotonic may especially benefit from either using inharmonic timbres in addition to harmonic ones or using a well-tempered version of 13edo adopted for this scale. The availability of certain consonances also varies greatly by mode: for example, only Dylathian, Ilarnekian and Sarnathian have a 5/4 on the tonic, and only Mnarian, Kadathian, Hlanithian and Sarnathian have an 11/8 on the tonic.

Modal harmony

Oneirotonic modes are named after cities in the Dreamlands.

1. Dylathian: LLsLLsLs (major with hints of Mixolydian and "#5")
2. Ilarnekian: LLsLsLLs (major with hints of "b6")
3. Celephaïsian: LsLLsLLs (the oneirotonic melodic minor. Very classical-sounding; Easley Blackwood's 13-note etude uses this as its home mode.)
4. Ultharian: LsLLsLsL (A Dorian analogue. Another Dorian analogue is the MODMOS LsLLLsLs)
5. Mnarian: LsLsLLsL
6. Kadathian: sLLsLLsL (another "Locrian")
7. Hlanithian: sLLsLsLL (closest Locrian analogue)
8. Sarnathian: sLsLLsLL (Darkest but ironically the most consonant. Here be dragons.)

Modes with sharp tritone

The brighter modes can be viewed as providing a distorted version of diatonic functional harmony. For example, in the Dylathian mode, the 4:5:9 triad on the sixth degree can sound like both "V" and "III of iv" depending on context. Basic chord progressions can move by minor fourths, thirds, or major seconds: for example, J major-M minor-P minor-Ob major-J major (in Ilarnekian) or J major-K major-O major-M major-J major (in Dylathian).

Modes with flat tritone

The darker modes are radically different in character than the brighter modes. Because of the consonant 11/8 minor tritone and the 13/8 minor sixth, 11/8 sounds more like a stable scale function and the relatively dissonant 21/16 minor fourth wants to be a major third resolving up to the 11/8.

Samples

‎(A rather classical-sounding 3-part harmonization of the ascending J Ilarnekian scale; tuning is 13edo)

(First 30 seconds is in Celephaïsian mode)

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.