5L 3s/Temperaments

Oneirotonic temperaments have a sort of analogy to diatonic temperaments superpyth and meantone in how they treat the large step. In diatonic the large step approximates 9/8 (a very good 9/8 in 12edo), but superpyth has 9/8 ~ 8/7, and meantone has 9/8 ~ 10/9. In oneirotonic the large step tends to approximate 10/9 (and is a very good 10/9 in 13edo which is the oneirotonic analogue to 12edo), but different oneiro temperaments do different things with it. In A-Team (13 & 18), 10/9 is equated with 9/8, making the major oneirothird a 5/4 (thus is "meantone" in that sense). In both Petrtri (13 & 21) and Tridec (21 & 29), 10/9 is equated with 11/10, making the major oneirothird a 11/9; and the perfect oneirofourth is equated to 13/10. So the compressed major triad add2 (R–M2–M3–M5, M5 = major oneirofifth = minor fifth in 13edo) is interpreted as 9:10:11:13 in petrtri, analogous to meantone's 8:9:10:12. Thus Petrtri and Tridec are the same temperament when you only care about the 9:10:11:13, or equivalently the 2.9/5.11/5.13/5 subgroup. This is one reason why Tridec can be viewed as the oneirotonic analogue of flattone—it's a flatter variant of the flat-of-13edo oneiro temperament on the 2.9/5.11/5.13/5 subgroup.

Vulture/Buzzard, in which four generators make a 3/1 (and three generators approximate an octave plus 8/7), is the only harmonic entropy minimum in the oneirotonic range. However, the rest of this region is still rich in notable subgroup temperaments.

Petrtri

Subgroup: 2.11/5.13/5

Comma: 2200/2197

Svalname: 3&5

POT2 generator: ~13/10 = 455.012

Gencom: [2 13/10; 2200/2197]

Gencom mapping: [<1 0 -1/3 0 -1/3 2/3|, <0 0 -4/3 0 5/3 -1/3|]

Mapping: [<1 0 1|, <0 3 1|]

EDOs: 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c

Tridec

Subgroup: 2.3.7/5.11/5.13/5

Period: 1\1

Optimal (POTE) generator: 455.2178

EDO generators: 8\21, 11\29, 14\37

Comma list: 196/195, 847/845, 1001/1000

Mapping: [⟨2: 1, 3: 5, 7/5: 2, 11/5: 0, 13/5: 1], ⟨2: 0, 3: -9, 7/5: -4, 11/5: 3, 13/5: 1]]

Mapping generators: ~2, ~13/10

Optimal ET sequence: 21, 29, 37

Intervals

Sortable table of intervals in the Dylathian mode and their Tridec interpretations:

Degree	Size in 21edo	Size in 29edo	Size in 37edo	Size in POTE tuning	Note name on Q	Approximate ratios	#Gens up
1	0\21, 0.00	0\29, 0.00	0\37, 0.00	0.00	Q	1/1	0
2	3\21, 171.43	4\29, 165.52	5\37, 163.16	165.65	J	11/10, 10/9	+3
3	6\21, 342.86	8\29, 331.03	10\37, 324.32	331.31	K	11/9, 6/5	+6
4	8\21, 457.14	11\29, 455.17	14\37, 454.05	455.17	L	13/10, 9/7	+1
5	11\21, 628.57	15\29, 620.69	19\37, 616.22	620.87	M	13/9, 10/7	+4
6	14\21, 800.00	19\29, 786.21	23\37, 778.38	786.52	N	11/7	+7
7	16\21, 914.29	22\29, 910.34	28\37, 908.11	910.44	O	22/13	+2
8	19\21, 1085.71	26\29, 1075.86	33\37, 1070.27	1076.09	P	13/7, 28/15	+5

Petrtri extension

Subgroup: 2.5.9.11.13.17

Period: 1\1

Optimal (POL2) generator: 459.1502

EDO generators: 5\13, 8\21, 13\34

Comma list: 100/99, 144/143, 170/169, 221/220

Mapping (for 2, 5, 9, 11, 13, 17): [⟨2: 1, 5: 5, 9: 7, 11: 5, 13: 6, 17: 6], ⟨2: 0, 5: -7, 9: -10, 11: -4, 13: -6, 17: -5]]

Mapping generators: ~2, ~13/10

Optimal ET sequence: 13, 21, 34

Intervals

Sortable table of intervals in the Dylathian mode and their Petrtri interpretations:

Degree	Size in 13edo	Size in 21edo	Size in 34edo	Size in POTE tuning	Note name on Q	Approximate ratios	#Gens up
1	0\13, 0.00	0\21, 0.00	0\34, 0.00	0.00	Q	1/1	0
2	2\13, 184.62	3\21, 171.43	5\34, 176.47	177.45	J	10/9, 11/10	+3
3	4\13, 369.23	6\21, 342.86	10\34, 352.94	354.90	K	11/9, 16/13	+6
4	5\13, 461.54	8\21, 457.14	13\34, 458.82	459.15	L	13/10, 17/13, 22/17	+1
5	7\13, 646.15	11\21, 628.57	18\34, 635.294	636.60	M	13/9, 16/11, 23/16 (esp. 21edo)	+4
6	9\13, 830.77	14\21, 800.00	23\34, 811.77	814.05	N	8/5	+7
7	10\13, 923.08	16\21, 914.29	26\34, 917.65	918.30	O	17/10	+2
8	12\13, 1107.69	19\21, 1085.71	31\34, 1094.12	1095.75	P	17/9, 32/17, 15/8	+5

A-Team

Subgroup: 2.5.9.21

Period: 1\1

Optimal (POL2) generator: 464.3865

EDO generators: 5\13, 7\18, 12\31, 17\44

Lookalike temperament: Dual-3 A-Team

Comma list: 81/80, 1029/1024

Mapping: [⟨2: 1, 5: 0, 9: 2, 21: 4], ⟨2: 0, 5: 6, 9: 3, 21: 1]]

Mapping generators: ~2, ~21/16

Optimal ET sequence: 13, 18, 31, 44

Intervals

Sortable table of intervals in the Dylathian mode and their A-Team interpretations:

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

* The ratio interpretations that are not valid for 18edo are italicized.

Buzzard

Subgroup: 2.3.5.7

Period: 1\1

Optimal (POTE) generator: ~21/16 = 475.636

EDO generators: 15\38, 17\43, 19\48, 21\53, 23\58, 25\63

Commas: 1728/1715, 5120/5103

Mapping: [<1 0 -6 4|, <0 4 21 -3|]

Mapping generators: ~2, ~21/16

Wedgie: <<4 21 -3 24 -16 -66||

Optimal ET sequence: 48, 53, 111, 164d, 275d

Badness: 0.0480

Intervals

Sortable table of intervals in the Dylathian mode and their Buzzard interpretations:

Degree	Size in 38edo	Size in 53edo	Size in 63edo	Size in POTE tuning	Note name on Q	Approximate ratios	#Gens up
1	0\38, 0.00	0\53, 0.00	0\63, 0.00	0.00	Q	1/1	0
2	7\38, 221.05	10\53, 226.42	12\63, 228.57	227.07	J	8/7	+3
3	14\38, 442.10	20\53, 452.83	24\63, 457.14	453.81	K	13/10, 9/7	+6
4	15\38, 473.68	21\53, 475.47	25\63, 476.19	475.63	L	21/16	+1
5	22\38, 694.73	31\53, 701.89	37\63, 704.76	702.54	M	3/2	+4
6	29\38, 915.78	41\53, 928.30	49\63, 933.33	929.45	N	12/7, 22/13	+7
7	30\38, 947.36	42\53, 950.94	50\63, 952.38	951.27	O	26/15	+2
8	37\38, 1168.42	52\53, 1177.36	62\63, 1180.95	1178.18	P	108/55, 160/81	+5