Petrtri

Petrtri is an oneirotonic-based temperament or harmonic framework, based on the oneirotonic MOS with period 1\1 and a generator chain with generator a subfourth between 21edo's 8\21 (457.14¢) and 13edo's 5\13 (461.54¢).

Notation

The notation used in this article is described in 5L 3s#Notation.

Tuning range

Petrtri tunings (with generator between 8\21 and 5\13) have less extreme step ratios than A-Team tunings, between 3/2 and 2/1. The 8\21-to-5\13 range of oneirotonic tunings remains relatively unexplored. In these tunings,

• the large step of oneirotonic tends to be intermediate in size between 10/9 and 11/10; the small step size is a semitone close to 17/16, about 92¢ to 114¢.
• The major mosthird (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342¢) to 4\13 (369¢), and the temperament interprets it as both 11/9 and 16/13.

The three major edos in this range, 13edo, 21edo and 34edo, all nominally support petrtri.

• 13edo nominally supports it, but its approximation of 9:10:11:13 is quite weak and tempers 11/9 to a 369¢ submajor third, which may not be desirable.
• 21edo is a much better petrtri tuning than 13edo, in terms of approximating 9:10:11:13. 21edo will serve those who like the combination of neogothic minor thirds (285.71¢) and Baroque diatonic semitones (114.29¢, close to quarter-comma meantone's 117.11¢).
• 34edo is close to optimal for the temperament, with a generator only 0.33¢ flat of the 2.5.9.11.13.17 POTE petrtri generator of 459.1502¢ and 0.73¢ sharp of the 2.9/5.11/5.13/5 POTE (i.e. optimal for the chord 9:10:11:13, spelled as R-M2-M3-M5 in oneirotonic intervals) petrtri generator of 458.0950¢.
• If you only care about optimizing 9:10:11:13, then 55edo's 21\55 (458.182¢) is even better, but 55 is a bit big for a usable edo.

The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings.

	13edo	21edo	34edo	Optimal (2.5.9.11.13.17 POTE) tuning	JI intervals represented (2.5.9.11.13.17 subgroup)
generator (g)	5\13, 461.54	8\21, 457.14	13\34, 458.82	459.15	13/10, 17/13, 22/17
L (3g - octave)	2\13, 184.62	3\21, 171.43	5\34, 176.47	177.45	10/9, 11/10
s (-5g + 2 octaves)	1\13, 92.31	2\21, 114.29	3\34, 105.88	104.25	18/17, 17/16

Temperament data

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

Basic theory

Primodal theory

Nejis

21nejis

1. 128:132:137:141:146:151:156:161:166:172:178:184:190:197:204:210:217:224:232:240:248:256