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 |