|
Tag: Redirect target changed |
| (7 intermediate revisions by one other user not shown) |
| Line 1: |
Line 1: |
| '''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¢).
| | #redirect [[Subgroup temperaments#Petrtri]] |
| == 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.
| |
| {| class="wikitable right-2 right-3 right-4 right-5"
| |
| |-
| |
| !
| |
| ! [[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:
| |
| {| class="wikitable right-2 right-3 right-4 right-5 sortable"
| |
| |-
| |
| ! Degree
| |
| ! Size in 13edo
| |
| ! Size in 21edo
| |
| ! Size in 34edo
| |
| ! Size in POTE tuning
| |
| ! Note name on Q
| |
| ! class="unsortable"| 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 ==
| |
| === Primodal chords ===
| |
| === Nejis ===
| |
| ==== 21nejis ====
| |
| # 128:132:137:141:146:151:156:161:166:172:178:184:190:197:204:210:217:224:232:240:248:256 | |
| [[Category:Oneirotonic]]
| |
| [[Category:Temperament]]
| |
| [[Category:Petrtri|*]]
| |