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| The notation used in this article is described in [[5L 3s#Notation]]. | | The notation used in this article is described in [[5L 3s#Notation]]. |
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| == Tuning range ==
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| 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,
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| * 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¢.
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| * 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]].
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| The three major edos in this range, [[13edo]], [[21edo]] and [[34edo]], all nominally support petrtri.
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| * [[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.
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| * [[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¢).
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| * [[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¢.
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| * 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.
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| The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings.
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| {| class="wikitable right-2 right-3 right-4 right-5"
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| |-
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| !
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| ! [[13edo]]
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| ! [[21edo]]
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| ! [[34edo]]
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| ! Optimal (2.5.9.11.13.17 [[POTE]]) tuning
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| ! JI intervals represented (2.5.9.11.13.17 subgroup)
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| |-
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| | generator (g)
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| | 5\13, 461.54
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| | 8\21, 457.14
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| | 13\34, 458.82
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| | 459.15
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| | 13/10, 17/13, 22/17
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| |-
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| | L (3g - octave)
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| | 2\13, 184.62
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| | 3\21, 171.43
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| | 5\34, 176.47
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| | 177.45
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| | 10/9, 11/10
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| |-
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| | s (-5g + 2 octaves)
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| | 1\13, 92.31
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| | 2\21, 114.29
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| | 3\34, 105.88
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| | 104.25
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| | 18/17, 17/16
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| |}
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| == Temperament data == | | == Temperament data == |
| === Intervals === | | === Intervals === |
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.
Temperament data
Intervals
Sortable table of intervals in the Dylathian mode and their Petrtri interpretations:
| Degree
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Size in 13edo
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Size in 21edo
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Size in 34edo
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Size in POTE tuning
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Note name on Q
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Approximate ratios
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#Gens up
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| 1
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0\13, 0.00
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0\21, 0.00
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0\34, 0.00
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0.00
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Q
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1/1
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0
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| 2
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2\13, 184.62
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3\21, 171.43
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5\34, 176.47
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177.45
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J
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10/9, 11/10
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+3
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| 3
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4\13, 369.23
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6\21, 342.86
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10\34, 352.94
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354.90
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K
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11/9, 16/13
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+6
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| 4
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5\13, 461.54
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8\21, 457.14
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13\34, 458.82
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459.15
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L
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13/10, 17/13, 22/17
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+1
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| 5
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7\13, 646.15
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11\21, 628.57
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18\34, 635.294
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636.60
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M
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13/9, 16/11, 23/16 (esp. 21edo)
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+4
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| 6
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9\13, 830.77
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14\21, 800.00
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23\34, 811.77
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814.05
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N
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8/5
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+7
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| 7
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10\13, 923.08
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16\21, 914.29
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26\34, 917.65
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918.30
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O
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17/10
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+2
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| 8
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12\13, 1107.69
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19\21, 1085.71
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31\34, 1094.12
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1095.75
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P
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17/9, 32/17, 15/8
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+5
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Basic theory
Primodal theory
Primodal chords
Nejis
21nejis
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