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| '''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]] |
| == 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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| [[Category:Oneirotonic]]
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| [[Category:Temperament]]
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| [[Category:Petrtri|*]] | |