5L 3s: Difference between revisions

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+==  Basic info about tunings ==
+=== A-Team (13&18) ===
+A-Team tunings (with generator between 5\13 and 7\18) have L/s ratios between 2/1 and 3/1.
+EDOs that support A-Team include [[13edo]], [[18edo]], and [[31edo]].
+* 18edo can be used for a large L/s ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic), or for nearly pure 9/8 and 7/6.
+* 31edo can be used to make the major mos3rd a near-just 5/4.
+A-Team can be tuned by ear, by tuning a chain of pure harmonic sevenths and taking every other note. This corresponds to using a generator of 64/49 = 462.34819 cents. A chain of fourteen 7/4's are needed to tune the 8-note oneirotonic MOS. This produces a tuning close to 13edo.
+The sizes of the generator, large step and small step of oneirotonic are as follows in various A-Team tunings.
+{| class="wikitable right-2 right-3 right-4 right-5 right-6"
+|-
+!
+! [[13edo]]
+! [[18edo]]
+! [[31edo]]
+! 64/49 generator
+! [[POTE tuning]]
+! JI intervals represented (2.9.5.21 subgroup)
+|-
+| generator (g)
+| 5\13, 461.54
+| 7\18, 466.67
+| 12\31, 464.52
+| 462.35
+| 464.14
+| 21/16
+|-
+| L (3g - octave)
+| 2\13, 184.62
+| 3\18, 200.00
+| 5\31, 193.55
+| 187.04
+| 192.42
+| 9/8, 10/9
+|-
+| s (-5g + 2 octaves)
+| 1\13, 92.31
+| 1\18, 66.66
+| 2\31, 77.42
+| 88.26
+| 79.30
+| 21/20
+|}
+=== Petrtri (13&21) ===
+Petrtri tunings (with generator between 8\21 and 5\13) have less extreme L-to-s ratios than A-Team tunings, between 3/2 and 2/1. The 8\21-to-5\13 range of oneirotonic tunings remains relatively unexplored.
+The three major edos in this range, [[13edo]], [[21edo]] and [[34edo]], all nominally support petrtri, but [[34edo]] is close to optimal for the temperament, with a generator only .33c flat of the optimal ([[POTE]]) petrtri generator of 459.1502c. Close-to-optimal petrtri tunings such as 34edo may be particularly useful for the Sarnathian mode, as Sarnathian in these tunings uniquely approximates four over-2 harmonics plausibly, namely 17/16, 5/4, 11/8, and 13/8.
+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]]
+! [[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
+|}
+Trivia: One petrtri tuning is golden oneirotonic, which uses (2-φ)*1200 cents = 458.3592135¢ as generator and has L/s = φ; it is the limit of taking generators in Fibonacci number edos 5\13, 8\21, 13\34, 21\55, 34\89,....
 == Oneirotonic rank-2 temperaments ==
 The only notable harmonic entropy minimum is Vulture/[[Hemifamity_temperaments|Buzzard]], in which four generators make a 3/1 (and three generators approximate an octave plus 8/7). The rest of this region does not approximate low-complexity JI harmony well, but can be melodically interesting due to the distorted diatonic scale structure (see also [[oneirotonic]]).