5L 3s: Difference between revisions

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== Tuning ranges and data ==

=== 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.

A-Team tunings (with generator between 5\13 and 7\18) have L/s ratios between 2/1 and 3/1. In these tunings, the large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]). Major mosthirds (made of two large steps) in these tunings stand in for [[meantone]]- or [[flattone]]-sized major thirds.

EDOs that support A-Team include [[13edo]], [[18edo]], and [[31edo]].

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=== 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.

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. 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 100c to 110c. The major mosthird (made of two large steps) in these tunings are more like neutral thirds, ranging from 6\21 (342c) to 4\13 (369c) 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, 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.

@@ Line 426: / Line 426: @@
 ==  Tuning ranges and data ==
 === 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.
+A-Team tunings (with generator between 5\13 and 7\18) have L/s ratios between 2/1 and 3/1. In these tunings, the large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]). Major mosthirds (made of two large steps) in these tunings stand in for [[meantone]]- or [[flattone]]-sized major thirds.
 EDOs that support A-Team include [[13edo]], [[18edo]], and [[31edo]].
@@ Line 467: / Line 467: @@
 === 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.
+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. 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 100c to 110c. The major mosthird (made of two large steps) in these tunings are more like neutral thirds, ranging from 6\21 (342c) to 4\13 (369c) 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, 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.