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Oneirotonic: Difference between revisions

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There is also [[Hemifamity_temperaments#Buzzard|Buzzard]], when the generator is between 471.42¢ (11\28) and 480¢ (2\5), but while this is a harmonically accurate temperament, with 4 generators reaching [[3/2]] and -3 generators [[7/4]], it is relatively weak melodically, as the optimum size of the small steps is around 20-25 cents, making it difficult to distinguish from equal pentatonic.
There is also [[Hemifamity_temperaments#Buzzard|Buzzard]], when the generator is between 471.42¢ (11\28) and 480¢ (2\5), but while this is a harmonically accurate temperament, with 4 generators reaching [[3/2]] and -3 generators [[7/4]], it is relatively weak melodically, as the optimum size of the small steps is around 20-25 cents, making it difficult to distinguish from equal pentatonic.
== Tunings ==
A-Team (13&18) tunings have a L/s ratio between 2/1 and 3/1.
It is possible to tune A-Team 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.
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"
|-
!
! [[13edo]]
! [[18edo]]
! [[31edo]]
! 64/49 generator
! JI intervals represented (2.9.5.21 subgroup)
|-
| generator
| 5\13, 461.54
| 7\18, 466.67
| 12\31, 464.52
| 462.35
| 21/16
|-
| L
| 2\13, 184.62
| 3\18, 200.00
| 5\31, 193.55
| 187.04
| 9/8, 10/9
|-
| s
| 1\13, 92.31
| 1\18, 66.66
| 2\31, 77.42
| 88.26
| 21/20
|}
Petrtri tunings have a less extreme L-to-s ratio than A-Team tunings, between 3/2 and 2/1.
The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings. (Golden oneirotonic uses 1200*(2-φ) = 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,....)
{| class="wikitable right-2 right-3 right-4"
|-
!
! [[13edo]]
! [[21edo]]
! [[34edo]]
! Golden oneirotonic
! JI intervals represented (2.5.9.11.13.17 subgroup)
|-
| generator (g)
| 5\13, 461.54
| 8\21, 457.14
| 13\34, 458.82
| 458.36
| 13/10, 17/13, 22/17
|-
| L (3g - octave)
| 2\13, 184.62
| 3\21, 171.43
| 5\34, 176.47
| 175.08
| 10/9, 11/10
|-
| s (-5g + 2 octaves)
| 1\13, 92.31
| 2\21, 114.29
| 3\34, 105.88
| 108.20
| 18/17, 17/16
|}


== Notation==
== Notation==
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* 18edo can be used if a large L/s ratio of 3 is desired (thus 18edo oneirotonic is distorted 17edo diatonic).
* 18edo can be used if a large L/s ratio of 3 is desired (thus 18edo oneirotonic is distorted 17edo diatonic).
* 31edo can be used if a near-just 5/4 is desired.
* 31edo can be used if a near-just 5/4 is desired.
A-Team tunings have a L/s ratio between 2/1 and 3/1.
It is possible to tune A-Team 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.
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"
|-
!
! [[13edo]]
! [[18edo]]
! [[31edo]]
! 64/49 generator
! JI intervals represented (2.9.5.21 subgroup)
|-
| generator
| 5\13, 461.54
| 7\18, 466.67
| 12\31, 464.52
| 462.35
| 21/16
|-
| L
| 2\13, 184.62
| 3\18, 200.00
| 5\31, 193.55
| 187.04
| 9/8, 10/9
|-
| s
| 1\13, 92.31
| 1\18, 66.66
| 2\31, 77.42
| 88.26
| 21/20
|}


=== Intervals ===
=== Intervals ===
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== Petrtri (13&21, 4:5:9:11:13:17) interpretation ==
== Petrtri (13&21, 4:5:9:11:13:17) interpretation ==
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 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.  
Petrtri tunings have a less extreme L-to-s ratio than A-Team tunings, between 3/2 and 2/1.
The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings. (Golden oneirotonic uses 1200*(2-φ) = 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,....)
{| class="wikitable right-2 right-3 right-4"
|-
!
! [[13edo]]
! [[21edo]]
! [[34edo]]
! Golden oneirotonic
! JI intervals represented (2.5.9.11.13.17 subgroup)
|-
| generator (g)
| 5\13, 461.54
| 8\21, 457.14
| 13\34, 458.82
| 458.36
| 13/10, 17/13, 22/17
|-
| L (3g - octave)
| 2\13, 184.62
| 3\21, 171.43
| 5\34, 176.47
| 175.08
| 10/9, 11/10
|-
| s (-5g + 2 octaves)
| 1\13, 92.31
| 2\21, 114.29
| 3\34, 105.88
| 108.20
| 18/17, 17/16
|}


=== Intervals ===
=== Intervals ===
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