5L 3s: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,004 edits
mNo edit summary
Tags: Mobile edit Mobile web edit
ArrowHead294 (talk | contribs)
14,512 edits
m substitute deprecated template
 
(637 intermediate revisions by 14 users not shown)
Line 1: Line 1:
5L 3s refers to the structure of moment of symmetry scales with generators ranging from 2\5 (two degrees of [[5edo]] = 480¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's).  
{{Interwiki
| en = 5L 3s
| de =
| es =
| ja =
| ko = 5L3s (Korean)
}}
{{Infobox MOS
| Neutral = 2L 6s
}}
: ''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
{{MOS intro}}
5L 3s can be seen as a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).


The term '''oneirotonic''' (/oʊnaɪrəˈtɒnɪk/ ''oh-ny-rə-TON-ik'' or /ənaɪrə-/ ''ə-ny-rə-'') is often used for the octave-equivalent MOS structure 5L 3s, whose brightest mode is LLsLLsLs. The name ''oneirotonic'' (from Greek ''oneiros'' 'dream') was coined by [[Cryptic Ruse]] after the Dreamlands in H.P. Lovecraft's Dream Cycle mythos. Oneirotonic is a distorted diatonic, because it has one extra small step compared to diatonic ([[5L 2s]]).
== Name ==
{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.


The generator size ranges from 450¢ (3\8) to 480¢ (2\5). Hence any edo with an interval between 450¢ and 480¢ has an oneirotonic scale. [[13edo]] is the smallest edo with a (non-degenerate) 5L3s oneirotonic scale and thus is the most commonly used oneirotonic tuning.
'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].


In terms of [[regular temperament]]s, there are at least two melodically viable ways to interpret oneirotonic (analogous to diatonic having multiple temperament interpretations depending on generator size):
== Scale properties ==
# When the generator is between 461.54¢ (5\13) and 466.67¢ (7\18): [[A-Team]] (13&18, a 4:5:9:21 or 2.9.5.21 temperament)
# When the generator is between 457.14¢ (8\21) and 461.54¢ (5\13): [[Chromatic_pairs#Petrtri|Petrtri]] (13&21, a 4:5:9:11:13:17 or 2.5.9.11.13.17 temperament)
[[13edo]] represents both temperaments.


More extreme oneirotonic temperaments include:
=== Intervals ===
* [[Chromatic pairs#Tridec|Tridec]] (a 5:7:11:13 or 2.7/5.11/5.13/5 subgroup temperament), when the generator is between 454.05c (14\37) and 457.14c (8\21). These have a L/s ratio of 5/4 to 3/2.
{{MOS intervals}}
* [[Hemifamity_temperaments#Buzzard|Buzzard]], when the generator is between 471.42¢ (11\28) and 480¢ (2\5). 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.


== Scale tree ==
=== Generator chain ===
{| class="wikitable" style="text-align:center;"
{{MOS genchain}}
|-
! colspan="5" | generator
! | tetrachord
! | g in cents
! | 2g
! | 3g
! | 4g
! | Comments
|-
| | 2\5
| |
| |
| |
| |
| | 1 0 1
|  | 480.000
|  | 960.000
|  | 240.00
|  | 720.000
|  |
|-
| | 21\53
| |
| |
| |
| |
|  | 10 1 10
|  | 475.472
|  | 950.943
|  | 226.415
|  | 701.887
|  | Vulture/Buzzard is around here
|-
| | 19\48
| |
| |
| |
| |
|  | 9 1 9
|  | 475
|  | 950
|  | 225
|  | 700
|  |
|-
| | 17\43
| |
| |
| |
| |
|  | 8 1 8
|  | 474.419
|  | 948.837
|  | 223.256
|  | 697.674
|  |
|-
| | 15\38
| |
| |
| |
| |
|  | 7 1 7
|  | 473.684
|  | 947.368
|  | 221.053
|  | 694.737
|  |
|-
| | 13\33
| |
| |
| |
| |
|  | 6 1 6
|  | 472.727
|  | 945.455
|  | 218.181
|  | 690.909
|  |
|-
| | 11\28
| |
| |
| |
| |
|  | 5 1 5
|  | 471.429
|  | 942.857
|  | 214.286
|  | 685.714
|  |
|-
| | 9\23
| |
| |
| |
| |
|  | 4 1 4
|  | 469.565
|  | 939.130
|  | 208.696
|  | 678.261
|  | L/s = 4
|-
| |
| |
| |
| |
| |
|  | pi 1 pi
|  | 467.171
|  | 934.3425
|  | 201.514
|  | 668.685
|  | L/s = pi
|-
| | 7\18
| |
| |
| |
| |
|  | 3 1 3
|  | 466.667
|  | 933.333
|  | 200.000
|  | 666.667
|  | L/s = 3<br/>[[A-Team]] starts around here...
|-
| |
| |
| |
| |
| |
|  | e 1 e
|  | 465.535
|  | 931.069
|  | 196.604
|  | 662.139
|  | L/s = e
|-
| |
| | 19\49
| |
| |
| |
|  | 8 3 8
|  | 465.306
|  | 930.612
|  | 195.918
|  | 661.2245
| |
|-
| |
| |
| | 50\129
| |
| |
|  | 21 8 21
|  | 465.116
|  | 930.233
|  | 195.349
|  | 660.465
| |
|-
| |
| |
| |
| | 131\338
| |
|  | 55 21 55
|  | 465.089
|  | 930.1775
|  | 195.266
|  | 660.335
| |
|-
| |
| |
| |
| |
| | 212\547
|  | 89 34 89
|  | 465.082
|  | 930.1645
|  | 195.247
|  | 660.329
| |
|-
| |
| |
| |
| | 81\209
| |
|  | 34 13 34
|  | 465.072
|  | 930.1435
|  | 195.215
|  | 660.287
| |
|-
| |
| |
| | 31\80
| |
| |
|  | 13 5 13
|  | 465
|  | 930
|  | 195
|  | 660
| |
|-
| |
| | 12\31
| |
| |
| |
|  | 5 2 5
|  | 464.516
|  | 929.032
|  | 193.549
|  | 658.065
|  |
|-
| | 5\13
| |
| |
| |
| |
|  | 2 1 2
|  | 461.538
|  | 923.077
|  | 184.615
|  | 646.154
|  | ...and ends here<br/>Boundary of propriety (generators smaller than this are proper)<br/>[[Petrtri]] starts here...
|-
| |
| |
| |
| |
| |
|  | √3 1 √3
|  | 459.417
|  | 918.8345
|  | 178.252
|  | 637.669
| |
|-
| |
| | 13\34
| |
| |
| |
|  | 5 3 5
|  | 458.824
|  | 917.647
|  | 176.471
|  | 635.294
|  |
|-
| |
| |
| | 34\89
| |
| |
|  | 13 8 13
|  | 458.427
|  | 916.854
|  | 175.281
|  | 633.708
|  |
|-
| |
| |
| |
| | 89\233
| |
|  | 34 21 34
|  | 458.369
|  | 916.738
|  | 175.107
|  | 633.473
|  |
|-
| |
| |
| |
| |
| | 233\610
|  | 89 55 89
|  | 458.361
|  | 916.721
|  | 175.082
|  | 633.443
|  | Golden father; generator is 2 octaves minus logarithmic [[phi]]
|-
| |
| |
| |
| | 144\377
| |
|  | 55 34 55
|  | 458.355
|  | 916.711
|  | 175.066
|  | 633.422
|  |
|-
| |
| |
| | 55\144
| |
| |
|  | 21 13 21
|  | 458.333
|  | 916.666
|  | 175
|  | 633.333
|  |
|-
| |
| | 21\55
| |
| |
| |
|  | 8 5 8
|  | 458.182
|  | 916.364
|  | 174.545
|  | 632.727
|  |
|-
| |
| |
| |
| |
| |
|  | pi 2 pi
|  | 457.883
|  | 915.777
|  | 173.665
|  | 631.553
| |
|-
| | 8\21
| |
| |
| |
| |
|  | 3 2 3
|  | 457.143
|  | 914.286
|  | 171.429
|  | 628.571
|  | ...and ends here<br/> Optimum rank range (L/s=3/2) father
|-
| | 11\29
| |
| |
| |
| |
|  | 4 3 4
|  | 455.172
|  | 910.345
|  | 165.517
|  | 620.690
|  | [[Tridec]] is around here
|-
| | 14\37
| |
| |
| |
| |
|  | 5 4 5
|  | 454.054
|  | 908.108
|  | 162.162
|  | 616.216
| |
|-
| | 17\45
| |
| |
| |
| |
|  | 6 5 6
|  | 453.333
|  | 906.667
|  | 160
|  | 613.333
| |
|-
| | 20\53
| |
| |
| |
| |
|  | 7 6 7
|  | 452.83
|  | 905.66
|  | 158.491
|  | 611.321
| |
|-
| | 23\61
| |
| |
| |
| |
|  | 8 7 8
|  | 452.459
|  | 904.918
|  | 157.377
|  | 609.836
| |
|-
| | 26\69
| |
| |
| |
| |
|  | 9 8 9
|  | 452.174
|  | 904.348
|  | 156.522
|  | 608.696
| |
|-
| | 29\77
| |
| |
| |
| |
|  | 10 9 10
|  | 451.948
|  | 903.896
|  | 155.844
|  | 607.792
| |
|-
| | 3\8
| |
| |
| |
| |
|  | 1 1 1
|  | 450.000
|  | 900.000
|  | 150.000
|  | 600.000
|  |
|}


== Tunings ==
=== Modes ===
=== A-Team (13&18) ===
{{MOS mode degrees}}
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]].
==== Proposed mode names ====
* 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.
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
* 31edo can be used to make the major mos3rd a near-just 5/4.
{{MOS modes
| Mode Names=
Dylathian $
Ilarnekian $
Celephaïsian $
Ultharian $
Mnarian $
Kadathian $
Hlanithian $
Sarnathian $
| Collapsed=1
}}


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.
== Tunings==
=== Simple tunings ===
The simplest tuning for 5L&nbsp;3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.


The sizes of the generator, large step and small step of oneirotonic are as follows in various A-Team tunings.
{{MOS tunings|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}
{| 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) ===
=== Hypohard tunings ===
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.  
[[Hypohard]] oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with [[meantone]] diatonic tunings:
* The large step is a "meantone", around the range of [[10/9]] to [[9/8]].
* The major 2-mosstep is a [[meantone]]- to [[flattone]]-sized major third, thus is a stand-in for the classical diatonic major third.


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.  
With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to [[7/6]].


The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings.
EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]], and are associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.
{| class="wikitable right-2 right-3 right-4 right-5"
* 13edo has characteristically small 1-mossteps of about 185{{c}}. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best [[11/8]] out of all hypohard tunings.
|-
* 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3{{c}}, a perfect 5-mosstep) and falling fifths (666.7{{c}}, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
!
* 31edo can be used to make the major 2-mosstep a near-just 5/4.
! [[13edo]]
* [[44edo]] (generator {{nowrap|17\44 {{=}} 463.64{{c}}}}), [[57edo]] (generator {{nowrap|22\57 {{=}} 463.16{{c}}}}), and [[70edo]] (generator 27\70 {{=}} 462.857{{c}}}}) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.
! [[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,....


== Notation==
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}
The notation used in this article is J Celephaïsian (LsLLsLLs) = JKLMNOPQJ, with reference pitch J = 360 Hz, unless specified otherwise. We denote raising and lowering by a chroma (L-s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)


Thus the [[13edo]] gamut is as follows:
=== Hyposoft tunings ===
[[Hyposoft]] oneirotonic tunings have step ratios between 3:2 and 2:1, which 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 92{{c}} to 114{{c}}.
* The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342{{c}}) to 4\13 (369{{c}}).


'''J''' J&/K@ '''K''' '''L''' L&/M@ '''M''' M&/N@ '''N''' '''O''' O&/P@ '''P''' P&/Q@ '''Q''' '''J'''
* [[21edo]]'s P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71{{c}}) and Baroque diatonic semitones (114.29{{c}}, close to quarter-comma meantone's 117.11{{c}}).
* [[34edo]]'s 9:10:11:13 is even better.


Note: N is close to standard C, since the reference pitch 360 Hz for J was chosen to be nearly a pure 11/8 above standard 12edo C.
This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri|petrtri]] temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" [[petrtri]] temperament is.)
== Intervals ==
{| class="wikitable center-all"
|-
! Generators
! Notation (1/1 = J)
! Octatonic interval category name
! Generators
! Notation of 2/1 inverse
! Octatonic interval category name
|-
| colspan="6" style="text-align:left" | The 8-note MOS has the following intervals (from some root):
|-
| 0
| J
| perfect unison
| 0
| J
| octave
|-
| 1
| M
| perfect mosfourth
| -1
| O
| perfect mossixth
|-
| 2
| P
| major mosseventh
| -2
| L
| minor mosthird
|-
| 3
| K
| major mossecond
| -3
| Q@
| minor moseighth
|-
| 4
| N
| major mosfifth
| -4
| N@
| minor mosfifth
|-
| 5
| Q
| major moseighth
| -5
| K@
| minor mossecond
|-
| 6
| L&
| major mosthird
| -6
| P@
| minor mosseventh
|-
| 7
| O&
| augmented sixth
| -7
| M@
| diminished fourth
|-
| colspan="6" style="text-align:left" | The chromatic 13-note MOS also has the following intervals (from some root):
|-
| 8
| J&
| augmented unison
| -8
| J@
| diminished octave
|-
| 9
| M&
| augmented mosfourth
| -9
| O@
| diminished mossixth
|-
| 10
| P&
| augmented mosseventh
| -10
| L@
| diminished mosthird
|-
| 11
| K&
| augmented mossecond
| -11
| Q@@
| diminished moseighth
|-
| 12
| N&
| augmented mosfifth
| -12
| N@@
| diminished mosfifth
|}
== Key signatures ==
Flat keys:
* J@ Celephaïsian, L@ Dylathian = Q@, N@, K@, P@, M@, J@, O@, L@
* M@ Celephaïsian, O@ Dylathian = Q@, N@, K@, P@, M@, J@, O@
* P@ Celephaïsian, J@ Dylathian = Q@, N@, K@, P@, M@, J@
* K@ Celephaïsian, M@ Dylathian = Q@, N@, K@, P@, M@
* N@ Celephaïsian, P@ Dylathian = Q@, N@, K@, P@
* Q@ Celephaïsian, K@ Dylathian = Q@, N@, K@
* L Celephaïsian, N@ Dylathian = Q@, N@
* O Celephaïsian, Q@ Dylathian = Q@
All-natural key signature:
* J Celephaïsian, L Dylathian = no sharps or flats
Sharp keys:
* M Celephaïsian, O Dylathian = L&
* P Celephaïsian, J Dylathian = L&, O&
* K Celephaïsian, M Dylathian = L&, O&, J&
* N Celephaïsian, P Dylathian = L&, O&, J&, M&
* Q Celephaïsian, K Dylathian = L&, O&, J&, M&, P&
** Enharmonic with J@ Celeph., L@ Dylath. in [[13edo]]
* L& Celephaïsian, N Dylathian = L&, O&, J&, M&, P&, K&
** Enharmonic with M@ Celeph., O@ Dylath. in 13edo
* O& Celephaïsian, Q Dylathian = L&, O&, J&, M&, P&, K&, N&
** Enharmonic with P@ Celeph., J@ Dylath. in 13edo
* J& Celephaïsian, L& Dylathian = L&, O&, J&, M&, P&, K&, N&, Q&
** Enharmonic with K@ Celeph., M@ Dylath. in 13edo


== Modes ==
{{MOS tunings
Oneirotonic modes are named after cities in the Dreamlands.
| Step Ratios = Hyposoft
| JI Ratios =  
1/1;
16/15;
10/9; 11/10;
13/11; 20/17;
11/9;
5/4;
13/10;
18/13; 32/23;
13/9; 23/16;
20/13;
8/5;
18/11;
22/13; 17/10;
9/5;
15/8;
2/1
}}


# Dylathian: LLSLLSLS
=== Parasoft and ultrasoft tunings ===
# Ilarnekian: LLSLSLLS
The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to [[porcupine]] temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. The chord 10:11:13 is very well approximated in 29edo.
# Celephaïsian: LSLLSLLS (Easley Blackwood's 13-note etude uses this as its home mode.)
# Ultharian: LSLLSLSL (A kinda-sorta Dorian analogue. Depending on your purposes, a better Dorian analogue may be the MODMOS LSLLLSLS; see the section on oneiro MODMOSes below.)
# Mnarian: LSLSLLSL
# Kadathian: SLLSLLSL
# Hlanithian: SLLSLSLL
# Sarnathian: SLSLLSLL


The modes on the white keys JKLMNOPQJ are:
{{MOS tunings
* J Celephaïsian
| Step Ratios = 6/5; 3/2; 4/3
* K Kadathian
| JI Ratios =
* L Dylathian
1/1;
* M Ultharian
14/13;
* N Hlanithian
11/10;
* O Ilarnekian
9/8;
* P Mnarian
15/13;
* Q Sarnathian
13/11;
14/11;
13/10;
4/3;
15/11;
7/5;
10/7;
22/15;
3/2;
20/13;
11/7;
22/13;
26/15;
16/9;
20/11;
13/7;
2/1
}}


The modes in 13edo edo steps and C-H notation:
=== Parahard tunings ===
23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).


[[File:Oneirotonic.png|alt=Oneirotonic.png|Oneirotonic.png]]
{{MOS tunings
| JI Ratios =  
1/1;
21/17;
17/16;
14/11;
6/5;
21/16;
21/17;
34/21;
32/21;
5/3;
11/7;
32/17;
34/21;
2/1
| Step Ratios = 4/1
}}


== Pseudo-diatonic theory ==
=== Ultrahard tunings ===
Oneirotonic is often used as distorted diatonic. Because distorted diatonic modal harmony and functional harmony both benefit from a recognizable major third, the following theory essentially assumes an [[A-Team]] tuning, i.e. an oneirotonic tuning with generator between 5\13 and 7\18 (or possibly an approximation of such a tuning, such as a [[neji]]). The reader should experiment and see how well these ideas work in other oneirotonic tunings.
{{Main|5L&nbsp;3s/Temperaments#Buzzard}}
=== Ana modes ===
We call modes with a major mos5th  ''ana modes'' (from Greek for 'up'), because the sharper 5th degree functions as a flattened melodic fifth when moving from the tonic up. The ana modes of the MOS are the 4 brightest modes, namely Dylathian, Ilarnekian, Celephaïsian and Ultharian.


The ana modes have squashed versions of the classical major and minor pentachords R-M2-M3-P4-P5 and R-M2-m3-P4-P5 and can be viewed as providing a distorted version of classical diatonic functional harmony and counterpoint. For example, in the Dylathian mode, the 4:5:9 triad on the sixth degree can sound like both "V" and "III of iv" depending on context.
[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.


In pseudo-classical functional harmony, the 6th scale degree (either an augmented mossixth or a perfect mossixth) could be treated as mutable. The perfect mossixth would be used when invoking the diatonic V-to-I trope by modulating by a perfect mosfourth from the sixth degree. The augmented mossixth would be used when a major key needs to be used on the fourth degree.
In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. [[23edo]], [[28edo]] and [[33edo]] can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. [[38edo]] & [[43edo]] are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but [[48edo]] is where it really comes into its own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[The_Archipelago|archipelago]] harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.  


==== Progressions ====
Beyond that, it's a question of which intervals you want to favor. [[53edo]] has an essentially perfect [[3/2]], [[58edo]] gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while [[63edo]] does the same for the basic 4:6:7 triad. You could in theory go up to [[83edo]] if you want to favor the [[7/4]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.
Some suggested basic ana functional harmony progressions, outlined very roughly (note: VI is the sharp 5th, etc.). "I" means either Imaj or Imin. "Natural" Roman numerals follow the Ilarnekian mode.


* I-IVmin-VImaj-I
{{MOS tunings
* Imaj-VIImin-IVmaj-Imaj
| JI Ratios =
* Imin-@IIImaj-VImaj-Imaj
1/1;
* Imin-@IIImaj-Vdim-VImaj-Imin
8/7;
* Imin-@VIIImin-IIImaj-VImaj-Imin
13/10;
* Imin-IVmin-@VIIImin-@IIImaj-VImaj-Imin
21/16;
* Imin-IVmin-IIdim-VImaj-Imin
3/2;
* Imin-IVmin-IIdim-@IIImaj-Imin
12/7, 22/13;
* I-VIImin-IImin-VImaj-I
26/15;
* Imaj-VIImin-IVmin-VImaj-Imaj
49/25, 160/81;
* Modulations by major mos2nd:
2/1
** I-IV-VII-II
| Step Ratios = 7/1; 10/1; 12/1
** I-IVmaj-II
| Tolerance = 30
** I-VIImin-II
}}
* Modulations by major mos3rd:
** Modulate up major mos2nd twice
** Imin-VImin-III (only in 13edo)
** Imaj-&VImin-III (only in 13edo)
* Modulations by minor mos3rd:
** I-VI-@III
** I-IVmin-VImin-@VIIImaj-@III
Another approach to oneirotonic chord progressions is to let the harmony emerge from counterpoint.


=== Kata modes ===
== Approaches ==
We call modes with a minor mos5th ''kata modes'' (from Greek for 'down'). The kata modes of the MOS are the 4 darkest modes, namely Mnarian, Kadathian, Hlanithian and Sarnathian. In kata modes, the melodically squashed fifth from the tonic downwards is the flatter 5th degree. Kata modes could be used to distort diatonic tropes that start from the tonic and work downwards or work upwards towards the tonic from below it. For example:
* [[5L&nbsp;3s/Temperaments]]
* Mnarian (LSLSLLSL) and Kadathian (SLLSLLSL) are kata-Mixolydians
* Hlanithian (SLLSLSLL) is a kata-melodic major (the 4th degree sounds like a major third; it's actually a perfect mosfourth.)
* Sarnathian (SLSLLSLL) is a kata-melodic minor (When starting from the octave above, the 4th degree sounds like a minor third; it's actually a diminished mosfourth.)


When used in an "ana" way, the kata modes are radically different in character than the brighter modes. Because the fifth and seventh scale degrees become the more consonant minor tritone and the minor sixth respectively, the flat tritone sounds more like a stable scale function. Hlanithian, in particular, is a lot like a more stable version of the Locrian mode in diatonic.
== Samples ==
[[File:The Angels' Library edo.mp3]] [[:File:The Angels' Library edo.mp3|The Angels' Library]] by [[Inthar]] in the Sarnathian (23233233) mode of 21edo oneirotonic ([[:File:The Angels' Library Score.pdf|score]])


=== MODMOSes ===
[[File:13edo Prelude in J Oneirominor.mp3]]
The most important oneirotonic MODMOS is LSLLLSLS (and its rotations), because it allows one to evoke certain ana or kata diatonic modes where three whole steps in a row are important (Dorian, Phrygian, Lydian or Mixo) in an octatonic context. The MOS would not always be able to do this because it has at most two consecutive large steps.


As with the MOS, this MODMOS has four ana and four kata rotations:
[[WT13C]] [[:File:13edo Prelude in J Oneirominor.mp3|Prelude II (J Oneirominor)]] ([[:File:13edo Prelude in J Oneirominor Score.pdf|score]]) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.
* LLLSLSLS: Dylathian &4: an ana-Lydian
* LLSLSLSL: Ilarnekian @8: an ana-Mixolydian
* LSLLLSLS: Celephaïsian &6: an ana-Dorian
* SLLLSLSL: Ultharian @2: an ana-Phrygian
* SLSLSLLL: Sarnathian @6: a kata-Locrian
* SLSLLLSL: Sarnathian &6: a kata-Dorian
* LSLSLLLS: Mnarian &8: a kata-Ionian
* LSLSLSLL: Hlanithian &2: a kata-Aeolian


Other potentially interesting oneirotonic MODMOSes (that do not use half-sharps or half-flats) are:
[[File:13edo_1MC.mp3]]
* the distorted harmonic minor LSLSLLSAS (A = aug 2nd = L + chroma)
* the distorted Freygish SASLSLLS


=== Chords ===
(13edo, first 30 seconds is in J Celephaïsian)
Chords are given in oneirotonic MOS interval notation. For example, M5 means major mosfifth (squashed fifth).
* R-M3-M5: Squashed Major Triad
* R-m3-M5: Squashed Minor Triad
* R-m3-m5: Squashed Dim Triad
* R-M3-A5: Squashed Aug Triad
* R-M3-M5-A6: Squashed Major Triad Add6
* R-m3-M5-A6: Squashed Minor Triad Add6
* R-M3-M5-M7: Oneiro Major Tetrad
* R-m3-M5-M7: Oneiro Minor Tetrad
* R-m3-m5-M7: Oneiro Half-Diminished Tetrad
* R-m3-m5-m7: Orwell Tetrad, Oneiro Diminished Tetrad
* R-M3-A6: Squashed 1st Inversion Minor Triad
* R-m3-P6: Squashed 1st Inversion Major Triad
* R-M3-M7: 1st Inversion Squashed Minor Triad (note the order of terms!)
* R-m3-m7: 1st Inversion Squashed Major Triad
* R-m5-M7: 2nd Inversion Squashed Major Triad
* R-m5-m7: 2nd Inversion Squashed Minor Triad
* R-M3-M8: Oneiro Major Seventh
* R-m3-M8: Oneiro Minor Major Seventh
* R-M3-(M2): Oneiro Major Add9
* R-m3-(M2): Oneiro Minor Add9
* R-M3-(M2)-(P4): Oneiro Major Add9 Sub11
* R-m3-(M2)-(P4): Oneiro Minor Add9 Sub11
* R-M2-P4: Oneiro Sus2 Sus4
* R-P4-M7: Oneiro Quartal Triad
* R-P4-M7-(M2): Oneiro Quartal Tetrad, Core Tetrad
* R-P4-M7-(M2)-(M5): Oneiro Quartal Pentad, Core Pentad
* R-P4-M7-(M2)-(M5)-(M8): Oneiro Quartal Hexad
* R-P4-M7-M8: Oneiro Quartal Seventh Tetrad
* R-M3-m7: Sephiroth Triad (approximates 8:10:13 in 13edo)
* R-M3-m7-m2-(P4): Sephiroth Triad Addmin9 Sub11
* R-M3-m7-(P4): Sephiroth Triad Sub11
* R-P4-m8
* R-m3-P4-m8
* R-m5-m8
* R-m5-m7-m8
== 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]]).
=== Tridec (21&29, 2.7/5.11/5.13/5) ===
=== Petrtri (13&21, 2.5.9.11.13.17) ===
=== A-Team (13&18, 2.5.9.21) ===


=== Buzzard (48&53, 2.3.5.7) ===
[[File:A Moment of Respite.mp3]]
Commas: 1728/1715, 5120/5103


[[POTE_tuning|POTE generator]]: ~320/243 = 475.636
(13edo, L Ilarnekian)


Map: [&lt;1 0 -6 4|, &lt;0 4 21 -3|]
[[File:Lunar Approach.mp3]]


Wedgie: &lt;&lt;4 21 -3 24 -16 -66||
(by [[Igliashon Jones]], 13edo, J Celephaïsian)


EDOs: 48, 53, 111, 164d, 275d
=== 13edo Oneirotonic Modal Studies ===
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian
* [[File:Inthar-13edo Oneirotonic Studies 2 Ultharian.mp3]]: Tonal Study in Ultharian
* [[File:Inthar-13edo Oneirotonic Studies 3 Hlanithian.mp3]]: Tonal Study in Hlanithian
* [[File:Inthar-13edo Oneirotonic Studies 4 Illarnekian.mp3]]: Tonal Study in Ilarnekian
* [[File:Inthar-13edo Oneirotonic Studies 5 Mnarian.mp3]]: Tonal Study in Mnarian
* [[File:Inthar-13edo Oneirotonic Studies 6 Sarnathian.mp3]]: Tonal Study in Sarnathian
* [[File:Inthar-13edo Oneirotonic Studies 7 Celephaisian.mp3]]: Tonal Study in Celephaïsian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian


Badness: 0.0480
== Scale tree ==
 
{{MOS tuning spectrum
== Tritave MOSes with the 5L 3s pattern ==
| 13/8 = Golden oneirotonic (458.3592{{c}})
By a weird coincidence, the other generator for this MOS will generate the same pattern within a tritave equivalence. By yet another weird coincidence, this MOS belongs to a temperament which has [[Bohlen-Pierce|Bohlen-Pierce]] as its index-2 subtemperament. In addition to being harmonious, this tuning of the MOS gives an L/s ratio between 3/1 and 3/2, which is squarely in the middle of the range, being thus neither too exaggerated nor too equalized to be recognizable as such, unlike in octaves, where the only notable harmonic entropy minimum is near a greatly exaggerated 10/1 L/s ratio.
| 13/5 = Golden A-Team (465.0841{{c}})
 
}}
{| class="wikitable" style="text-align:center;"
|-
! colspan="5" |\
! | tetrachord
! | g in cents
hekts
! | 2g
! | 3g
! | 4g
! | Comments
|-
|  | 2\5
|  |
|  |
|  |
|  |
|  | 1 0 1
|  | 760.782
520
|  | 1521.564
1040
|  | 380.391
260
|  | 1141.173
780
|  |
|-
|  | 27\68
|  |
|  |
|  |
|  |
|  | 13 1 13
|  | 755.188
516.1765
|  | 1510.376
1032.353
|  | 363.609
248.529
|  | 1118.797
764.706
|  | 2g=12/5 minus quarter comma near here
|-
|  | 25\63
|  |
|  |
|  |
|  |
|  | 12 1 12
|  | 754.744
515.873
|  | 1509.488
1031.746
|  | 362.277
247.619
|  | 1117.021
763.492
|  |
|-
|  | 23\58
|  |
|  |
|  |
|  |
|  | 11 1 11
|  | 754.2235
515.517
|  | 1508.447
1031.0345
|  | 360.716
246.551
|  | 1114.939
762.069
|  |
|-
|  | 21\53
|  |
|  |
|  |
|  |
|  | 10 1 10
|  | 753.605
515.094
|  | 1507.21
1030.189
|  | 358.859
245.283
|  | 1112.464
760.378
|  |
|-
|  | 19\48
|  |
|  |
|  |
|  |
|  | 9 1 9
|  | 752.857
514.583
|  | 1505.714
1029.167
|  | 356.617
243.75
|  | 1109.474
758.333
|  |
|-
|  | 17\43
|  |
|  |
|  |
|  |
|  | 8 1 8
|  | 751.936
513.9535
|  | 1503.871
1027.907
|  | 353.852
241.8605
|  | 1105.788
755.814
|  |
|-
|  | 15\38
|  |
|  |
|  |
|  |
|  | 7 1 7
|  | 750.771
513.158
|  | 1501.543
1026.316
|  | 350.36
239.474
|  | 1101.132
752.632
|  |
|-
|  |
|  | 28/71
|  |
|  |
|  |
|  | 13 2 13
|  | 750.067
512.676
|  | 1500.1335
1025.352
|  | 348.245
238.028
|  | 1098.312
750.704
|  |
|-
|  |
|  | 41\104
|  |
|  |
|  |
|  | 19 3 19
|  | 749.809
512.5
|  | 1499.618
1025
|  | 347.4725
237.5
|  | 1097.282
750
|  | 3g=11/3 near here
|-
|  | 13\33
|  |
|  |
|  |
|  |
|  | 6 1 6
|  | 749.255
512.121
|  | 1498.51
1024.242
|  | 345.81
236.364
|  | 1095.065
748.485
|  |
|-
|  |
|  | 24\61
|  |
|  |
|  |
|  | 11 2 11
|  | 748.31
511.475
|  | 1496.62
1022.951
|  | 342.976
234.426
|  | 1091.286
745.902
|  |
|-
|  |
|  | 35\89
|  |
|  |
|  |
|  | 16 3 16
|  | 747.96
511.236
|  | 1495.92
1022.472
|  | 341.924
233.708
|  | 1089.884
744.944
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | 5+√29 2 5+√29
|  | 747.648
511.023
|  | 1495.297
1022.046
|  | 340.99
233.069
|  | 1088.638
744.092
|  |4g=45/8 near here
|-
|  | 11\28
|  |
|  |
|  |
|  |
|  | 5 1 5
|  | 747.197
510.714
|  | 1494.393
1021.429
|  | 339.635
232.143
|  | 1086.831
742.857
|  |
|-
|  |
|  | 20\51
|  |
|  |
|  |
|  | 9 2 9
|  | 745.865
509.804
|  | 1491.729
1019.608
|  | 335.639
229.412
|  | 1081.504
739.216
|  |
|-
|  |
|  | 29\74
|  |
|  |
|  |
|  | 13 3 13
|  | 745.361
509.4595
|  | 1490.721
1018.919
|  | 334.127
228.378
|  | 1079.488
737.838
|  |
|-
|  |
|  | 38/97
|  |
|  |
|  |
|  | 17 4 17
|  | 745.096
509.278
|  | 1490.192
1018.557
|  | 333.332
227.835
|  | 1078.428
737.113
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | 2+√5 1 2+√5
|  | 754.051
509.2475
|  | 1490.101
1018.495
|  | 333.197
227.742
|  | 1078.247
736.99
|  |
|-
|  |
|  | 47\120
|  |
|  |
|  |
|  | 21 5 21
|  | 744.932
509.167
|  | 1489.865
1018.333
|  | 332.842
227.5
|  | 1077.7745
736.667
|  |
|-
|  | 9\23
|  |
|  |
|  |
|  |
|  | 4 1 4
|  | 744.243
508.696
|  | 1488.487
1017.391
|  | 330.775
226.087
|  | 1075.018
734.783
|  | L/s = 4
|-
|  |
|  | 34\87
|  |
|  |
|  |
|  | 15 4 15
|  | 743.293
508.046
|  | 1486.586
1016.092
|  | 327.923
224.138
|  | 1071.216
732.184
|  | 4g=39/7 near here
|-
|  |
|  | 25\64
|  |
|  |
|  |
|  | 11 3 11
|  | 742.951
507.8125
|  | 1485.902
1015.625
|  | 326.899
223.4375
|  | 1069.85
731.25
|  |
|-
|  |
|  | 16\41
|  |
|  |
|  |
|  | 7 2 7
|  | 742.226
507.317
|  | 1484.453
1014.634
|  | 324.724
221.951
|  | 1066.95
728.268
|  |
|-
|  |
|  | 23\59
|  |
|  |
|  |
|  | 10 3 10
|  | 741.44
506.78
|  | 1482.88
1013.56
|  | 322.365
220.34
|  | 1063.805
727.12
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | 3+√13 2 3+√13
|  | 741.289
506.676
|  | 1482.577
1013.352
|  | 321.911
220.028
|  | 1063.2
726.705
|  |
|-
|  |
|  | 30\77
|  |
|  |
|  |
|  | 13 4 13
|  | 741.021
506.4935
|  | 1482.043
1012.987
|  | 321.109
219.4805
|  | 1062.131
725.974
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | pi 1 pi
|  | 740.449
506.102
|  | 1480.898
1012.204
|  | 319.392
218.3065
|  | 1056.841
724.409
|  | L/s = pi
|-
|  | 7\18
|  |
|  |
|  |
|  |
|  | 3 1 3
|  | 739.649
505.556
|  | 1479.298
1011.111
|  | 316.992
216.667
|  | 1056.642
722.222
|  | L/s = 3
|-
|  |
|  | 68\175
|  |
|  |
|  |
|  | 29 10 29
|  | 739.045
505.143
|  | 1478.091
1010.286
|  | 315.181
215.429
|  | 1054.227
720.571
|  |3g=18/5 near here
|-
|  |
|  | 61/157
|  |
|  |
|  |
|  | 26 9 26
|  | 738.976
505.0955
|  | 1477.952
1010.191
|  | 314.973
215.287
|  | 1053.949
720.382
|  |
|-
|  |
|  | 54\139
|  |
|  |
|  |
|  | 23 8 23
|  | 738.889
505.036
|  | 1477.778
1010.072
|  | 314.712
215.108
|  | 1053.601
720.144
|  |
|-
|  |
|  | 47\121
|  |
|  |
|  |
|  | 20 7 20
|  | 738.776
504.959
|  | 1477.552
1009.917
|  | 314.373
214.876
|  | 1053.149
719.835
|  |
|-
|  |
|  | 40\103
|  |
|  |
|  |
|  | 17 6 17
|  | 738.623
504.854
|  | 1477.247
1009.709
|  | 313.915
214.563
|  | 1052.538
719.4175
|  |
|-
|  |
|  | 33\85
|  |
|  |
|  |
|  | 14 5 14
|  | 738.406
504.706
|  | 1476.812
1009.412
|  | 313.263
214.1765
|  | 1051.669
718.882
|  |
|-
|  |
|  | 26\67
|  |
|  |
|  |
|  | 11 4 11
|  | 738.072
504.478
|  | 1476.144
1008.955
|  | 312.261
213.433
|  | 1050.333
717.91
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | e 1 e
|  | 737.855
504.329
|  | 1475.71
1008.6585
|  | 311.61
212.988
|  | 1049.465
717.317
|  | L/s = e
|-
|  |
|  | 19\49
|  |
|  |
|  |
|  | 8 3 8
|  | 737.493
504.082
|  | 1474.986
1008.163
|  | 310.523
212.245
|  | 1048.016
716.3265
|  | 3g=18/5 minus quarter comma near here
|-
|  |
|  |
|  | 50\129
|  |
|  |
|  | 21 8 21
|  | 737.192
503.876
|  | 1474.384
1007.752
|  | 309.621
211.628
|  | 1046.812
715.504
|  |
|-
|  |
|  |
|  |
|  | 131\338
|  |
|  | 55 21 55
|  | 737.148
503.846
|  | 1474.296
1007.692
|  | 309.49
211.5385
|  | 1046.638
715.385
|  |
|-
|  |
|  |
|  |
|  |
|  | 212\547
|  | 89 34 89
|  | 737.138
503.839
|  | 1474.276
1007.678
|  | 309.459
211.517
|  | 1046.597
715.3565
|  |
|-
|  |
|  |
|  |
|  | 81\209
|  |
|  | 34 13 34
|  | 737.121
503.828
|  | 1474.243
1007.6555
|  | 309.409
211.483
|  | 1046.53
715.311
|  |
|-
|  |
|  |
|  | 31\80
|  |
|  |
|  | 13 5 13
|  | 737.008
503.75
|  | 1474.015
1007.5
|  | 309.068
211.25
|  | 1046.075
715
|  |
|-
|  |
|  | 12\31
|  |
|  |
|  |
|  | 5 2 5
|  | 736.241
503.226
|  | 1472.481
1006.452
|  | 306.767
209.677
|  | 1043.007
712.903
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | 1+√2 1 1+√2
|  | 735.542
502.748
|  | 1471.084
1005.497
|  | 304.6715
208.245
|  | 1040.214
710.994
|  | Silver false father
|-
|  |
|  | 17\44
|  |
|  |
|  |
|  | 7 3 7
|  | 734.846
502.273
|  | 1469.693
1004.5455
|  | 302.584
206.818
|  | 1037.41
709.091
|  |
|-
|  |
|  | 22\57
|  |
|  |
|  |
|  | 9 4 9
|  | 734.088
501.754
|  | 1468.176
1003.509
|  | 300.309
205.263
|  | 1034.397
707.0175
|  |
|-
|  |
|  | 27\70
|  |
|  |
|  |
|  | 11 5 11
|  | 733.611
501.429
|  | 1467.222
1002.857
|  | 298.879
204.286
|  | 1032.49
705.714
|  |
|-
|  |
|  | 32\83
|  |
|  |
|  |
|  | 13 6 13
|  | 733.284
501.205
|  | 1466.568
1002.41
|  | 297.897
203.6145
|  | 1031.181
704.819
|  | 2g=7/3 near here
|-
|  | 5\13
|  |
|  |
|  |
|  |
|  | 2 1 2
|  | 731.521
500
|  | 1463.042
1000
|  | 292.609
200
|  | 1024.13
700
|  |
|-
|  |
|  | 48\125
|  |
|  |
|  |
|  | 19 10 19
|  | 730.35
499.2
|  | 1460.701
998.4
|  | 289.097
197.6
|  | 1019.448
696.8
|  | 3g=39/11 near here
|-
|  |
|  | 43\112
|  |
|  |
|  |
|  | 17 9 17
|  | 730.215
499.107
|  | 1460.43
998.214
|  | 288.69
197.321
|  | 1018.905
696.429
|  |
|-
|  |
|  | 38\99
|  |
|  |
|  |
|  | 15 8 15
|  | 730.043
498.99
|  | 1460.087
997.98
|  | 288.175
196.97
|  | 1018.218
695.96
|  |
|-
|  |
|  | 33\86
|  |
|  |
|  |
|  | 13 7 13
|  | 729.82
498.837
|  | 1459.64
997.674
|  | 287.505
196.512
|  | 1017.325
695.349
|  | 4g=27/5 near here
|-
|  |
|  | 28\73
|  |
|  |
|  |
|  | 11 6 11
|  | 729.547
498.63
|  | 1459.034
997.26
|  | 286.596
195.89
|  | 1016.113
694.5205
|  |
|-
|  |
|  | 23\60
|  |
|  |
|  |
|  | 9 5 9
|  | 729.083
498.333
|  | 1458.1655
996.667
|  | 285.293
195
|  | 1014.376
693.333
|  |
|-
|
|
|41\107
|
|
|16 9 16
|728.7865
498.131
|1457.563
996.262
|284.4045
194.3925
|1013.191
692.523
|
|-
|  |
|  |
|  | 59\154
|  |
|  |
|  | 23 13 23
|  | 728.671
498.052
|  | 1457.342
996.104
|  | 284.058
194.156
|  | 1012.729
692.208
|  | 3g=99/28 near here
|-
|  |
|  |
|  | 77\201
|  |
|  |
|  | 30 17 30
|  | 728.61
498.01
|  | 1457.219
996.02
|  | 283.874
194.03
|  | 1012.483
692.04
|  |
|-
|  |
|  |
|  | 95\248
|  |
|  |
|  | 37 21 37
|  | 728.5715
497.984
|  | 1457.143
995.968
|  | 283.7145
193.952
|  | 1012.286
691.9355
|  | Golden BP is index-2 near here
|-
|  |
|  | 18\47
|  |
|  |
|  |
|  | 7 4 7
|  | 728.408
497.872
|  | 1456.817
995.745
|  | 283.27
193.617
|  | 1011.678
691.49
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | √3 1 √3
|  | 728.159
497.702
|  | 1456.318
995.404
|  | 282.522
193.106
|  | 1010.6815
690.808
|  | 4g=27/5 minus third comma near here
|-
|  |
|  |
|  | 31\81
|  |
|  |
|  | 12 7 12
|  | 727.909
497.531
|  | 1455.817
995.062
|  | 281.771
192.593
|  | 1009.68
690.1235
|  |
|-
|  |
|  | 13\34
|  |
|  |
|  |
|  | 5 3 5
|  | 727.218
497.059
|  | 1454.436
994.118
|  | 279.699
191.1765
|  | 1006.917
688.235
|  |
|-
|  |
|  |
|  | 34\89
|  |
|  |
|  | 13 8 13
|  | 726.59
496.629
|  | 1453.179
993.258
|  | 277.814
189.888
|  | 1004.403
686.517
|  |
|-
|  |
|  |
|  |
|  | 89\233
|  |
|  | 34 21 34
|  | 726.498
496.5665
|  | 1452.996
993.133
|  | 277.538
189.7
|  | 1004.036
686.266
|  |
|-
|  |
|  |
|  |
|  |
|  | 233\610
|  | 89 55 89
|  | 726.4845
496.557
|  | 1452.969
993.115
|  | 277.4985
189.672
|  | 1003.983
686.2295
|  | Golden false father
|-
|  |
|  |
|  |
|  | 144\377
|  |
|  | 55 34 55
|  | 726.476
496.552
|  | 1452.952
993.104
|  | 277.473
189.655
|  | 1003.95
686.207
|  |
|-
|  |
|  |
|  | 55\144
|  |
|  |
|  | 21 13 21
|  | 726.441
496.528
|  | 1452.882
993.056
|  | 277.368
189.583
|  | 1003.809
686.111
|  |
|-
|  |
|  | 21\55
|  |
|  |
|  |
|  | 8 5 8
|  | 726.201
496.364
|  | 1452.402
992.727
|  | 276.468
189.091
|  | 1002.849
685.4545
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | pi 2 pi
|  | 725.736
496.046
|  | 1451.472
992.091
|  | 275.252
188.137
|  | 1000.988
684.183
|  |
|-
|  | 8\21
|  |
|  |
|  |
|  |
|  | 3 2 3
|  | 724.554
495.238
|  | 1449.109
990.476
|  | 271.708
185.714
|  | 996.226
680.952
|  | Optimum rank range (L/s=3/2) false father
4g=16/3 near here
|-
|  |
|  | 27\71
|  |
|  |
|  |
|  | 10 7 10
|  | 723.279
494.366
|  | 1446.557
988.732
|  | 267.881
183.099
|  | 991.16
677.465
|  |
|-
|  |
|  |
|  | 46\121
|  |
|  |
|  | 17 12 17
|  | 723.057
494.215
|  | 1446.115
988.43
|  | 267.217
182.645
|  | 990.274
676.8595
|  |
|-
|  |
|  | 19\50
|  |
|  |
|  |
|  | 7 5 7
|  | 722.743
494
|  | 1445.486
988
|  | 266.274
182
|  | 989.017
676
|  |3g=7/2 near here
|-
|  | 11\29
|  |
|  |
|  |
|  |
|  | 4 3 4
|  | 721.431
493.103
|  | 1442.862
986.207
|  | 262.338
179.31
|  | 983.77
672.414
|  |
|-
|  |
|  | 25\66
|  |
|  |
|  |
|  | 9 7 9
|  | 720.4375
492.424
|  | 1440.875
984.8485
|  | 259.3575
177.273
|  | 979.795
669.697
|  |
|-
|  |
|  |
|  | 64\169
|  |
|  |
|  | 23 18 23
|  | 720.267
492.308
|  | 1440.534
984.615
|  | 258.848
176.923
|  | 979.113
669.231
|  |
|-
|  |
|  |
|  |
|  | 167\441
|  |
|  | 60 47 60
|  | 720.2415
492.29
|  | 1440.483
984.5805
|  | 258.7965
176.871
|  | 979.001
669.161
|  |
|-
|  |
|  |
|  |
|  |
|  | 437\1154
|  | 157 123 157
|  | 720.238
492.288
|  | 1440.475
984.575
|  | 258.758
176.863
|  | 978.996
669.151
|  |
|-
|  |
|  |
|  |
|  | 270\713
|  |
|  | 97 76 97
|  | 720.235
492.286
|  | 1440.471
984.572
|  | 258.751
176.858
|  | 978.987
669.1445
|  |
|-
|  |
|  |
|  | 103\272
|  |
|  |
|  | 37 29 37
|  | 720.226
492.279
|  | 1440.451
984.558
|  | 258.722
176.837
|  | 978.947
669.116
|  |
|-
|  |
|  | 39\103
|  |
|  |
|  |
|  | 14 11 14
|  | 720.158
492.233
|  | 1440.315
984.466
|  | 258.518
176.699
|  | 978.676
668.932
|  |
|-
|  | 14\37
|  |
|  |
|  |
|  |
|  | 5 4 5
|  | 719.659
491.892
|  | 1439.317
983.784
|  | 257.021
175.676
|  | 976.679
667.568
|  |
|-
|  |
|  | 31\82
|  |
|  |
|  |
|  | 11 9 11
|  | 719.032
491.463
|  | 1438.064
982.927
|  | 255.14
174.39
|  | 974.172
665.844
|  |
|-
|  |
|  |
|  | 79\209
|  |
|  |
|  | 28 23 28
|  | 718.921
491.388
|  | 1437.842
982.775
|  | 254.807
174.163
|  | 973.728
665.55
|  |
|-
|  |
|  |
|  |
|  | 206\545
|  |
|  | 73 60 73
|  | 718.904
491.376
|  | 1437.808
982.752
|  | 254.757
174.138
|  | 973.661
665.505
|  |
|-
|  |
|  |
|  |
|  |
|  | 539\1426
|  | 191 117 191
|  | 718.902
491.3745
|  | 1437.803
982.749
|  | 254.75
174.123
|  | 973.652
665.498
|  |
|-
|  |
|  |
|  |
|  | 333\881
|  |
|  | 118 97 118
|  | 718.9
491.373
|  | 1437.8
982.747
|  | 254.745
174.12
|  | 973.6455
665.494
|  |
|-
|  |
|  |
|  | 127\336
|  |
|  |
|  | 45 37 45
|  | 718.893
491.369
|  | 1437.787
982.738
|  | 254.726
174.107
|  | 973.619
665.476
|  |
|-
|  |
|  | 48\127
|  |
|  |
|  |
|  | 17 14 17
|  | 718.849
491.339
|  | 1437.698
982.677
|  | 254.592
174.016
|  | 973.441
665.354
|  |
|-
|  | 17\45
|  |
|  |
|  |
|  |
|  | 6 5 6
|  | 718.516
491.111
|  | 1437.032
982.222
|  | 253.549
173.333
|  | 972.11
664.444
|  |
|-
|  | 20\53
|  |
|  |
|  |
|  |
|  | 7 6 7
|  | 717.719
490.566
|  | 1435.438
981.132
|  | 251.202
171.698
|  | 968.9205
662.264
|  |4g=21/4 near here
|-
|  | 23\61
|  |
|  |
|  |
|  |
|  | 8 7 8
|  | 717.131
490.164
|  | 1434.261
980.328
|  | 249.437
170.492
|  | 966.567
660.656
|  |
|-
|
|49\130
|
|
|
|17 15 17
|716.891
490
|1433.7815
980
|248.717
170
|965.608
660
|4g=quarter-comma meantone 21/4 near here
 
6g=12 near here
|-
|  | 26\69
|  |
|  |
|  |
|  |
|  | 9 8 9
|  | 716.679
489.855
|  | 1433.357
979.71
|  | 248.081
169.565
|  | 964.76
659.42
|  |
|-
|  | 29\77
|  |
|  |
|  |
|  |
|  | 10 9 10
|  | 716.321
489.61
|  | 1432.641
979.221
|  | 247.007
168.831
|  | 963.328
658.442
|  |
|-
|  | 32\85
|  |
|  |
|  |
|  |
|  | 11 10 11
|  | 716.03
489.412
|  | 1432.06
978.8235
|  | 246.135
168.235
|  | 962.1655
657.647
|  |
|-
|  | 35\93
|  |
|  |
|  |
|  |
|  | 12 11 12
|  | 715.7895
489.247
|  | 1431.579
978.495
|  | 245.4135
167.742
|  | 961.203
656.989
|  |
|-
|  | 38/101
|  |
|  |
|  |
|  |
|  | 13 12 13
|  | 715.587
489.109
|  | 1431.174
978.218
|  | 244.806
167.327
|  | 960.393
656.436
|  | 2g=16\7 near here
|-
|  | 3\8
|  |
|  |
|  |
|  |
|  | 1 1 1
|  | 713.233
487.5
|  | 1426.466
975
|  | 237.744
162.5
|  | 950.9775
650
|  |
|}


[[Category:Abstract MOS patterns]][[Category:Oneirotonic]]
[[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
[[Category:Pages with internal sound examples]]

Latest revision as of 13:59, 5 May 2025

↖ 4L 2s ↑ 5L 2s 6L 2s ↗
← 4L 3s 5L 3s 6L 3s →
↙ 4L 4s ↓ 5L 4s 6L 4s ↘ 
┌╥╥┬╥╥┬╥┬┐
│║║│║║│║││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLLsLs
sLsLLsLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 3\8 to 2\5 (450.0 ¢ to 480.0 ¢)
Dark 3\5 to 5\8 (720.0 ¢ to 750.0 ¢)
TAMNAMS information
Name oneirotonic
Prefix oneiro-
Abbrev. onei
Related MOS scales
Parent 3L 2s
Sister 3L 5s
Daughters 8L 5s, 5L 8s
Neutralized 2L 6s
2-Flought 13L 3s, 5L 11s
Equal tunings
Equalized (L:s = 1:1) 3\8 (450.0 ¢)
Supersoft (L:s = 4:3) 11\29 (455.2 ¢)
Soft (L:s = 3:2) 8\21 (457.1 ¢)
Semisoft (L:s = 5:3) 13\34 (458.8 ¢)
Basic (L:s = 2:1) 5\13 (461.5 ¢)
Semihard (L:s = 5:2) 12\31 (464.5 ¢)
Hard (L:s = 3:1) 7\18 (466.7 ¢)
Superhard (L:s = 4:1) 9\23 (469.6 ¢)
Collapsed (L:s = 1:0) 2\5 (480.0 ¢)
For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (3/1-equivalent).

5L 3s, named oneirotonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 450 ¢ to 480 ¢, or from 720 ¢ to 750 ¢. 5L 3s can be seen as a warped diatonic scale, because it has one extra small step compared to diatonic (5L 2s).

Name

TAMNAMS suggests the temperament-agnostic name oneirotonic as the name of 5L 3s. The name was originally used as a name for the 5L 3s scale in 13edo. 'Oneiro' is sometimes used as a shortened form.

'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as father is technically an abstract regular temperament (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate 3L 2s.

Scale properties

Intervals

Intervals of 5L 3s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-oneirostep Perfect 0-oneirostep P0oneis 0 0.0 ¢
1-oneirostep Minor 1-oneirostep m1oneis s 0.0 ¢ to 150.0 ¢
Major 1-oneirostep M1oneis L 150.0 ¢ to 240.0 ¢
2-oneirostep Minor 2-oneirostep m2oneis L + s 240.0 ¢ to 300.0 ¢
Major 2-oneirostep M2oneis 2L 300.0 ¢ to 480.0 ¢
3-oneirostep Diminished 3-oneirostep d3oneis L + 2s 240.0 ¢ to 450.0 ¢
Perfect 3-oneirostep P3oneis 2L + s 450.0 ¢ to 480.0 ¢
4-oneirostep Minor 4-oneirostep m4oneis 2L + 2s 480.0 ¢ to 600.0 ¢
Major 4-oneirostep M4oneis 3L + s 600.0 ¢ to 720.0 ¢
5-oneirostep Perfect 5-oneirostep P5oneis 3L + 2s 720.0 ¢ to 750.0 ¢
Augmented 5-oneirostep A5oneis 4L + s 750.0 ¢ to 960.0 ¢
6-oneirostep Minor 6-oneirostep m6oneis 3L + 3s 720.0 ¢ to 900.0 ¢
Major 6-oneirostep M6oneis 4L + 2s 900.0 ¢ to 960.0 ¢
7-oneirostep Minor 7-oneirostep m7oneis 4L + 3s 960.0 ¢ to 1050.0 ¢
Major 7-oneirostep M7oneis 5L + 2s 1050.0 ¢ to 1200.0 ¢
8-oneirostep Perfect 8-oneirostep P8oneis 5L + 3s 1200.0 ¢

Generator chain

Generator chain of 5L 3s
Bright gens Scale degree Abbrev.
12 Augmented 4-oneirodegree A4oneid
11 Augmented 1-oneirodegree A1oneid
10 Augmented 6-oneirodegree A6oneid
9 Augmented 3-oneirodegree A3oneid
8 Augmented 0-oneirodegree A0oneid
7 Augmented 5-oneirodegree A5oneid
6 Major 2-oneirodegree M2oneid
5 Major 7-oneirodegree M7oneid
4 Major 4-oneirodegree M4oneid
3 Major 1-oneirodegree M1oneid
2 Major 6-oneirodegree M6oneid
1 Perfect 3-oneirodegree P3oneid
0 Perfect 0-oneirodegree
Perfect 8-oneirodegree		 P0oneid
P8oneid
−1 Perfect 5-oneirodegree P5oneid
−2 Minor 2-oneirodegree m2oneid
−3 Minor 7-oneirodegree m7oneid
−4 Minor 4-oneirodegree m4oneid
−5 Minor 1-oneirodegree m1oneid
−6 Minor 6-oneirodegree m6oneid
−7 Diminished 3-oneirodegree d3oneid
−8 Diminished 8-oneirodegree d8oneid
−9 Diminished 5-oneirodegree d5oneid
−10 Diminished 2-oneirodegree d2oneid
−11 Diminished 7-oneirodegree d7oneid
−12 Diminished 4-oneirodegree d4oneid

Modes

Scale degrees of the modes of 5L 3s
UDP Cyclic
order 		Step
pattern 		Scale degree (oneirodegree)
0 1 2 3 4 5 6 7 8
7|0 1 LLsLLsLs Perf. Maj. Maj. Perf. Maj. Aug. Maj. Maj. Perf.
6|1 4 LLsLsLLs Perf. Maj. Maj. Perf. Maj. Perf. Maj. Maj. Perf.
5|2 7 LsLLsLLs Perf. Maj. Min. Perf. Maj. Perf. Maj. Maj. Perf.
4|3 2 LsLLsLsL Perf. Maj. Min. Perf. Maj. Perf. Maj. Min. Perf.
3|4 5 LsLsLLsL Perf. Maj. Min. Perf. Min. Perf. Maj. Min. Perf.
2|5 8 sLLsLLsL Perf. Min. Min. Perf. Min. Perf. Maj. Min. Perf.
1|6 3 sLLsLsLL Perf. Min. Min. Perf. Min. Perf. Min. Min. Perf.
0|7 6 sLsLLsLL Perf. Min. Min. Dim. Min. Perf. Min. Min. Perf.

Proposed mode names

The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.

Modes of 5L 3s
UDP Cyclic
order		 Step
pattern		 Mode names
7|0 1 LLsLLsLs Dylathian
6|1 4 LLsLsLLs Ilarnekian
5|2 7 LsLLsLLs Celephaïsian
4|3 2 LsLLsLsL Ultharian
3|4 5 LsLsLLsL Mnarian
2|5 8 sLLsLLsL Kadathian
1|6 3 sLLsLsLL Hlanithian
0|7 6 sLsLLsLL Sarnathian

Tunings

Simple tunings

The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.


Simple Tunings of 5L 3s
Scale degree Abbrev. Basic (2:1)
13edo 		Hard (3:1)
18edo 		Soft (3:2)
21edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\13 0.0 0\18 0.0 0\21 0.0
Minor 1-oneirodegree m1oneid 1\13 92.3 1\18 66.7 2\21 114.3
Major 1-oneirodegree M1oneid 2\13 184.6 3\18 200.0 3\21 171.4
Minor 2-oneirodegree m2oneid 3\13 276.9 4\18 266.7 5\21 285.7
Major 2-oneirodegree M2oneid 4\13 369.2 6\18 400.0 6\21 342.9
Diminished 3-oneirodegree d3oneid 4\13 369.2 5\18 333.3 7\21 400.0
Perfect 3-oneirodegree P3oneid 5\13 461.5 7\18 466.7 8\21 457.1
Minor 4-oneirodegree m4oneid 6\13 553.8 8\18 533.3 10\21 571.4
Major 4-oneirodegree M4oneid 7\13 646.2 10\18 666.7 11\21 628.6
Perfect 5-oneirodegree P5oneid 8\13 738.5 11\18 733.3 13\21 742.9
Augmented 5-oneirodegree A5oneid 9\13 830.8 13\18 866.7 14\21 800.0
Minor 6-oneirodegree m6oneid 9\13 830.8 12\18 800.0 15\21 857.1
Major 6-oneirodegree M6oneid 10\13 923.1 14\18 933.3 16\21 914.3
Minor 7-oneirodegree m7oneid 11\13 1015.4 15\18 1000.0 18\21 1028.6
Major 7-oneirodegree M7oneid 12\13 1107.7 17\18 1133.3 19\21 1085.7
Perfect 8-oneirodegree P8oneid 13\13 1200.0 18\18 1200.0 21\21 1200.0

Hypohard tunings

Hypohard oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with meantone diatonic tunings:

  • The large step is a "meantone", around the range of 10/9 to 9/8.
  • The major 2-mosstep is a meantone- to flattone-sized major third, thus is a stand-in for the classical diatonic major third.

With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to 7/6.

EDOs that are in the hypohard range include 13edo, 18edo, and 31edo, and are associated with A-Team temperament.

  • 13edo has characteristically small 1-mossteps of about 185 ¢. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best 11/8 out of all hypohard tunings.
  • 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3 ¢, a perfect 5-mosstep) and falling fifths (666.7 ¢, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
  • 31edo can be used to make the major 2-mosstep a near-just 5/4.
  • 44edo (generator 17\44 = 463.64 ¢), 57edo (generator 22\57 = 463.16 ¢), and 70edo (generator 27\70 = 462.857 ¢}}) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.


Hypohard Tunings of 5L 3s
Scale degree Abbrev. Basic (2:1)
13edo 		Semihard (5:2)
31edo 		Hard (3:1)
18edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\13 0.0 0\31 0.0 0\18 0.0
Minor 1-oneirodegree m1oneid 1\13 92.3 2\31 77.4 1\18 66.7
Major 1-oneirodegree M1oneid 2\13 184.6 5\31 193.5 3\18 200.0
Minor 2-oneirodegree m2oneid 3\13 276.9 7\31 271.0 4\18 266.7
Major 2-oneirodegree M2oneid 4\13 369.2 10\31 387.1 6\18 400.0
Diminished 3-oneirodegree d3oneid 4\13 369.2 9\31 348.4 5\18 333.3
Perfect 3-oneirodegree P3oneid 5\13 461.5 12\31 464.5 7\18 466.7
Minor 4-oneirodegree m4oneid 6\13 553.8 14\31 541.9 8\18 533.3
Major 4-oneirodegree M4oneid 7\13 646.2 17\31 658.1 10\18 666.7
Perfect 5-oneirodegree P5oneid 8\13 738.5 19\31 735.5 11\18 733.3
Augmented 5-oneirodegree A5oneid 9\13 830.8 22\31 851.6 13\18 866.7
Minor 6-oneirodegree m6oneid 9\13 830.8 21\31 812.9 12\18 800.0
Major 6-oneirodegree M6oneid 10\13 923.1 24\31 929.0 14\18 933.3
Minor 7-oneirodegree m7oneid 11\13 1015.4 26\31 1006.5 15\18 1000.0
Major 7-oneirodegree M7oneid 12\13 1107.7 29\31 1122.6 17\18 1133.3
Perfect 8-oneirodegree P8oneid 13\13 1200.0 31\31 1200.0 18\18 1200.0

Hyposoft tunings

Hyposoft oneirotonic tunings have step ratios between 3:2 and 2:1, which 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 92 ¢ to 114 ¢.
  • The major 2-mosstep (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 ¢).
  • 21edo's P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 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 ¢).
  • 34edo's 9:10:11:13 is even better.

This set of JI identifications is associated with petrtri temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" petrtri temperament is.)


Hyposoft Tunings of 5L 3s
Scale degree Abbrev. Soft (3:2)
21edo 		Semisoft (5:3)
34edo 		Basic (2:1)
13edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\21 0.0 0\34 0.0 0\13 0.0
Minor 1-oneirodegree m1oneid 2\21 114.3 3\34 105.9 1\13 92.3
Major 1-oneirodegree M1oneid 3\21 171.4 5\34 176.5 2\13 184.6
Minor 2-oneirodegree m2oneid 5\21 285.7 8\34 282.4 3\13 276.9
Major 2-oneirodegree M2oneid 6\21 342.9 10\34 352.9 4\13 369.2
Diminished 3-oneirodegree d3oneid 7\21 400.0 11\34 388.2 4\13 369.2
Perfect 3-oneirodegree P3oneid 8\21 457.1 13\34 458.8 5\13 461.5
Minor 4-oneirodegree m4oneid 10\21 571.4 16\34 564.7 6\13 553.8
Major 4-oneirodegree M4oneid 11\21 628.6 18\34 635.3 7\13 646.2
Perfect 5-oneirodegree P5oneid 13\21 742.9 21\34 741.2 8\13 738.5
Augmented 5-oneirodegree A5oneid 14\21 800.0 23\34 811.8 9\13 830.8
Minor 6-oneirodegree m6oneid 15\21 857.1 24\34 847.1 9\13 830.8
Major 6-oneirodegree M6oneid 16\21 914.3 26\34 917.6 10\13 923.1
Minor 7-oneirodegree m7oneid 18\21 1028.6 29\34 1023.5 11\13 1015.4
Major 7-oneirodegree M7oneid 19\21 1085.7 31\34 1094.1 12\13 1107.7
Perfect 8-oneirodegree P8oneid 21\21 1200.0 34\34 1200.0 13\13 1200.0

Parasoft and ultrasoft tunings

The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to porcupine temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a porcupine generator. The chord 10:11:13 is very well approximated in 29edo.


Soft Tunings of 5L 3s
Scale degree Abbrev. 6:5
45edo 		Supersoft (4:3)
29edo 		Soft (3:2)
21edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\45 0.0 0\29 0.0 0\21 0.0
Minor 1-oneirodegree m1oneid 5\45 133.3 3\29 124.1 2\21 114.3
Major 1-oneirodegree M1oneid 6\45 160.0 4\29 165.5 3\21 171.4
Minor 2-oneirodegree m2oneid 11\45 293.3 7\29 289.7 5\21 285.7
Major 2-oneirodegree M2oneid 12\45 320.0 8\29 331.0 6\21 342.9
Diminished 3-oneirodegree d3oneid 16\45 426.7 10\29 413.8 7\21 400.0
Perfect 3-oneirodegree P3oneid 17\45 453.3 11\29 455.2 8\21 457.1
Minor 4-oneirodegree m4oneid 22\45 586.7 14\29 579.3 10\21 571.4
Major 4-oneirodegree M4oneid 23\45 613.3 15\29 620.7 11\21 628.6
Perfect 5-oneirodegree P5oneid 28\45 746.7 18\29 744.8 13\21 742.9
Augmented 5-oneirodegree A5oneid 29\45 773.3 19\29 786.2 14\21 800.0
Minor 6-oneirodegree m6oneid 33\45 880.0 21\29 869.0 15\21 857.1
Major 6-oneirodegree M6oneid 34\45 906.7 22\29 910.3 16\21 914.3
Minor 7-oneirodegree m7oneid 39\45 1040.0 25\29 1034.5 18\21 1028.6
Major 7-oneirodegree M7oneid 40\45 1066.7 26\29 1075.9 19\21 1085.7
Perfect 8-oneirodegree P8oneid 45\45 1200.0 29\29 1200.0 21\21 1200.0

Parahard tunings

23edo oneiro combines the sound of neogothic tunings like 46edo and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as 46edo's neogothic major second, and is both a warped 22edo superpyth diatonic and a warped 24edo semaphore semiquartal (and both nearby scales are superhard MOSes).


Superhard Tuning of 5L 3s
Scale degree Abbrev. Superhard (4:1)
23edo
Steps ¢
Perfect 0-oneirodegree P0oneid 0\23 0.0
Minor 1-oneirodegree m1oneid 1\23 52.2
Major 1-oneirodegree M1oneid 4\23 208.7
Minor 2-oneirodegree m2oneid 5\23 260.9
Major 2-oneirodegree M2oneid 8\23 417.4
Diminished 3-oneirodegree d3oneid 6\23 313.0
Perfect 3-oneirodegree P3oneid 9\23 469.6
Minor 4-oneirodegree m4oneid 10\23 521.7
Major 4-oneirodegree M4oneid 13\23 678.3
Perfect 5-oneirodegree P5oneid 14\23 730.4
Augmented 5-oneirodegree A5oneid 17\23 887.0
Minor 6-oneirodegree m6oneid 15\23 782.6
Major 6-oneirodegree M6oneid 18\23 939.1
Minor 7-oneirodegree m7oneid 19\23 991.3
Major 7-oneirodegree M7oneid 22\23 1147.8
Perfect 8-oneirodegree P8oneid 23\23 1200.0

Ultrahard tunings

Main article: 5L 3s/Temperaments § Buzzard

Buzzard is a rank-2 temperament in the pseudocollapsed range. It represents the only harmonic entropy minimum of the oneirotonic spectrum.

In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. 23edo, 28edo and 33edo can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. 38edo & 43edo are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but 48edo is where it really comes into its own in terms of harmonies, providing not only an excellent 3/2, but also 7/4 and archipelago harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.

Beyond that, it's a question of which intervals you want to favor. 53edo has an essentially perfect 3/2, 58edo gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while 63edo does the same for the basic 4:6:7 triad. You could in theory go up to 83edo if you want to favor the 7/4 above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.


Ultrahard Tunings of 5L 3s
Scale degree Abbrev. 7:1
38edo 		10:1
53edo 		12:1
63edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\38 0.0 0\53 0.0 0\63 0.0
Minor 1-oneirodegree m1oneid 1\38 31.6 1\53 22.6 1\63 19.0
Major 1-oneirodegree M1oneid 7\38 221.1 10\53 226.4 12\63 228.6
Minor 2-oneirodegree m2oneid 8\38 252.6 11\53 249.1 13\63 247.6
Major 2-oneirodegree M2oneid 14\38 442.1 20\53 452.8 24\63 457.1
Diminished 3-oneirodegree d3oneid 9\38 284.2 12\53 271.7 14\63 266.7
Perfect 3-oneirodegree P3oneid 15\38 473.7 21\53 475.5 25\63 476.2
Minor 4-oneirodegree m4oneid 16\38 505.3 22\53 498.1 26\63 495.2
Major 4-oneirodegree M4oneid 22\38 694.7 31\53 701.9 37\63 704.8
Perfect 5-oneirodegree P5oneid 23\38 726.3 32\53 724.5 38\63 723.8
Augmented 5-oneirodegree A5oneid 29\38 915.8 41\53 928.3 49\63 933.3
Minor 6-oneirodegree m6oneid 24\38 757.9 33\53 747.2 39\63 742.9
Major 6-oneirodegree M6oneid 30\38 947.4 42\53 950.9 50\63 952.4
Minor 7-oneirodegree m7oneid 31\38 978.9 43\53 973.6 51\63 971.4
Major 7-oneirodegree M7oneid 37\38 1168.4 52\53 1177.4 62\63 1181.0
Perfect 8-oneirodegree P8oneid 38\38 1200.0 53\53 1200.0 63\63 1200.0

Approaches

Samples

The Angels' Library by Inthar in the Sarnathian (23233233) mode of 21edo oneirotonic (score)

WT13C Prelude II (J Oneirominor) (score) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.

(13edo, first 30 seconds is in J Celephaïsian)

(13edo, L Ilarnekian)

(by Igliashon Jones, 13edo, J Celephaïsian)

13edo Oneirotonic Modal Studies

Scale tree

Scale tree and tuning spectrum of 5L 3s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
3\8 450.000 750.000 1:1 1.000 Equalized 5L 3s
17\45 453.333 746.667 6:5 1.200
14\37 454.054 745.946 5:4 1.250
25\66 454.545 745.455 9:7 1.286
11\29 455.172 744.828 4:3 1.333 Supersoft 5L 3s
30\79 455.696 744.304 11:8 1.375
19\50 456.000 744.000 7:5 1.400
27\71 456.338 743.662 10:7 1.429
8\21 457.143 742.857 3:2 1.500 Soft 5L 3s
29\76 457.895 742.105 11:7 1.571
21\55 458.182 741.818 8:5 1.600
34\89 458.427 741.573 13:8 1.625 Golden oneirotonic (458.3592 ¢)
13\34 458.824 741.176 5:3 1.667 Semisoft 5L 3s
31\81 459.259 740.741 12:7 1.714
18\47 459.574 740.426 7:4 1.750
23\60 460.000 740.000 9:5 1.800
5\13 461.538 738.462 2:1 2.000 Basic 5L 3s
Scales with tunings softer than this are proper
22\57 463.158 736.842 9:4 2.250
17\44 463.636 736.364 7:3 2.333
29\75 464.000 736.000 12:5 2.400
12\31 464.516 735.484 5:2 2.500 Semihard 5L 3s
31\80 465.000 735.000 13:5 2.600 Golden A-Team (465.0841 ¢)
19\49 465.306 734.694 8:3 2.667
26\67 465.672 734.328 11:4 2.750
7\18 466.667 733.333 3:1 3.000 Hard 5L 3s
23\59 467.797 732.203 10:3 3.333
16\41 468.293 731.707 7:2 3.500
25\64 468.750 731.250 11:3 3.667
9\23 469.565 730.435 4:1 4.000 Superhard 5L 3s
20\51 470.588 729.412 9:2 4.500
11\28 471.429 728.571 5:1 5.000
13\33 472.727 727.273 6:1 6.000
2\5 480.000 720.000 1:0 → ∞ Collapsed 5L 3s
Retrieved from "https://en.xen.wiki/index.php?title=5L_3s&oldid=195291"