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
ArrowHead294 (talk | contribs)
14,512 edits
m substitute deprecated template
 
(475 intermediate revisions by 14 users not shown)
Line 1: Line 1:
:''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (tritave-equivalent)]].''
{{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]]).


'''5L 3s''' refers to the structure of [[MOS]] 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).  
== Name ==
{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.


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]]): for example, the Ionian diatonic mode LLsLLLs can be distorted to the Dylathian oneirotonic mode LLsLLsLs.
'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]].


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.
== Scale properties ==


In terms of [[regular temperament]]s, there are at least two melodically viable ways to interpret oneirotonic:
=== Intervals ===
# When the generator is between 457.14¢ (8\21) and 461.54¢ (5\13): [[5L_3s#Petrtri_.2813.2621.2C_2.5.9.11.13.17.29|Petrtri]] (13&21, a 4:5:9:11:13:17 or 2.5.9.11.13.17 temperament)
{{MOS intervals}}
# 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)
In a sense, these two temperaments represent the middle of the oneirotonic spectrum (with the L/s ratio ranging from 3/2 to 3/1); [[13edo]] represents both temperaments, with a L/s ratio of 2/1. This is analogous to how in the diatonic spectrum, the [[19edo]]-to-[[17edo]]-range has the least extreme ratio of large to small step sizes, with [[12edo]] representing both [[meantone]] (19edo to 12edo) and [[pythagorean]]/[[neogothic]] (12edo to 17edo).


More extreme oneirotonic temperaments include:
=== Generator chain ===
* [[Chromatic pairs#Tridec|Tridec]] (a 5:7:11:13 or 2.7/5.11/5.13/5 subgroup temperament), when the generator is between 453.33c (17\45) and 457.14c (8\21). These have near-equal L/s ratios of 6/5 to 3/2.
{{MOS genchain}}
* [[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 ==
=== Modes ===
{| class="wikitable" style="text-align:center;"
{{MOS mode degrees}}
|-
! 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
|-
| | 7\18
| |
| |
| |
| |
|  | 3 1 3
|  | 466.667
|  | 933.333
|  | 200.000
|  | 666.667
|  | L/s = 3<br/>[[A-Team]] starts around here...
|-
| |
| | 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/>[[5L_3s#Petrtri_.2813.2621.2C_2.5.9.11.13.17.29|Petrtri]] starts here...
|-
| |
| | 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 oneirotonic; 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
|  |
|-
| | 8\21
| |
| |
| |
| |
|  | 3 2 3
|  | 457.143
|  | 914.286
|  | 171.429
|  | 628.571
|  | ...and ends here<br/> Optimum rank range (L/s=3/2) oneirotonic
|-
| | 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
|  |
|}


== Tuning ranges and data ==
==== Proposed mode names ====
=== A-Team (13&18) ===
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
A-Team tunings (with generator between 5\13 and 7\18) have L/s ratios between 2/1 and 3/1. A-Team tunings share the following features with classical meantone tunings:
{{MOS modes
* The large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]).
| Mode Names=
* The major mosthird (made of two large steps) is a [[meantone]]- to [[flattone]]-sized major third, thus is a stand-in for the classical diatonic major third.
Dylathian $
Ilarnekian $
Celephaïsian $
Ultharian $
Mnarian $
Kadathian $
Hlanithian $
Sarnathian $
| Collapsed=1
}}


EDOs that support A-Team include [[13edo]], [[18edo]], and [[31edo]].
== Tunings==
* 18edo can be used for a large L/s ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic), or for its nearly pure 9/8 and 7/6. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
=== Simple tunings ===
* 31edo is very close to the POTE tuning, and can be used to make the major mos3rd a near-just 5/4.
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"
|-
!
! [[13edo]]
! [[18edo]]
! [[31edo]]
! Optimal ([[POTE]]) tuning
! JI intervals represented (2.9.5.21 subgroup)
|-
| generator (g)
| 5\13, 461.54
| 7\18, 466.67
| 12\31, 464.52
| 464.39
| 21/16
|-
| L (3g - octave)
| 2\13, 184.62
| 3\18, 200.00
| 5\31, 193.55
| 193.16
| 9/8, 10/9
|-
| s (-5g + 2 octaves)
| 1\13, 92.31
| 1\18, 66.66
| 2\31, 77.42
| 78.07
| 21/20
|}


Trivia: 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.
=== 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.


=== Petrtri (13&21) ===
With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to [[7/6]].
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 tends to be more of a neutral third, 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.  
EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]], and are associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.
* 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.
* [[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.


The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings.
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[13edo]]
! [[21edo]]
! [[34edo]]
! Optimal ([[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
|}


=== Tridec (29&37) ===
=== Hyposoft tunings ===
In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps make a [[4/3]] perfect fourth. [This identification may come in handy since some altered oneirotonic modes have three large steps.] Based on the JI interpretations of the [[29edo]] and [[37edo]] tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a [[Non-over-2 temperament|non-over-2 temperament]] that approximates the chord 5:7:11:13:15. The optimal generator is 455.2178c, which is very close to 29edo's 11\29 (455.17c), but we could accept any generator between 17\45 (453.33c) and 8\21 (457.14c), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.
[[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}}).


The sizes of the generator, large step and small step of oneirotonic are as follows in various tridec tunings.
* [[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}}).
{| class="wikitable right-2 right-3 right-4 right-5"
* [[34edo]]'s 9:10:11:13 is even better.
|-
!
! [[21edo]]
! [[29edo]]
! [[37edo]]
! Optimal ([[POTE]]) tuning
! JI intervals represented (2.3.7/5.11/5.13/5 subgroup)
|-
| generator (g)
| 8\21, 457.14
| 11\29, 455.17
| 14\37, 454.05
| 455.22
| 13/10
|-
| L (3g - octave)
| 3\21, 171.43
| 4\29, 165.52
| 5\37, 162.16
| 165.65
| 11/10
|-
| s (-5g + 2 octaves)
| 2\21, 114.29
| 3\29, 124.14
| 4\37, 129.73
| 123.91
| 14/13, 15/14
|}


=== Buzzard (48&53) ===
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.)
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 A-Team 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 it's 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.  


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.
{{MOS tunings
| 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
}}


The sizes of the generator, large step and small step of oneirotonic are as follows in various buzzard tunings.
=== Parasoft and ultrasoft tunings ===
{| class="wikitable right-2 right-3 right-4 right-5"
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.
|-
!
! [[38edo]]
! [[53edo]]
! [[63edo]]
! Optimal ([[POTE]]) tuning
! JI intervals represented (2.3.5.7.13 subgroup)
|-
| generator (g)
| 15\38, 473.68
| 21\53, 475.47
| 25\63, 476.19
| 475.69
| 3/2 21/16
|-
| L (3g - octave)
| 7/38, 221.04
| 10/53, 226.41
| 12/63, 228.57
| 227.07
| 8/7
|-
| s (-5g + 2 octaves)
| 1/38 31.57
| 1/53 22.64
| 1/63 19.05
| 21.55
| 55/54 81/80 91/90
|}


== Notation==
{{MOS tunings
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 &minus; s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)
| Step Ratios = 6/5; 3/2; 4/3
| JI Ratios =  
1/1;
14/13;
11/10;
9/8;
15/13;
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
}}


Thus the [[13edo]] gamut is as follows:
=== 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).


'''J/Q&''' J&/K@ '''K/L@''' '''L/K&''' L&/M@ '''M''' M&/N@ '''N/O@''' '''O/N&''' O&/P@ '''P''' P&/Q@ '''Q/J@''' '''J'''
{{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
}}


The [[18edo]] gamut is notated as follows:
=== Ultrahard tunings ===
{{Main|5L&nbsp;3s/Temperaments#Buzzard}}


'''J''' Q&/K@ J&/L@ '''K''' '''L''' K&/M@ L& '''M''' N@ M&/O@ '''N''' '''O''' P@ O& '''P''' Q@ P&/J@ '''Q''' '''J'''
[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.


The [[21edo]] gamut:
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.


'''J''' J& K@ '''K''' K&/L@ '''L''' L& M@ '''M''' M& N@ '''N''' N&/O@ '''O''' O& P@ '''P''' P& Q@ '''Q''' Q&/J@ '''J'''
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.
 
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.
 
== 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 (aka minor fourth)
| -1
| O
| perfect mossixth (aka major fifth)
|-
| 2
| P
| major mosseventh
| -2
| L
| minor mosthird
|-
| 3
| K
| major mossecond
| -3
| Q@
| minor moseighth
|-
| 4
| N
| major mosfifth (aka minor fifth)
| -4
| N@
| minor mosfifth (aka major fourth)
|-
| 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 ==
Oneirotonic modes are named after cities in the Dreamlands. (The names are by Cryptic Ruse.)
 
# Dylathian: LLSLLSLS
# Illarnekian: LLSLSLLS
# 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:
* J Celephaïsian
* K Kadathian
* L Dylathian
* M Ultharian
* N Hlanithian
* O Illarnekian
* P Mnarian
* Q Sarnathian
 
The modes in 13edo edo steps and C-H notation (table by Cryptic Ruse):
 
[[File:Oneirotonic.png|alt=Oneirotonic.png|Oneirotonic.png]]
 
== Pseudo-diatonic theory ==
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 is encouraged to experiment and see what ideas work for other oneirotonic tunings.
=== 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, Illarnekian, 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. 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.
 
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 "dominant". The augmented mossixth would be used when a major key needs to be used on the fourth degree "subdominant".
 
==== Functional harmony ====
Some suggested basic ana functional harmony progressions, outlined very roughly Note that VI, VII and VIII are sharp 5th, 6th-like and 7th-like degrees respectively. A Roman numeral without maj or min means either major or minor. The "Natural" Roman numerals follow the Illarnekian mode.
 
* I-IVmin-VImaj-I
* Imaj-VIImin-IVmin-Imaj
* Imin-@IIImaj-VImaj-Imaj
* Imin-@IIImaj-Vdim-VImaj-Imin
* Imin-@VIIImin-IIImaj-VImaj-Imin
* Imin-IVmin-@VIIImin-@IIImaj-VImaj-Imin
* Imin-IVmin-IIdim-VImaj-Imin
* Imin-IVmin-IIdim-@IIImaj-Imin
* I-VIImin-IImin-VImaj-I
* Imaj-VIImin-IVmin-VImaj-Imaj
* Modulations by major mos2nd:
** I-IV-VII-II
** I-IVmaj-II
** 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 ===
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:
* 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.
 
=== Alterations ===
The most important oneirotonic [[MODMOS]] (MOS with one or more alterations) is LSLLLSLS together with 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:
* LLLSLSLS: Dylathian &4: an ana-Lydian
* LLSLSLSL: Illarnekian @8: an ana-Mixolydian
* LSLLLSLS: Celephaïsian &6: an ana-Dorian
* SLLLSLSL: Ultharian @2: an ana-Phrygian
* SLSLSLLL: Sarnathian @6: a kata-Locrian
* SLSLLLSL: Sarnathian &7: 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:
* the distorted harmonic minor LSLSLLSAS (A = aug 2nd = L + chroma)
* the distorted Freygish SASLSLLS
 
=== Chords ===
Chords are given in oneirotonic MOS interval notation. For example, M5 means major mosfifth (squashed fifth).
 
[Todo: clean up naming. 5ths should be optional except in squashed tertian triads and oneiro tetrads]
* 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; Sephiroth Triad (approximates 8:10:13 in 13edo and 31edo)
* R-M3-A6-(M2)-(P4): Sephiroth Triad Add9 Sub11
* R-M3-A6-(P4): Sephiroth Triad Sub11
* R-m3-P6: Squashed 1st Inversion Major Triad
* R-M3-M7: 1st Inversion Squashed Minor Triad (note the order of terms!)
* R-m3-M7: Minor Add6 Triad
* 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 Triad
* R-m3-M8: Oneiro Minor Major Seventh Triad
* R-M3-M5-M8: Oneiro Major Seventh Tetrad
* R-m3-M5-M8: Oneiro Minor Seventh Tetrad
* R-M3-M7-M8: Oneiro Major Seventh Add6
* R-m3-M7-M8: Oneiro Minor Major Seventh Add6
* R-M3-P6-M8: Oneiro Major Seventh Add Major Fifth
* R-m3-P6-M8: Oneiro Minor Major Seventh Add Major Fifth
* R-M3-(M2): Oneiro Major Add9
* R-m3-(M2): Oneiro Minor Add9
* R-M3-M5-(M2): Squashed Major Triad Add9
* R-m3-M5-(M2): Squashed Minor Triad Add9
* R-M3-(M2)-(P4): Oneiro Major Add9 Sub11
* R-m3-(M2)-(P4): Oneiro Minor Add9 Sub11
* R-m3-P6-M7-(M2)-(P4)-(A6)-(M8)
* R-M2-P4: Squashed Sus24 No5
* R-M2-M5: Squashed Sus2 Triad
* R-P4-M5: Squashed Sus4 Triad
* R-M2-P4-M5: Squashed Sus24
* 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-P4-m8: Expanding Quartal Triad
* R-M2-P4-m8: Expanding Quartal Triad add2
* R-m3-P4-m8: Expanding Quartal Triad Addm3
* R-m5-m8: Contracting Quartal Triad
* R-m5-m7-m8: Contracting Quartal Triad Addm7
* R-M3-M5-m8: Squashed Major Triad addm7
 
== Primodal theory ==
A-Team oneirotonic may be a particularly good place to bring to bear [[primodality]]'s high harmonic series chords, as A-Team temperament doesn't yield many low-complexity chords.
 
18edo may be a better basis for a style of oneirotonic primodality using comma sharp and comma flat fifths than 13edo (in particular diesis sharp and diesis flat fifths; diesis is a category with a central region of 32 to 40c). In 18edo both the major fifth (+31.4c) and the minor fifth (-35.3c) are about a diesis off from a just perfect fifth. In 13edo only the major fifth is a diesis sharp, and it is +36.5c off from just; so there's less wiggle room for a [[neji]] if you want every major fifth to be at most a diesis sharp).
 
31nejis and 34nejis (though 34edo is not an A-Team tuning) also provide opportunities to use dieses directly, since 1\31 (38.71c) and 1\34 (35.29c) are both dieses.
=== Primodal chords ===
Some relatively low-complexity oneirotonic-inspired primodal chords. They are grouped by [[prime family]].
==== /13 ====
*13:15:19 Tridecimal Squashed Minor Triad
*13:16:19 Tridecimal Squashed Major Triad
*13:17:19 Tridecimal Naiadic Maj2
*13:17:20 Tridecimal Squashed 2nd Inversion Minor Triad
*13:17:21 Tridecimal Squashed 2nd Inversion Major Triad
*13:16:19:22 Tridecimal Oneiro Major Tetrad
*26:29:38 Tridecimal Squashed Sus2 Triad
*26:29:34:38 Tridecimal Squashed Sus24
 
==== /17 ====
*17:20:25 Septendecimal Squashed Minor Triad
*17:21:25 Septen Squashed Major Triad
*17:20:26 Septen Squashed 1st Inversion Major Triad
*17:20:25:29 Septen Minor Oneiro Tetrad
*17:21:25:29 Septen Major Oneiro Tetrad
*17:20:26:29 Septen Squashed 1st Inversion Major Triad addM6
*34:43:50 Septen Squashed Supermajor Triad
*34:40:47:55 Septen Orwell Tetrad
*34:40:52:58:76:89:102:129 (Celephaïsian + P5; R-min3-sup5-M6-M9-sub11-P12(fc)-M14)
*34:40:52:58:76:89:102:110:129 (Celephaïsian + P5; R-min3-sup5-M6-M9-sub11-P12(fc)-supmin13-M14)
*34:40:50:58:89:102:129 (R-min3-sub5-M6-M9-sub11-P12(rc)-M14)
*34:40:50:58:89:102:110:129 (R-min3-sub5-M6-M9-sub11-P12(rc)-supmin13-M14)
*34:40:50:58:76:89:110:129 (R-m3-sub5-M6-M9-sub11-supm13-M7)
*34:40:50:58:76:89:102:110:129:208 (R-m3-sub5-M6-M9-sub11-P12(rc)-supm13-M14-sup19(rc^2))
 
==== /19 ====
The notes 38:41:43:46:48:50:52:54:56:58:60:63:65:68:70:73:76 provide the best low complexity fit to oneirotonic (in particular, 18edo) in the [[prime family]] /19.
*19:24:28 Novemdecimal Squashed Major Triad
*19:23:28 Novem Squashed Neutral Triad
*19:22:28 Novem Squashed Minor Triad
*19:24:29 Novem Semiaugmented Triad
*19:24:30 Novem Augmented Triad
*19:24:43 Novem Oneiro Major add9
*19:24:43:50 Novem Oneiro Major add9sub11
*19:24:28:43:50 Novem Squashed Major Triad add9 sub11
*19:24:29:43:50 Novem Semiaug Triad add9 sub11
*19:25:34 Novem Expanding Quartal
*19:26:34 Novem Contracting Quartal
*38:48:56:65 Novem Oneiro Major Tetrad
*38:48:73 Novem Oneiro Major Seventh Triad
*38:48:63 Novem Squashed 1st Inversion Minor Triad
*38:50:65 Novem Oneiro Quartal Triad
*38:50:65:73 Novem Oneiro Quartal Seventh Tetrad
*38:50:65:86 Novem Oneiro Core Tetrad
*38:50:65:86:112 Novem Oneiro Core Pentad
*38:50:65:86:112:146 Novem Oneiro Core Hexad
*38:50:63 Novem Squashed First Inversion Neutral Triad
 
==== /23 ====
23(2:4) has many oneiro pitches, some close to 13edo, and some close to 18edo:
46:48:50:51:52:54:56:57:58:60:63:65:67:68:70:73:74:76:79:82:83:85:87:88:92
 
*23:27:30 Vicesimotertial Squashed Min4
*23:27:30:35:44 Vice Squashed Min4 addM5,M7
*23:27:37 Vice Orwell Tetrad no4
*23:29:34 Vice Squashed Major Triad
*46:54:68 Vice Squashed Minor Triad
*46:54:60:67 Vice Squashed Min4
*46:54:63 Vice Squashed Dim
*46:54:63:68 Vice Oneiro Half-diminished Tetrad
*46:54:63:74 Vice Orwell Tetrad
*46:54:67 Vice Squashed Minor Triad
*46:54:67:78 Vice Oneiro Minor Tetrad
*46:54:60:67:78 Vice Oneiro Minor Tetrad Add Min4
*46:60:67 Vice Squashed Sus4
*46:54:60:67 Vice Squashed Sus4 Min5
 
==== /29 ====
*29:34:38 Vicesimononal Squashed Sus4
*29:34:42 Vicenon Squashed Minor Triad
*29:36:42 Vicenon Squashed Major Triad
*29:34:40:47 Vicenon Orwell Tetrad
*29:38:65:84:99 Vicenon Oneiro Core Pentad
*29:38:65:84:99:110 Vicenon Oneiro Core Hexad
*58:65:72:80:84:94:99:110:116 Vicenon Dylathian &4
*58:65:72:76:84:94:99:110:116 Vicenon Dylathian
*58:65:72:76:84:89:99:110:116 Vicenon Illarnekian
*58:65:72:76:84:89:99:104:116 Vicenon Illarnekian @8
*58:65:68:76:84:94:99:110:116 Vicenon Celephaïsian &6
*58:65:68:76:84:89:99:110:116 Vicenon Celephaïsian
*58:65:68:76:84:89:99:104:116 Vicenon Ultharian
*58:65:68:76:80:89:99:104:116 Vicenon Mnarian
*58:65:68:76:80:89:99:110:116 Vicenon Mnarian &8
*58:65:68:76:80:89:94:104:116 Vicenon Hlanithian &2
*58:61:68:76:80:89:99:104:116 Vicenon Kadathian
*58:61:68:76:84:89:99:104:116 Vicenon Ultharian @2
*58:61:68:76:80:89:94:104:116 Vicenon Hlanithian
*58:61:68:72:80:89:99:104:116 Vicenon Sarnathian &7
*58:61:68:72:80:89:94:104:116 Vicenon Sarnathian
*58:61:68:72:80:84:94:104:116 Vicenon Sarnathian @6
 
==== Over small prime multiples ====
 
=== Some oneirotonic nejis ===
The reader is encouraged to tweak these nejis and add more nejis that they like.
*'''58''':61:65:'''68''':72:'''76''':80:84:89:94:100:'''104''':110:116 - A low-complexity 13neji (Could use 64 instead of 65); has /13, /17, /19, and /29 prime modes
*'''46''':48:50:'''52''':54:56:'''58''':60:63:65:'''68''':70:73:'''76''':79:82:85:89:92 - A low-complexity 18neji; has /13, /17, /19, /23 and /29 prime modes (bolded)
*'''92''':96:100:'''104''':108:112:'''116''':120:''125'':130:'''136''':''141'':146:'''152''':158:164:170:''177'':184 - a more equal 18neji that still keeps the prime modes (changes italicized)
 
== 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). However, the rest of this region still has a couple notable subgroup temperaments.
 
Todo: Add temperament data
=== Tridec (21&29, 2.7/5.11/5.13/5) ===
Period: 1\1
 
Optimal ([[POTE]] generator: 454.5555
 
EDO generators: [[21edo|8\21]], [[29edo|11\29]], [[37edo|14\37]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
[[Comma]] list: 847/845, 1001/1000
 
[[Mapping]] (for 2, 7/5, 11/5, 13/5): [{{val|1 2 0 1}}, {{val|0 -4 3 1}}]
 
Mapping generators: ~2, ~13/10
 
{{Vals|legend=1| 21, 29, 37 }}
 
</div></div>
=== Petrtri (13&21, 2.5.9.11.13.17) ===
Period: 1\1
 
Optimal ([[POTE]]) generator: 459.1502


EDO generators: [[13edo|5\13]], [[21edo|8\21]], [[34edo|13\34]]
{{MOS tunings
| JI Ratios =
1/1;
8/7;
13/10;
21/16;
3/2;
12/7, 22/13;
26/15;
49/25, 160/81;
2/1
| Step Ratios = 7/1; 10/1; 12/1
| Tolerance = 30
}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
== Approaches ==
<div style="line-height:1.6;">Technical data</div>
* [[5L&nbsp;3s/Temperaments]]
<div class="mw-collapsible-content">
 
[[Comma]] list: 100/99, 144/143, 170/169, 221/220
 
[[Mapping]] (for 2, 5, 9, 11, 13, 17): [{{val|1 5 7 5 6 6}}, {{val|0 -7 -10 -4 -6 -5}}]
 
Mapping generators: ~2, ~13/10
 
{{Vals|legend=1| 13, 21, 34 }}
 
</div></div>
==== Intervals ====
Sortable table of intervals in the Dylathian mode and their Petrtri interpretations:
{| class="wikitable right-2 right-3 right-4 right-5 sortable"
|-
! Degree
! Size in 13edo
! Size in 21edo
! Size in 34edo
! Size in POTE tuning
! Note name on L
! class="unsortable"| Approximate ratios
! #Gens up
|-
| 1
| 0\13, 0.00
| 0\21, 0.00
| 0\34, 0.00
| 0.00
| L
| 1/1
| 0
|-
| 2
| 2\13, 184.62
| 3\21, 171.43
| 5\34, 176.47
| 177.45
| M
| 10/9, 11/10
| +3
|-
| 3
| 4\13, 369.23
| 6\21, 342.86
| 10\34, 352.94
| 354.90
| N
| 11/9, 16/13
| +6
|-
| 4
| 5\13, 461.54
| 8\21, 457.14
| 13\34, 458.82
| 459.15
| O
| 13/10, 17/13, 22/17
| +1
|-
| 5
| 7\13, 646.15
| 11\21, 628.57
| 18\34, 635.294
| 636.60
| P
| 13/9, 16/11
| +4
|-
| 6
| 9\13, 830.77
| 14\21, 800.00
| 23\34, 811.77
| 814.05
| Q
| 8/5
| +7
|-
| 7
| 10\13, 923.08
| 16\21, 914.29
| 26\34, 917.65
| 918.30
| J
| 17/10
| +2
|-
| 8
| 12\13, 1107.69
| 19\21, 1085.71
| 31\34, 1094.12
| 1095.75
| K
| 17/9, 32/17
| +5
|}
 
=== A-Team (13&18, 2.5.9.21) ===
Period: 1\1
 
Optimal ([[POTE]]) generator: 464.3865
 
EDO generators: [[13edo|5\13]], [[18edo|7\18]], [[31edo|12\31]], [[44edo|17\44]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
[[Comma]] list: 81/80, 1029/1024
 
[[Mapping]] (for 2, 5, 9, 21): [{{val|1 0 2 4}}, {{val|0 6 3 1}}]
 
Mapping generators: ~2, ~21/16
 
{{Vals|legend=1| 13, 18, 31, 44 }}
 
</div></div>
==== Intervals ====
Sortable table of intervals in the Dylathian mode and their A-Team interpretations:
 
{| class="wikitable right-2 right-3 right-4 sortable"
|-
! Degree
! Size in 13edo
! Size in 18edo
! Size in 31edo
! Note name on L
! class="unsortable"| Approximate ratios<ref>The harmonics over 1/1 are in bold. The ratio interpretations that are not valid for 18edo are italicized.</ref>
! #Gens up
|-
| 1
| 0\13, 0.00
| 0\18, 0.00
| 0\31, 0.00
| L
| '''1/1'''
| 0
|-
| 2
| 2\13, 184.62
| 3\18, 200.00
| 5\31, 193.55
| M
| '''9/8''', 10/9
| +3
|-
| 3
| 4\13, 369.23
| 6\18, 400.00
| 10\31, 387.10
| N
| '''5/4'''
| +6
|-
| 4
| 5\13, 461.54
| 7\18, 466.67
| 12\31, 464.52
| O
| '''21/16''', ''13/10''
| +1
|-
| 5
| 7\13, 646.15
| 10\18, 666.66
| 17\31, 658.06
| P
| ''13/9'', ''16/11''
| +4
|-
| 6
| 9\13, 830.77
| 13\18, 866.66
| 22\31, 851.61
| Q
| '''''13/8''''', ''18/11''
| +7
|-
| 7
| 10\13, 923.08
| 14\18, 933.33
| 24\31, 929.03
| J
| 12/7
| +2
|-
| 8
| 12\13, 1107.69
| 17\18, 1133.33
| 29\31, 1122.58
| K
|
| +5
|}
<references/>
 
=== Buzzard (48&53, 2.3.5.7) ===
Period: 1\1
 
Optimal ([[POTE]]) generator: ~21/16 = 475.636
 
EDO generators: [[38edo|15\38]], [[43edo|17\43]], [[48edo|19\48]], [[53edo|21\53]], [[58edo|23\58]], [[63edo|25\63]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Commas: 1728/1715, 5120/5103
 
Map: [&lt;1 0 -6 4|, &lt;0 4 21 -3|]
 
Mapping generators: ~2, ~21/16
 
Wedgie: &lt;&lt;4 21 -3 24 -16 -66||
 
{{Vals| 48, 53, 111, 164d, 275d}}
 
Badness: 0.0480
</div></div>
 
==== Intervals ====
Sortable table of intervals in the Dylathian mode and their Buzzard interpretations:
 
{| class="wikitable right-2 right-3 right-4 right-5 sortable"
|-
! Degree
! Size in 38edo
! Size in 53edo
! Size in 63edo
! Size in POTE tuning
! Note name on L
! class="unsortable"| Approximate ratios
! #Gens up
|-
| 1
| 0\38, 0.00
| 0\53, 0.00
| 0\63, 0.00
| 0.00
| L
| 1/1
| 0
|-
| 2
| 7\38, 221.05
| 10\53, 226.42
| 12\63, 228.57
| 227.07
| M
| 8/7
| +3
|-
| 3
| 14\38, 442.10
| 20\53, 452.83
| 24\63, 457.14
| 453.81
| N
| 13/10, 9/7
| +6
|-
| 4
| 15\38, 473.68
| 21\53, 475.47
| 25\63, 476.19
| 475.63
| O
| 21/16
| +1
|-
| 5
| 22\38, 694.73
| 31\53, 701.89
| 37\63, 704.76
| 702.54
| P
| 3/2
| +4
|-
| 6
| 29\38, 915.78
| 41\53, 928.30
| 49\63, 933.33
| 929.45
| Q
| 12/7, 22/13
| +7
|-
| 7
| 30\38, 947.36
| 42\53, 950.94
| 50\63, 952.38
| 951.27
| J
| 26/15
| +2
|-
| 8
| 37\38, 1168.42
| 52\53, 1177.36
| 62\63, 1180.95
| 1178.18
| K
| 108/55, 160/81
| +5
|}


== Samples ==
== 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]])


[[File:Oneirotonic 3 part sample.mp3]]
[[File:13edo Prelude in J Oneirominor.mp3]]


‎(A rather classical-sounding 3-part harmonization of the ascending J Illarnekian scale; tuning is 13edo)
[[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.


[[File:13edo_1MC.mp3]]
[[File:13edo_1MC.mp3]]  


(13edo, first 30 seconds is in J Celephaïsian)
(13edo, first 30 seconds is in J Celephaïsian)
Line 1,324: Line 188:
[[File:A Moment of Respite.mp3]]
[[File:A Moment of Respite.mp3]]


(13edo, L Illarnekian)
(13edo, L Ilarnekian)


[[File:Lunar Approach.mp3]]
[[File:Lunar Approach.mp3]]
Line 1,330: Line 194:
(by [[Igliashon Jones]], 13edo, J Celephaïsian)
(by [[Igliashon Jones]], 13edo, J Celephaïsian)


[[Category:Scales]]
=== 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
 
== Scale tree ==
{{MOS tuning spectrum
| 13/8 = Golden oneirotonic (458.3592{{c}})
| 13/5 = Golden A-Team (465.0841{{c}})
}}
 
[[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
[[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
[[Category:Mos]]
[[Category:Pages with internal sound examples]]
[[Category:MOS scales]]
 
[[Category:Abstract MOS patterns]][[Category:Oneirotonic]]

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"