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:''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (tritave-equivalent)]].''
{{Interwiki
| en = 5L 3s
| de =
| es =
| ja = 5L 3s
| 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). 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.
== Name ==
{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.
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]].
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 ==
5L 2s has a pentatonic MOS subset [[3L 2s]] (SLSLL), and in this context we call this the ''oneiro-pentatonic''. When viewed as a chord (with undetermined voicing) we call it the Oneiro Core Pentad. (Note: [[3L 5s]] scales also have 3L 2s subsets.)
=== Intervals ===
{{MOS intervals}}
In terms of [[Tour of Regular Temperaments|regular temperament]]s, there are at least two melodically viable ways to interpret oneirotonic:
=== Generator chain ===
# 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 2.5.9.11.13.17 temperament that approximates the harmonic series chord 4:5:9:11:13:17)
{{MOS genchain}}
# When the generator is between 461.54¢ (5\13) and 466.67¢ (7\18): [[A-Team]] (13&18, a 2.9.5.21 temperament that approximates 4:5:9:21)
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).
(In the past, 5L 3s has been viewed as a MOS of the low-accuracy 5-limit temperament [[father]]. This viewpoint is increasingly considered obsolete, but "father" is still sometimes used for both the 5L 3s oneirotonic and the 3L 2s oneiro-pentatonic.)
=== Modes ===
{{MOS mode degrees}}
More extreme oneirotonic temperaments include:
==== Proposed mode names ====
* [[Chromatic pairs#Tridec|Tridec]] (a 2.3.7/5.11/5.13/5 subgroup temperament that approximates 5:7:11:13:15), when the generator is between 453.33¢ (17\45) and 457.14¢ (8\21). These have near-equal L/s ratios of 6/5 to 3/2.
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
* [[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.
{{MOS modes
| Mode Names=
Dylathian $
Ilarnekian $
Celephaïsian $
Ultharian $
Mnarian $
Kadathian $
Hlanithian $
Sarnathian $
| Collapsed=1
}}
== Scale tree ==
== Tunings==
{| class="wikitable" style="text-align:center;"
=== 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.
! 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
A-Team tunings (with generator between 5\13 and 7\18) have L/s ratios between 2/1 and 3/1.
A short definition of A-Team is "meantone oneirotonic". This is because A-Team tunings share the following features with [[meantone]] diatonic tunings:
=== Hypohard tunings ===
* The large step is a "meantone", somewhere between near-10/9 (as in [[13edo]]) and near-9/8 (as in [[18edo]]). Thus A-Team tempers out [[81/80]] like meantone does.
[[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 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.
* 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.
EDOs that support A-Team include [[13edo]], [[18edo]], and [[31edo]].
With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to [[7/6]].
* 13edo has characteristically small major mosseconds of about 185c. 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 A-team 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. It also makes rising fifths (733.3c, a perfect mossixth) and falling fifths (666.7c, a major mosfifth) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* 31edo is very close to the 2.9.5.21 POTE tuning, and can be used to make the major mos3rd 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.
The sizes of the generator, large step and small step of oneirotonic are as follows in various A-Team 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.
* 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.
! [[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.
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}
=== Petrtri (13&21) ===
=== Hyposoft 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. In these 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¢. In particular, [[21edo]]'s small step (114.29¢) is close to the diatonic semitone in quarter-comma meantone (117.11¢), which may help give the tuning a Baroque vibe despite the fact that the major mosthirds are now flat neutral thirds.
* 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 mosthird (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¢), and the temperament interprets it as both [[11/9]] and [[16/13]].
* 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 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 0.33¢ flat of the 2.5.9.11.13.17 [[POTE]] petrtri generator of 459.1502¢ and 0.73¢ sharp of the 2.9/5.11/5.13/5 POTE (i.e. optimal for the chord 9:10:11:13, spelled as R-M2-M3-M5 in oneirotonic intervals) petrtri generator of 458.0950¢. If you only care about optimizing 9:10:11:13, then [[55edo]]'s 21\55 (458.182¢) is even better, but 55 is a bit big for a usable edo. 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.
* [[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.
The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings.
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.)
! 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) ===
{{MOS tunings
In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a [[4/3]] perfect fourth. [This identification may come in handy since many altered oneirotonic modes have three consecutive 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-1 temperament|non-over-1 temperament]] that approximates the chord 5:7:11:13:15. Since it is the same as Petrtri when you only care about the 9:10:11:13 (R-M2-M3-M5), it can be regarded as a flatter variant of Petrtri (analogous to how septimal meantone and flattone are the same when you only consider how it maps 8:9:10:12).
| 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 optimal generator is 455.2178¢, which is very close to 29edo's 11\29 (455.17¢), but we could accept any generator between 17\45 (453.33¢) and 8\21 (457.14¢), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.
=== 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. The generator is close to the optimal range for [[tridec]] temperament, which approximates the 5:7:11:13 chord.
Based on the EDOs that support it, Tridec is essentially the same as 13-limit [[Ammonite]].
{{MOS tunings
| 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
}}
The sizes of the generator, large step and small step of oneirotonic are as follows in various tridec 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).
|-
!
! [[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) ===
{{MOS tunings
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.
| 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
}}
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 ===
{{Main|5L 3s/Temperaments#Buzzard}}
The sizes of the generator, large step and small step of oneirotonic are as follows in various buzzard tunings.
[[Buzzard]] is a rank-2 temperament in the [[Step ratio|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]] and [[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 perfect fifth in four it obviously also divides it in two as well.
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:
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.
== 5L 3s as diatonic with a fifth sharper than 5edo ==
You may have heard how [[2L 5s]] can be viewed as diatonic ([[5L 2s]]) generated by a fifth flatter than [[7edo]], in which case the large and small steps flip sizes. Similar to that, 5L 3s can be viewed as diatonic generated by a fifth sharper than [[5edo]]. To understand how this works and how you suddenly get an extra small step, let's first start with fifths slightly flat of 5edo and 5edo's fifth itself.
The [[21edo]] gamut:
In diatonic scales generated by fifths slightly flat of [[5edo]]'s fifth, the small step (found by 8 octaves - 5 fifths) becomes increasingly narrow, for example in [[27edo]] (which has a fifth of 711.1 cents) the diatonic small step is 44.4 cents, and in [[32edo]] (which has a fifth of 712.5 cents) the diatonic small step is even smaller at 37.5 cents. The fifth-generated major scale in 32edo in [[cent]]s is 225-450-487.5-712.5-937.5-1162.5-1200 – notice how the "major third" (450) and "perfect fourth" (487.5) are quite close to each other, as are the "major seventh" (1162.5) and octave (1200). As the fifth gets closer and closer to 5edo's fifth, the small step becomes more and more narrow.
In [[5edo]] itself, if you attempt to generate a diatonic scale by stacking 5edo's 720-centfifth, the diatonic small step becomes 0 cents, which means two pairs of notes in the diatonic scale will be the same, collapsing it into a 5-note scale (this is why a 1:0 step ratio is known as "collapsed" in [[TAMNAMS]]). For example, attempting to create the major scale in 5edo results in (in cents 240-480-480-720-960-1200-1200 – now the "major third" and "perfect fourth" are ''identical'', as are the "major seventh" and "octave".
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.
But what happens if you were to make the fifth even sharper than 5edo's? Well now the small step is going to become even smaller than 0 cents, that is, it will become negative in size. Let's try to create a major scale by stacking 28edo's 728.6-cent fifth, which is just slightly larger than 5edo's: 257.1-514.3-471.4-728.6-985.7-1242.9-1200. Now the so-called "major third" is ''larger'' than the "perfect fourth", and likewise for the "major seventh" being larger than the octave.
== Intervals ==
Obviously it is quite nonsensical to have an octave-repeating scale with the notes out of order, so let's put these in order and octave-reduce the 1242.9-cent "major seventh" to 42.9 cents: 42.9-257.1-471.4-514.3-728.6-985.7-1200. This scale is now in order but it is not a MOS because there are 3 step sizes, because there is now the 42.9-cent small step, the 214.3-cent large step and a single 257.1-cent "extra large" step (between 728.6 and 985.7). If you want to turn it back into a MOS scale, then the extra-large step can be simply split into one large step and one small step, which adds an extra note. This results in the "Celephaïsian" mode of 28edo's oneirotonic scale, 42.9-257.1-471.4-514.3-728.6-942.9-985.7-1200. This approach will work for any fifth between 720 and 750 cents (the generator range of oneirotonic). So the oneirotonic scale can be seen as the result of a diatonic scale with fifths even sharper than that of 5edo, just with an adjustment to make it into a normal properly-ordered MOS scale.
{| 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, falling fourth)
| -1
| O
| perfect mossixth (aka major fifth, rising fifth)
|-
| 2
| P
| major mosseventh
| -2
| L
| minor mosthird
|-
| 3
| K
| major mossecond
| -3
| Q@
| minor moseighth
|-
| 4
| N
| major mosfifth (aka minor fifth, falling fifth)
| -4
| N@
| minor mosfifth (aka major fourth, rising 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 ==
== Samples ==
Flat keys:
[[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]])
** 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
{| class="wikitable"
|-
|+ Table of modes (based on J, from brightest to darkest)
|-
! Mode
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! (9)
|-
| III. Dylathian
| J
| K
| L&
| M
| N
| O&
| P
| Q
| (J)
|-
| VI. Illarnekian
| J
| K
| L&
| M
| N
| O
| P
| Q
| (J)
|-
| I. Celephaïsian
| J
| K
| L
| M
| N
| O
| P
| Q
| (J)
|-
| IV. Ultharian
| J
| K
| L
| M
| N
| O
| P
| Q@
| (J)
|-
| VII. Mnarian
| J
| K
| L
| M
| N@
| O
| P
| Q@
| (J)
|-
| II. Kadathian
| J
| K@
| L
| M
| N@
| O
| P
| Q@
| (J)
|-
| V. Hlanithian
| J
| K@
| L
| M
| N@
| O
| P@
| Q@
| (J)
|-
| VIII. Sarnathian
| J
| K@
| L
| M@
| N@
| O
| P@
| Q@
| (J)
|}
== 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".
The ''Oneiro Falling Suspended Pentatonic'', i.e. P1-M2-P4-M5-M7 (on J, J-K-M-N-P), is also an important subset in ana modes: it roughly implies the "least" tonality (In particular, it only implies ana-ness, not major or minor tonality), and it sounds floaty, suspended and un-tonicized, much like suspended and quartal chords do in diatonic contexts. The ''Oneiro Rising Suspended Pentatonic'' P1-M2-P4-P6-M7 (J-K-M-O-P) can be used for similar effect.
==== 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 Western-classical-influenced approach to oneirotonic chord progressions is to let the harmony emerge from counterpoint.
===== Samples =====
[[File:Oneiro Baroque Exercises 13edo.mp3]]
(A short contrapuntal 13edo keyboard exercise, meant to be played in all 13 keys. The first part is in Oneiromajor, i.e. Illarnekian with mutable 6th degree, and the second part is in Oneirominor, i.e. Celephaïsian with mutable 7th degree.)
[[File:Oneiro Baroque Exercises 18edo.mp3]]
([[18edo]] for comparison)
[[File:Oneiro Baroque Exercises 31edo.mp3]]
([[31edo]] for comparison)
[[File:Oneirotonic 3 part sample.mp3]]
(A rather classical-sounding 3-part harmonization of the ascending J Illarnekian scale; tuning is 13edo)
=== 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 ana modes. Particularly in 13edo and tunings close to it, the fifth and seventh scale degrees become the more concordant 11/8 and quasi-13/8 respectively, so they may sound more like stable scale functions. Hlanithian, in particular, may be like a more stable version of the Locrian mode in diatonic.
[[File:13edo Prelude in J Oneirominor.mp3]]
=== Alterations ===
[[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.
==== Archeodim ====
We call the LSLLLSLS pattern (independently of modal rotation) '''archeodim''', because the "LLL" resembles the [[archeotonic]] mode in 13edo and the "LSLSLS" resembles the diminished scale. Archeodim is the most important oneirotonic [[MODMOS]] pattern (a MODMOS is a MOS with one or more alterations), 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, archeodim has four ana and four kata rotations:
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 40¢). In 18edo both the major fifth (+31.4¢) and the minor fifth (-35.3¢) are about a diesis off from a just perfect fifth. In 13edo only the major fifth is a diesis sharp, and it is +36.5¢ 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 (38.71¢) and 1\34 (35.29¢) are both dieses.
=== Primodal chords ===
Some relatively low-complexity oneirotonic-inspired primodal chords. They are grouped by [[prime family]].
==== /13 ====
* 13:15:19 Tridecimal Falling Ultraminor Triad
* 13:16:19 Tridecimal Falling Submajor Triad
* 13:16:21 Tridecimal Squashed 1st Inversion Minor Triad
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 Falling Bright Major Triad
* 19:23:28 Novem Falling Supraminor Triad
* 19:22:28 Novem Falling Ultraminor Triad
* 19:24:29 Novem Rising Major Triad
* 19:24:30 Novem Augmented Triad
* 19:24:43 Novem Major no5 add9
* 19:24:43:50 Novem Major no5 add9sub11
* 19:24:28:43:50 Novem Falling Major Triad add9 sub11
* 19:24:29:43:50 Novem Rising Major Triad add9 sub11
* 19:25:34 Novem Expanding Quartal
* 19:26:34 Novem Contracting Quartal
* 38:43:56 Novem Falling Minor Triad
* 38:45:56 Novemdecimal Falling Dark Major Triad
* 38:48:56:65 Novem Falling Major Tetrad
* 38:48:73 Novem Major Seventh no5
* 38:48:63 Novem Falling Major 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:
# '''92''':96:100:'''104''':108:112:'''116''':120:125:130:'''136''':141:146:'''152''':158:164:170:177:184 - 18neji with /13, /17, /19, /23, and /29 prime modes
Oneirotonic temperaments have a sort of analogy to diatonic temperaments superpyth and meantone in how they treat the large step. In diatonic the large step is 9/8 (a very good 9/8 in 12edo), but superpyth has 9/8 ~ 8/7, and meantone has 9/8 ~ 10/9. In oneirotonic the large step tends to approximate 10/9 (and is a very good 10/9 in 13edo which is the oneirotonic analogue to 12edo), but different oneiro temperaments do different things with it.
in A-Team (13&18), 10/9 is equated with 9/8, making the major oneirothird a 5/4 (thus is "meantone" in that sense). In both Petrtri (13&21) and Tridec (21&29), 10/9 is equated with 11/10, making the major oneirothird a 11/9; and the perfect oneirofourth is equated to 13/10. So the compressed major triad add2 (R-M2-M3-M5, M5 = major oneirofifth = minor fifth in 13edo) is interpreted as 9:10:11:13 in petrtri, analogous to meantone's 8:9:10:12. Thus Petrtri and Tridec are the same temperament when you only care about the 9:10:11:13, or equivalently the 2.9/5.11/5.13/5 subgroup. This is one reason why Tridec can be viewed as oneirotonic analogue of [[flattone]]: it's a flatter variant of the flat-of-13edo oneiro temperament on the 2.9/5.11/5.13/5 subgroup.
Vulture/[[Hemifamity_temperaments|Buzzard]], in which four generators make a 3/1 (and three generators approximate an octave plus 8/7), is the only [[harmonic entropy]] minimum in the oneirotonic range. However, the rest of this region is still rich in notable subgroup temperaments.
=== Tridec ===
Subgroup: 2.3.7/5.11/5.13/5
Period: 1\1
Optimal ([[POTE]]) generator: 455.2178
EDO generators: [[21edo|8\21]], [[29edo|11\29]], [[37edo|14\37]]
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).
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.
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.
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 ¢).
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.)
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. The generator is close to the optimal range for tridec temperament, which approximates the 5:7:11:13 chord.
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 22edosuperpythdiatonic and a warped 24edosemaphoresemiquartal (and both nearby scales are superhard MOSes).
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 and 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 perfect fifth in four 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.
You may have heard how 2L 5s can be viewed as diatonic (5L 2s) generated by a fifth flatter than 7edo, in which case the large and small steps flip sizes. Similar to that, 5L 3s can be viewed as diatonic generated by a fifth sharper than 5edo. To understand how this works and how you suddenly get an extra small step, let's first start with fifths slightly flat of 5edo and 5edo's fifth itself.
In diatonic scales generated by fifths slightly flat of 5edo's fifth, the small step (found by 8 octaves - 5 fifths) becomes increasingly narrow, for example in 27edo (which has a fifth of 711.1 cents) the diatonic small step is 44.4 cents, and in 32edo (which has a fifth of 712.5 cents) the diatonic small step is even smaller at 37.5 cents. The fifth-generated major scale in 32edo in cents is 225-450-487.5-712.5-937.5-1162.5-1200 – notice how the "major third" (450) and "perfect fourth" (487.5) are quite close to each other, as are the "major seventh" (1162.5) and octave (1200). As the fifth gets closer and closer to 5edo's fifth, the small step becomes more and more narrow.
In 5edo itself, if you attempt to generate a diatonic scale by stacking 5edo's 720-centfifth, the diatonic small step becomes 0 cents, which means two pairs of notes in the diatonic scale will be the same, collapsing it into a 5-note scale (this is why a 1:0 step ratio is known as "collapsed" in TAMNAMS). For example, attempting to create the major scale in 5edo results in (in cents 240-480-480-720-960-1200-1200 – now the "major third" and "perfect fourth" are identical, as are the "major seventh" and "octave".
But what happens if you were to make the fifth even sharper than 5edo's? Well now the small step is going to become even smaller than 0 cents, that is, it will become negative in size. Let's try to create a major scale by stacking 28edo's 728.6-cent fifth, which is just slightly larger than 5edo's: 257.1-514.3-471.4-728.6-985.7-1242.9-1200. Now the so-called "major third" is larger than the "perfect fourth", and likewise for the "major seventh" being larger than the octave.
Obviously it is quite nonsensical to have an octave-repeating scale with the notes out of order, so let's put these in order and octave-reduce the 1242.9-cent "major seventh" to 42.9 cents: 42.9-257.1-471.4-514.3-728.6-985.7-1200. This scale is now in order but it is not a MOS because there are 3 step sizes, because there is now the 42.9-cent small step, the 214.3-cent large step and a single 257.1-cent "extra large" step (between 728.6 and 985.7). If you want to turn it back into a MOS scale, then the extra-large step can be simply split into one large step and one small step, which adds an extra note. This results in the "Celephaïsian" mode of 28edo's oneirotonic scale, 42.9-257.1-471.4-514.3-728.6-942.9-985.7-1200. This approach will work for any fifth between 720 and 750 cents (the generator range of oneirotonic). So the oneirotonic scale can be seen as the result of a diatonic scale with fifths even sharper than that of 5edo, just with an adjustment to make it into a normal properly-ordered MOS scale.
