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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 =
| ko = 5L3s (Korean)
}}
{{Infobox MOS
{{Infobox MOS
| Name = oneirotonic
| Neutral = 2L 6s
| Periods = 1
| nLargeSteps = 5
| nSmallSteps = 3
| Equalized = 3
| Paucitonic = 2
| Pattern = LLsLLsLs
}}
}}
: ''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''' or '''oneirotonic''' (/oʊnaɪrəˈtɒnɪk/ ''oh-ny-rə-TON-ik'' or /ənaɪrə-/ ''ə-ny-rə-'') refers to the structure of octave-equivalent [[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 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 3s 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 (see also [[5L 3s#Tuning_ranges|Tuning ranges]]):
=== 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 mainly approximates the harmonic series chord 5:9:11:13)
{{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 where two major mosseconds or "whole tones" approximate a [[5/4]] classical major third)
In a sense, these two temperaments represent the middle of the oneirotonic spectrum (with the [[step ratio]] (L/s) ranging from 3/2 to 3/1); [[13edo]] represents both temperaments, with a step 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:
=== Modes ===
* [[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 step ratios of 6/5 to 3/2.
{{MOS mode degrees}}
* [[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.
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.
==== Proposed mode names ====
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
{{MOS modes
| Mode Names=
Dylathian $
Ilarnekian $
Celephaïsian $
Ultharian $
Mnarian $
Kadathian $
Hlanithian $
Sarnathian $
| Collapsed=1
}}
== Notation==
== Tunings==
The notation used in this article is J Ultharian (LsLLsLsL) = JKLMNOPQJ, with reference pitch N = 261.6255653 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".) Ultharian has been chosen as the default mode because we want to carry over the diatonic idea of sharpening the second-to-last degree to get the leading tone for minor keys and the sharpened "Vmaj", and we also have the "sharp V" for the oneiromajor tonality by default.
=== 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.
The chain of oneirofourths becomes: ... P@ K@ N@ Q L O J M P K N Q& L& O& ...
[[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]].
The [[18edo]] gamut is notated as follows:
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.
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}
The [[21edo]] gamut:
=== Hyposoft tunings ===
[[Hyposoft]] oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,
* The large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92{{c}} to 114{{c}}.
* The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342{{c}}) to 4\13 (369{{c}}).
* [[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.
== Scale tree ==
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.)
{| class="wikitable" style="text-align:center;"
|-
! 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
|-
|
|16\41
|
|
|
|7 2 7
|468.293
|936.585
|204.878
|673.171
|Barbad is around here
|-
| | 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 ==
{{MOS tunings
=== A-Team (13&18) ===
| Step Ratios = Hyposoft
:''Main article: [[A-Team]]''
| JI Ratios =
A-Team tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.
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
}}
A short definition of A-Team is "meantone oneirotonic". This is because A-Team tunings share the following features with [[meantone]] diatonic tunings:
=== Parasoft and ultrasoft 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.
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 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.
EDOs that support A-Team include [[13edo]], [[18edo]], and [[31edo]].
{{MOS tunings
* 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.
| Step Ratios = 6/5; 3/2; 4/3
* 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 (733.3c, 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.
| JI Ratios =
* 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.
1/1;
* [[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.
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 A-Team 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).
|-
!
! [[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, 66.67
| 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
| 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
}}
=== Petrtri (13&21) ===
=== Ultrahard tunings ===
Petrtri tunings (with generator between 8\21 and 5\13) have less extreme step 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,
{{Main|5L 3s/Temperaments#Buzzard}}
* 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 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 three major edos in this range, [[13edo]], [[21edo]] and [[34edo]], all nominally support petrtri.
[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.
* [[13edo]] nominally supports it, but its approximation of 9:10:11:13 is quite weak and tempers 11/9 to a 369¢ submajor third, which may not be desirable.
* [[21edo]] is a much better petrtri tuning than 13edo, in terms of approximating 9:10:11:13. 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]] 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.
The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri 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 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.
! 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) ===
In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a [[4/3]] perfect fourth, i.e. equates the oneirotonic large step to a [[porcupine]] generator. [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).
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.
Based on the EDOs that support it, Tridec is essentially the same as 13-limit [[Ammonite]].
The sizes of the generator, large step and small step of oneirotonic are as follows in various tridec tunings.
! 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) ===
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.
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.
The sizes of the generator, large step and small step of oneirotonic are as follows in various buzzard tunings.
** Enharmonic with K@ Oneirominor, M@ Oneiromajor in 13edo
== Modes ==
Oneirotonic modes are named after cities in the Dreamlands. (The names are by Cryptic Ruse.)
# Dylathian (də-LA(H)TH-iən): LLSLLSLS
# Illarnekian (ill-ar-NEK-iən): LLSLSLLS
# Celephaïsian (kel-ə-FAY-zhən): LSLLSLLS
# Ultharian (ul-THA(I)R-iən): LSLLSLSL
# Mnarian (mə-NA(I)R-iən): LSLSLLSL
# Kadathian (kə-DA(H)TH-iən): SLLSLLSL
# Hlanithian (lə-NITH-iən): SLLSLSLL
# Sarnathian (sar-NA(H)TH-iən): SLSLLSLL
The modes on the white keys JKLMNOPQJ are:
* J Ultharian
* K Hlanithian
* L Illarnekian
* M Mnarian
* N Sarnathian
* O Celephaïsian
* P Kadathian
* Q Dylathian
{| class="wikitable"
|-
|+ Table of modes (based on J, from brightest to darkest)
|-
! Mode
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! (9)
|-
| Dylathian
| J
| K
| L&
| M
| N
| O&
| P
| Q&
| (J)
|-
| Illarnekian
| J
| K
| L&
| M
| N
| O
| P
| Q&
| (J)
|-
| Celephaïsian
| J
| K
| L
| M
| N
| O
| P
| Q&
| (J)
|-
| Ultharian
| J
| K
| L
| M
| N
| O
| P
| Q
| (J)
|-
| Mnarian
| J
| K
| L
| M
| N@
| O
| P
| Q
| (J)
|-
| Kadathian
| J
| K@
| L
| M
| N@
| O
| P
| Q
| (J)
|-
| Hlanithian
| J
| K@
| L
| M
| N@
| O
| P@
| Q
| (J)
|-
| Sarnathian
| J
| K@
| L
| M@
| N@
| O
| P@
| Q
| (J)
|}
For classical-inspired functional harmony, we propose the terms ''(Functional) Oneiromajor'' and ''(Functional) Oneirominor'': Oneiromajor for Illarnekian where the 6th degree (the rising fifth) can be sharpened, and Oneirominor for Ultharian where the 8th degree (the leading tone) can be sharpened. The respective purposes of these alterations are:
# in Oneiromajor, to have both major (requiring a sharpened 6th degree) on the flat fourth "subdominant" and the sharp fifth as "dominant"
# in Oneirominor, to have both the flat 8th degree as the dominant of the "mediant" (relative major) and the sharp 8th degree as leading tone
In key signatures, Oneirominor should be treated as Ultharian and Oneiromajor should be treated as Illarnekian. Note that Oneiromajor and Oneirominor still have the relative major-minor relationship; they are related by a major mosthird, just like diatonic major/minor.
=== Alterations ===
==== Archeodim ====
We call the LSLLLSLS pattern (independently of modal rotation) '''archeodim''', because the "LLL" resembles the [[archeotonic]] scale 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. Archeodim modes exist in all oneirotonic tunings, since they use the same large and small steps as the oneirotonic scale itself.
As with the MOS, archeodim has four ana and four kata rotations:
* Ana:
** LLLSLSLS: Dylathian &4, Dylydian
** LLSLSLSL: Illarnekian @8, Illarmixian
** LSLLLSLS: Celephaïsian &6, Celdorian
** SLLLSLSL: Ultharian @2, Ulphrygian
* Kata:
** LSLSLLLS: Mnarian &8, Mnionian
** SLSLLLSL: Sarnathian &7, Sardorian
** LSLSLSLL: Mnarian @7, Mnaeolian
** SLSLSLLL: Sarnathian @6, Sarlocrian
{| class="wikitable"
|-
|+ Table of archeodim modes (based on J)
|-
! Mode
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! (9)
|-
| Dylydian
| J
| K
| L&
| M&
| N
| O&
| P
| Q&
| (J)
|-
| Illarmixian
| J
| K
| L&
| M
| N
| O
| P
| Q
| (J)
|-
| Celdorian
| J
| K
| L
| M
| N
| O&
| P
| Q&
| (J)
|-
| Ulphrygian
| J
| K@
| L
| M
| N
| O
| P
| Q
| (J)
|-
| Mnionian
| J
| K
| L
| M
| N@
| O
| P
| Q&
| (J)
|-
| Sardorian
| J
| K@
| L
| M@
| N@
| O
| P
| Q
| (J)
|-
| Mnaeolian
| J
| K
| L
| M
| N@
| O
| P@
| Q
| (J)
|-
| Sarlocrian
| J
| K@
| L
| M@
| N@
| O@
| P@
| Q
| (J)
|}
==== Other MODMOSes ====
Other potentially interesting oneirotonic MODMOSes (that do not use half-sharps or half-flats) are:
* the distorted harmonic minor LSLLSALS (A = aug 2nd = L + chroma)
* the distorted Freygish SASLSLLS
* Celephaïsian &4 &6 LsAsLsLs
== Rank-2 temperaments ==
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 approximates 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 the 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.
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 & 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.