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5L 3s

Revision as of 21:00, 4 February 2021 by Inthar (talk | contribs) (→Modes: attribution for mode names)
For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (tritave-equivalent).

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. 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.

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.

In terms of regular temperaments, there are at least two melodically viable ways to interpret oneirotonic:

  1. When the generator is between 457.14¢ (8\21) and 461.54¢ (5\13): Petrtri (13&21, a 4:5:9:11:13:17 or 2.5.9.11.13.17 temperament)
  2. 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:

  • Tridec (a 5:7:11:13 or 2.7/5.11/5.13/5 subgroup temperament), when the generator is between 454.05c (14\37) and 457.14c (8\21). These have near-equal L/s ratios of 5/4 to 3/2.
  • Buzzard, when the generator is between 471.42¢ (11\28) and 480¢ (2\5). While this is a harmonically accurate temperament, with 4 generators reaching 3/2 and -3 generators 7/4, it is relatively weak melodically, as the optimum size of the small steps is around 20-25 cents, making it difficult to distinguish from equal pentatonic.

Scale tree

generator 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
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
Boundary of propriety (generators smaller than this are proper)
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
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

A-Team (13&18)

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:

  • The large step is a "meantone", somewhere between near-10/9 (as in 13edo) and near-9/8 (as in 18edo).
  • 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.

  • 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.
  • 31edo is very close to the POTE tuning, and can be used to make the major mos3rd a near-just 5/4.

The sizes of the generator, large step and small step of oneirotonic are as follows in various A-Team tunings.

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.

Petrtri (13&21)

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.

The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings.

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

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 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.

The sizes of the generator, large step and small step of oneirotonic are as follows in various buzzard tunings.

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

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:

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

The 18edo gamut is notated as follows:

J Q&/K@ J&/L@ K L K&/M@ L& M N@ M&/O@ N O P@ O& P Q@ P&/J@ Q J

The 21edo gamut:

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

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

Generators Notation (1/1 = J) Octatonic interval category name Generators Notation of 2/1 inverse Octatonic interval category name
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
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.)

  1. Dylathian: LLSLLSLS
  2. Illarnekian: LLSLSLLS
  3. Celephaïsian: LSLLSLLS (Easley Blackwood's 13-note etude uses this as its home mode.)
  4. 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.)
  5. Mnarian: LSLSLLSL
  6. Kadathian: SLLSLLSL
  7. Hlanithian: SLLSLSLL
  8. 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):

 

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

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/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: 8\21, 11\29, 14\37

Technical data

Comma list: 847/845, 1001/1000

Mapping (for 2, 7/5, 11/5, 13/5): [⟨1 2 0 1], ⟨0 -4 3 1]]

Mapping generators: ~2, ~13/10

Optimal ET sequence: 21, 29, 37

Petrtri (13&21, 2.5.9.11.13.17)

Period: 1\1

Optimal (POTE) generator: 459.1502

EDO generators: 5\13, 8\21, 13\34

Technical data

Comma list: 100/99, 144/143, 170/169, 221/220

Mapping (for 2, 5, 9, 11, 13, 17): [⟨1 5 7 5 6 6], ⟨0 -7 -10 -4 -6 -5]]

Mapping generators: ~2, ~13/10

Optimal ET sequence: 13, 21, 34

Intervals

Sortable table of intervals in the Dylathian mode and their Petrtri interpretations:

Degree Size in 13edo Size in 21edo Size in 34edo Size in POTE tuning Note name on L 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: 5\13, 7\18, 12\31, 17\44

Technical data

Comma list: 81/80, 1029/1024

Mapping (for 2, 5, 9, 21): [⟨1 0 2 4], ⟨0 6 3 1]]

Mapping generators: ~2, ~21/16

Optimal ET sequence: 13, 18, 31, 44

Intervals

Sortable table of intervals in the Dylathian mode and their A-Team interpretations:

Degree Size in 13edo Size in 18edo Size in 31edo Note name on L Approximate ratios[1] #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
  1. ↑ The harmonics over 1/1 are in bold. The ratio interpretations that are not valid for 18edo are italicized.

Buzzard (48&53, 2.3.5.7)

Period: 1\1

Optimal (POTE) generator: ~21/16 = 475.636

EDO generators: 15\38, 17\43, 19\48, 21\53, 23\58, 25\63

Technical data

Commas: 1728/1715, 5120/5103

Map: [<1 0 -6 4|, <0 4 21 -3|]

Mapping generators: ~2, ~21/16

Wedgie: <<4 21 -3 24 -16 -66||

48, 53, 111, 164d, 275d

Badness: 0.0480

Intervals

Sortable table of intervals in the Dylathian mode and their Buzzard interpretations:

Degree Size in 38edo Size in 53edo Size in 63edo Size in POTE tuning Note name on L 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

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

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

(13edo, L Illarnekian)

(by Igliashon Jones, 13edo, J Celephaïsian)

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