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Oneirotonic

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The term oneirotonic (/oʊnaɪrəˈtɒnɪk/ oh-ny-rə-TON-ik or /ənaɪrə-/ ə-ny-rə-) is used for the 8-note 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).

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 (analogous to diatonic having multiple temperament interpretations depending on generator size):

  1. 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)
  2. 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)

13edo represents both temperaments.

There is also Buzzard, when the generator is between 471.42¢ (11\28) and 480¢ (2\5), but 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.

Tunings

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.

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 nearly pure 9/8 and 7/6.
  • 31edo can be used if a near-just 5/4 is desired.

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.

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

13edo 18edo 31edo 64/49 generator POTE tuning JI intervals represented (2.9.5.21 subgroup)
generator (g) 5\13, 461.54 7\18, 466.67 12\31, 464.52 462.35 464.14 21/16
L (3g - octave) 2\13, 184.62 3\18, 200.00 5\31, 193.55 187.04 192.42 9/8, 10/9
s (-5g + 2 octaves) 1\13, 92.31 1\18, 66.66 2\31, 77.42 88.26 79.30 21/20

Petrtri (13&21)

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.

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 POTE tuning JI intervals represented (2.5.9.11.13.17 subgroup)
generator (g) 5\13, 461.54 8\21, 457.14 13\34, 458.82 459.15 13/10, 17/13, 22/17
L (3g - octave) 2\13, 184.62 3\21, 171.43 5\34, 176.47 177.45 10/9, 11/10
s (-5g + 2 octaves) 1\13, 92.31 2\21, 114.29 3\34, 105.88 104.25 18/17, 17/16

Trivia: One petrtri tuning is golden oneirotonic, which uses (2-φ)*1200 cents = 458.3592135¢ as generator and has L/s = φ; it is the limit of taking generators in Fibonacci number edos 5\13, 8\21, 13\34, 21\55, 34\89,....

Notation

The notation used in this article is J Celephaïsian (LsLLsLLs) = JKLMNOPQJ (with J ≈ 180 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 J&/K@ K L L&/M@ M M&/N@ N O O&/P@ P P&/Q@ Q J

Note: N is close to standard C, since the reference pitch 180 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 -1 O perfect mossixth
2 P major mosseventh -2 L minor mosthird
3 K major mossecond -3 Q@ minor moseighth
4 N major mosfifth -4 N@ minor mosfifth
5 Q major moseighth -5 K@ minor mossecond
6 L& major mosthird -6 P@ minor mosseventh
7 O& augmented sixth -7 M@ diminished fourth
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.

  1. Dylathian: LLSLLSLS
  2. Ilarnekian: 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 Ilarnekian
  • P Mnarian
  • Q Sarnathian

The modes in 13edo edo steps and C-H notation:

 

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 should experiment and see how well these ideas work in 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, Ilarnekian, Celephaïsian and Ultharian.

The ana modes have squashed versions of the classical major and minor pentachords R-M2-M3-P4-P5 and R-M2-m3-P4-P5 and can be viewed as providing a distorted version of classical diatonic functional harmony and counterpoint. For example, in the Dylathian mode, the 4:5:9 triad on the sixth degree can sound like both "V" and "III of iv" depending on context.

In pseudo-classical functional harmony, the 6th scale degree (either an augmented mossixth or a perfect mossixth) could be treated as mutable. The perfect mossixth would be used when invoking the diatonic V-to-I trope by modulating by a perfect mosfourth from the sixth degree. The augmented mossixth would be used when a major key needs to be used on the fourth degree.

Progressions

Some basic functional harmony progressions, outlined very roughly (note: VI is the sharp 5th, etc.). "I" means either Imaj or Imin. Unmodified Roman numerals follow the Ilarnekian mode.

  • I-IVmin-VImaj-I
  • Imaj-VIImin-IVmaj-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
  • Modulation by minor mos3rd: I-IVmin-VImin-@VIIImaj-@III

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.

MODMOSes

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

  • LLLSLSLS: Dylathian &4: Ana-Lydian
  • LLSLSLSL: Ilarnekian @8: Ana-Mixolydian
  • LSLLLSLS: Celephaïsian &6: Ana-Dorian
  • SLLLSLSL: Ultharian @2: Ana-Phrygian
  • SLSLSLLL: Sarnathian @6: Kata-Locrian
  • SLSLLLSL: Sarnathian &6: Kata-Dorian
  • LSLSLLLS: Mnarian &8: Kata-Ionian
  • LSLSLSLL: Hlanithian &2: 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).

  • R-M3-M5: Squashed Major Triad
  • R-m3-M5: Squashed Minor Triad
  • R-m3-m5: Squashed Dim Triad
  • R-M3-A5: Squashed Aug Triad
  • R-M3-M5-A6: Squashed Major Triad Add6
  • R-m3-M5-A6: Squashed Minor Triad Add6
  • R-M3-M5-M7: Oneiro Major Tetrad
  • R-m3-M5-M7: Oneiro Minor Tetrad
  • R-m3-m5-M7: Oneiro Half-Diminished Tetrad
  • R-m3-m5-m7: Orwell Tetrad, Oneiro Diminished Tetrad
  • R-M3-A6: Squashed 1st Inversion Minor Triad
  • R-m3-P6: Squashed 1st Inversion Major Triad
  • R-M3-M7: 1st Inversion Squashed Minor Triad (note the order of terms!)
  • R-m3-m7: 1st Inversion Squashed Major Triad
  • R-m5-M7: 2nd Inversion Squashed Major Triad
  • R-m5-m7: 2nd Inversion Squashed Minor Triad
  • R-M3-M8: Oneiro Major Seventh
  • R-m3-M8: Oneiro Minor Major Seventh
  • R-M3-(M2): Oneiro Major Add9
  • R-m3-(M2): Oneiro Minor Add9
  • R-M3-(M2)-(P4): Oneiro Major Add9 Sub11
  • R-m3-(M2)-(P4): Oneiro Minor Add9 Sub11
  • R-M2-P4: Oneiro Sus2 Sus4
  • R-P4-M7: Oneiro Quartal Triad
  • R-P4-M7-(M2): Oneiro Quartal Tetrad, Core Tetrad
  • R-P4-M7-(M2)-(M5): Oneiro Quartal Pentad, Core Pentad
  • R-P4-M7-(M2)-(M5)-(M8): Oneiro Quartal Hexad
  • R-P4-M7-M8: Oneiro Quartal Seventh
  • R-M3-m7: Sephiroth Triad
  • R-M3-m7-m2-(P4): Sephiroth Triad Addmin9 Sub11
  • R-M3-m7-(P4): Sephiroth Triad Sub11
  • R-P4-m8
  • R-m3-P4-m8
  • R-m5-m8
  • R-m5-m7-m8

"Oneirotonic maqam"

"Oneirotonic maqam" is based on the idea "If maqam is loosely an extension of diatonic that uses neutral intervals, what is the oneirotonic counterpart that uses oneirotonic neutral intervals?" or "What if we distorted maqam scales similarly to how oneirotonic distorts diatonic scales?" The following assumes an edo with A-Team oneirotonic scales and neutral mosseconds (i.e. half of an oneirotonic minor mosthird) such as 18edo and 26edo. In rank-2 temperament terms, this requires a loosely 18&26 structure.

  • 26edo can be used if you want neutral mosseconds and minor mosthirds closer to their 24edo counterparts. In 26edo these are 138c and 277c respectively, but in 18edo these are 133c and 267c.
  • 18edo can be used if you want neutral mosthirds (neutral mos2nd + major mos2nd) closer to conventional neutral thirds. The neutral mos3rd is 333c in 18edo and 323c in 26edo.

Zheanist theory

A-Team oneirotonic may be a particularly good place to bring to bear Zheanism's high harmonic chords, as A-Team temperament doesn't yield many low-complexity chords.

18edo may be a better basis for a style of oneirotonic Zheanism 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 (-35.3) are about a diesis off from a just perfect fifth. In 13edo only the major fifth is a diesis sharp, and it is +36.5c 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 also provide opportunities to use dieses directly, since 1\31 (38.71c) and 1\34 (35.29c) are both dieses.

Primodal chords

These are just oneirotonic-inspired chords, they aren't guaranteed to fit in your neji.

/13

  • 13:16:19 Tridecimal Squashed Major Triad
  • 13:17:19 Tridecimal Squashed Sus4
  • 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

/17

  • 17:20:25 Septen Squashed Minor 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
  • 34:40:47:55 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))

/23

  • 23:27:30 Vice Squashed Min4
  • 23:27:30:35:44 Vice Squashed Min4 addM5,M7
  • 23:27:37 Vice Orwell Tetrad no5
  • 46:54:63:76 Vice Orwell Tetrad
  • 46:54:67:78 Vice Minor Oneiro Tetrad
  • 46:54:60:67:78 Vice Min4 Oneiro Pentad

/29

  • 29:34:38 Vicenon 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 Ilarnekian
  • 58:65:72:76:84:89:99:104:116 Vicenon Ilarnekian @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 &6
  • 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

  • 58:61:65:68:72:76:80:84:89:94:99:104:110:116 A very low-complexity 13neji; not optimized for transposability.

Temperament interpretations

A-Team (13&18, 4:5:9:21)

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.

Petrtri (13&21, 4:5:9:11:13:17)

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

Samples

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

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

(13edo, L Ilarnekian)

(by Igliashon Jones, 13edo, J Celephaïsian)

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