5L 3s

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User:IlL/Template:RTT restriction

For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (tritave-equivalent).
↖ 4L 2s ↑ 5L 2s 6L 2s ↗
← 4L 3s 5L 3s 6L 3s →
↙ 4L 4s ↓ 5L 4s 6L 4s ↘ 
┌╥╥┬╥╥┬╥┬┐
│║║│║║│║││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLLsLs
sLsLLsLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 3\8 to 2\5 (450.0 ¢ to 480.0 ¢)
Dark 3\5 to 5\8 (720.0 ¢ to 750.0 ¢)
TAMNAMS information
Name oneirotonic
Prefix oneiro-
Abbrev. onei
Related MOS scales
Parent 3L 2s
Sister 3L 5s
Daughters 8L 5s, 5L 8s
Neutralized 2L 6s
2-Flought 13L 3s, 5L 11s
Equal tunings
Equalized (L:s = 1:1) 3\8 (450.0 ¢)
Supersoft (L:s = 4:3) 11\29 (455.2 ¢)
Soft (L:s = 3:2) 8\21 (457.1 ¢)
Semisoft (L:s = 5:3) 13\34 (458.8 ¢)
Basic (L:s = 2:1) 5\13 (461.5 ¢)
Semihard (L:s = 5:2) 12\31 (464.5 ¢)
Hard (L:s = 3:1) 7\18 (466.7 ¢)
Superhard (L:s = 4:1) 9\23 (469.6 ¢)
Collapsed (L:s = 1:0) 2\5 (480.0 ¢)

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.

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.

Any edo with an interval between 450¢ and 480¢ has an oneirotonic scale. 13edo is the smallest edo with a (non-degenerate) 5L 3s oneirotonic scale and thus is the most commonly used oneirotonic tuning.

5L 3s has a pentatonic MOS subset 3L 2s (SLSLL), and in this context we call this the oneiro-pentatonic or oneiro[5]. When viewed as a chord (with undetermined voicing) we call it the Oneiro Core Pentad. (Note: 3L 5s scales also have 3L 2s subsets.)

Notation

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.

The chain of oneirofourths becomes: ... P@ K@ N@ Q L O J M P K N Q& L& O& ...

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 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

Intervals

The table of oneirotonic intervals below takes the flat fourth as the generator. Given the size of the subfourth generator g, any oneirotonic interval can easily be found by noting what multiple of g it is, and multiplying the size by the number of generators it takes to reach the interval and reducing mod 1200 if necessary (The % sign can be used for the modulo operation on many search engines). For example, since the major oneirothird is reached by six subfourth generators, 18edo's major oneirothird is 6*466.67 mod 1200 = 2800 mod 1200 = 400¢, same as the 12edo major third.

# generators up Notation (1/1 = J) Octatonic interval category name Abbrev. # generators up Notation of 2/1 inverse Octatonic interval category name Abbrev.
The 8-note MOS has the following intervals (from some root):
0 J perfect unison P1 0 J octave Po9
1 M perfect oneirofourth (aka minor fourth, falling fourth) Po4 -1 O perfect oneirosixth (aka major fifth, rising fifth) Po6
2 P major oneiroseventh Mo7 -2 L minor oneirothird mo3
3 K major oneirosecond Mo2 -3 Q minor oneiroeighth mo8
4 N major oneirofifth (aka minor fifth, falling fifth) Mo5 -4 N@ minor oneirofifth (aka major fourth, rising fourth) mo4
5 Q& major oneiroeighth Mo8 -5 K@ minor oneirosecond mo2
6 L& major oneirothird Mo3 -6 P@ minor oneiroseventh mo7
7 O& augmented oneirosixth Ao6 -7 M@ diminished oneirofourth d-o4
The chromatic 13-note MOS (either 5L 8s, 8L 5s, or 13edo) also has the following intervals (from some root):
8 J& augmented oneirounison (aka oneirochroma) Ao1 -8 J@ diminished oneirooctave (aka diminished oneironinth) d-o9
9 M& augmented oneirofourth Ao4 -9 O@ diminished oneirosixth d-o6
10 P& augmented oneiroseventh Ao7 -10 L@ diminished oneirothird d-o3
11 K& augmented oneirosecond Ao2 -11 Q@ diminished oneiroeighth d-o8
12 N& augmented oneirofifth Ao5 -12 N@@ diminished oneirofifth d-o5

Tuning ranges

Hypohard

Hypohard oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.

Hypohard oneirotonic can be considered "meantone oneirotonic". This is because these tunings share the following features with meantone diatonic 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 are in the hypohard range include 13edo, 18edo, and 31edo.

  • 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 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 (733.3c, 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 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 hypohard oneiro tunings.

13edo (basic) 18edo (hard) 31edo (semihard)
generator (g) 5\13, 461.54 7\18, 466.67 12\31, 464.52
L (3g - octave) 2\13, 184.62 3\18, 200.00 5\31, 193.55
s (-5g + 2 octaves) 1\13, 92.31 1\18, 66.67 2\31, 77.42

Intervals

Sortable table of major and minor intervals in hypohard oneiro tunings:

Degree Size in 13edo (basic) Size in 18edo (hard) Size in 31edo (semihard) Note name on J Approximate ratios[1] #Gens up
unison 0\13, 0.00 0\18, 0.00 0\31, 0.00 J 1/1 0
min. o2nd 1\13, 92.31 1\18, 66.67 2\31, 77.42 K@ 21/20, 22/21 -5
maj. o2nd 2\13, 184.62 3\18, 200.00 5\31, 193.55 K 9/8, 10/9 +3
min. o3rd 3\13, 276.92 4\18, 266.67 7\31, 270.97 L 7/6 -2
maj. o3rd 4\13, 369.23 6\18, 400.00 10\31, 387.10 L& 5/4 +6
dim. o4th 4\13, 369.23 5\18, 333.33 9\31, 348.39 M@ 16/13, 11/9 -7
perf. o4th 5\13, 461.54 7\18, 466.67 12\31, 464.52 M 21/16, 13/10, 17/13 +1
min. o5th 6\13, 553.85 8\18, 533.33 14\31, 541.94 N@ 11/8 -4
maj. o5th 7\13, 646.15 10\18, 666.66 17\31, 658.06 N 13/9, 16/11 +4
perf. o6th 8\13, 738.46 11\18, 733.33 19\31, 735.48 O 26/17 -1
aug. o6th 9\13, 830.77 13\18, 866.66 22\31, 851.61 O& 13/8, 18/11 +7
min. o7th 9\13, 830.77 12\18, 800.00 21\31, 812.90 P@ 8/5 -6
maj. o7th 10\13, 923.08 14\18, 933.33 24\31, 929.03 P 12/7 +2
min. o8th 11\13, 1015.39 15\18, 1000.00 26\31, 1006.45 Q 9/5, 16/9 -3
maj. o8th 12\13, 1107.69 17\18, 1133.33 29\31, 1122.58 Q& +5
  1. ↑ The ratio interpretations that are not valid for 18edo are italicized.

Hyposoft

Hyposoft oneirotonic tunings (with generator between 8\21 and 5\13) have step ratios 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 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¢).
  • 21edo's P1-Mo2-Mo3-Mo5 approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71¢) and Baroque diatonic semitones (114.29¢, close to quarter-comma meantone's 117.11¢).
  • 34edo's 9:10:11:13 is even better.

The sizes of the generator, large step and small step of oneirotonic are as follows in various hyposoft oneiro tunings (13edo not shown).

21edo (soft) 34edo (semisoft)
generator (g) 8\21, 457.14 13\34, 458.82
L (3g - octave) 3\21, 171.43 5\34, 176.47
s (-5g + 2 octaves) 2\21, 114.29 3\34, 105.88

Intervals

Sortable table of major and minor intervals in hyposoft tunings (13edo not shown):

Degree Size in 21edo (soft) Size in 34edo (semisoft) Note name on J Approximate ratios #Gens up
unison 0\21, 0.00 0\34, 0.00 J 1/1 0
min. o2nd 2\21, 114.29 3\34, 105.88 K@ 16/15 -5
maj. o2nd 3\21, 171.43 5\34, 176.47 K 10/9, 11/10 +3
min. o3rd 5\21, 285.71 8\34, 282.35 L 13/11, 20/17 -2
maj. o3rd 6\21, 342.86 10\34, 352.94 L& 11/9 +6
dim. o4th 7\21, 400.00 11\34, 388.24 M@ 5/4 -7
perf. o4th 7\18, 457.14 12\31, 458.82 M 13/10 +1
min. o5th 10\21, 571.43 16\34, 564.72 N@ 18/13, 32/23 -4
maj. o5th 11\21, 628.57 18\34, 635.29 N 13/9, 23/16 +4
perf. o6th 13\21, 742.86 21\34, 741.18 O 20/13 -1
aug. o6th 14\21, 800.00 23\34, 811.77 O& 8/5 +7
min. o7th 15\21, 857.14 24\34, 847.06 P@ 18/11 -6
maj. o7th 16\21, 914.29 26\34, 917.65 P 22/13, 17/10 +2
min. o8th 18\21, 1028.57 29\34, 1023.53 Q 9/5 -3
maj. o8th 19\21, 1085.71 31\34, 1094.12 Q& 15/8 +5

Parasoft to ultrasoft tunings

The range of oneirotonic tunings of step ratio between 6/5 and 3/2 (thus in the parasoft to ultrasoft range) may be of interest because it 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. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.]

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

29edo (supersoft) 37edo
generator (g) 11\29, 455.17 14\37, 454.05
L (3g - octave) 4\29, 165.52 5\37, 162.16
s (-5g + 2 octaves) 3\29, 124.14 4\37, 129.73

Intervals

The intervals of the extended generator chain (-15 to +15 generators) are as follows in various softer-than-soft oneirotonic tunings.

Degree Size in 29edo (supersoft) Note name on J Approximate ratios (29edo) #Gens up
unison 0\29, 0.00 J 1/1 0
oneirochroma 1\29, 41.3 J& +8
dim. o2nd 2\29, 82.8 K@@ -13
min. o2nd 3\29, 124.1 K@ 14/13 -5
maj. o2nd 4\29, 165.5 K 11/10 +3
aug. o2nd 5\29, 206.9 K& 9/8 +11
dim. o3rd 6\29, 248.3 L@ 15/13 -10
min. o3rd 7\29, 289.7 L 13/11 -2
maj. o3rd 8\29, 331.0 L& +6
aug. o3rd 9\29, 372.4 L&& +14
doubly dim. o4th 9\29, 372.4 M@@ -15
dim. o4th 10\29, 413.8 M@ 14/11 -7
perf. o4th 11\29, 455.2 M 13/10 +1
aug. o4th 12\29, 496.6 M& 4/3 +9
dim. o5th 13\29, 537.9 N@@ 15/11 -12
min. o5th 14\29, 579.3 N@ 7/5 -4
maj. o5th 15\29 620.7 N 10/7 +4
aug. o5th 16\29 662.1 N& 22/15 +12
dim. o6th 17\29, 703.4 O@ 3/2 -9
perf. o6th 18\29, 755.2 O 20/13 -1
aug. o6th 19\29, 786.2 O& 11/7 +7
doubly aug. o7th 20\29 827.6 O&& +15
dim. o7th 20\29 827.6 P@@ -14
min. o7th 21\29 869.0 P@ -6
maj. o7th 22\29, 910.3 P 22/13 +2
aug. o7th 23\29, 951.7 P& 26/15 +8
dim. o8th 24\29, 993.1 Q@ 16/9 -11
min. o8th 25\29, 1034.5 Q 20/11 -3
maj. o8th 26\29, 1075.9 Q& 13/7 +5
aug. o8th 27\29, 1117.2 Q&& +13
dim. o9th 28\29, 1158.6 J@ -8


Key signatures

Flat keys:

  • J@ Oneirominor, L@ Oneiromajor = N@, K@, P@, M@, J@, O@, L@, Q@
  • M@ Oneirominor, O@ Oneiromajor = N@, K@, P@, M@, J@, O@, L@
  • P@ Oneirominor, J@ Oneiromajor = N@, K@, P@, M@, J@, O@
  • K@ Oneirominor, M@ Oneiromajor = N@, K@, P@, M@, J@
  • N@ Oneirominor, P@ Oneiromajor = N@, K@, P@, M@
  • Q Oneirominor, K@ Oneiromajor = N@, K@, P@
  • L Oneirominor, N@ Oneiromajor = N@, K@
  • O Oneirominor, Q Oneiromajor = N@

All-natural key signature:

  • J Oneirominor, L Oneiromajor = no sharps or flats

Sharp keys:

  • M Oneirominor, O Oneiromajor = Q&
  • P Oneirominor, J Oneiromajor = Q&, L&
  • K Oneirominor, M Oneiromajor = Q&, L&, O&
  • N Oneirominor, P Oneiromajor = Q&, L&, O&, J&
  • Q& Oneirominor, K Oneiromajor = Q&, L&, O&, J&, M&
    • Enharmonic with J@ Oneirominor, L@ Oneiromajor in 13edo
  • L& Oneirominor, N Oneiromajor = Q&, L&, O&, J&, M&, P&
    • Enharmonic with M@ Oneirominor, O@ Oneiromajor in 13edo
  • O& Oneirominor, Q& Oneiromajor = Q&, L&, O&, J&, M&, P&, K&
    • Enharmonic with P@ Oneirominor, J@ Oneiromajor in 13edo
  • J& Oneirominor, L& Oneiromajor = Q&, L&, O&, J&, M&, P&, K&, N&
    • Enharmonic with K@ Oneirominor, M@ Oneiromajor in 13edo

Modes

Oneirotonic modes are named after cities in the Dreamlands. (The names are by Cryptic Ruse.)

  1. Dylathian (də-LA(H)TH-iən): LLSLLSLS
  2. Illarnekian (ill-ar-NEK-iən): LLSLSLLS
  3. Celephaïsian (kel-ə-FAY-zhən): LSLLSLLS
  4. Ultharian (ul-THA(I)R-iən): LSLLSLSL
  5. Mnarian (mə-NA(I)R-iən): LSLSLLSL
  6. Kadathian (kə-DA(H)TH-iən): SLLSLLSL
  7. Hlanithian (lə-NITH-iən): SLLSLSLL
  8. 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
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 use 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:

  1. in Oneiromajor, to have both major (requiring a sharpened 6th degree) on the flat fourth "subdominant" and the sharp fifth as "dominant"
  2. 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
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 oneiro2nd = L + chroma)
  • the distorted Freygish SASLSLLS
  • Celephaïsian &4 &6 LsAsLsLs

Hypohard oneiro 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 a hypohard 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".

Pentatonic subsets

The Oneiro Falling Suspended Pentatonic, i.e. R-Mo2-Po4-Mo5-Mo7 (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, and suspended, much like suspended and quartal chords do in diatonic contexts. The Oneiro Rising Suspended Pentatonic R-Mo2-Po4-Po6-Mo7 (J-K-M-O-P) can be used for similar effect.

Modes of the oneiro-pentatonic MOS:

  1. R-Mo2-Po4-Mo5-Mo7 Oneiro Falling Suspended Pentatonic
  2. R-Mo2-Po4-Po6-Mo7 Oneiro Rising Suspended Pentatonic
  3. R-mo3-Po4-Po6-Mo7 Oneiro Symmetrical Pentatonic
  4. R-mo3-Po4-Po6-mo8 Oneiro Expanding Quartal Pentatonic
  5. R-mo3-mo5-Po6-mo8 Oneiro Diminished Pentatonic

Functional harmony

Oneiro has at least two different types of "V-to-I" resolution because of the two fifth sizes:

  1. One uses the sharp fifth as the "V" and uses a true major third. The sharp "V" voiceleads naturally to the flat fifth in the resolved falling tonic triad on the I: e.g. P6-M8-P2 > M5-P1-(M/m)3.
  2. One uses the flat fifth as the "V" and the chord on the "V" is a "false major triad" R-P4-P6 (root-falling 4th-rising 5th).

Some suggested basic ana functional harmony progressions are listed below, 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. This would allow, for example, using the perfect oneirofourth and minor oneirofifth (instead of the major oneirothird and the perfect oneirofourth) as stand-ins for major thirds and fourths in neobaroque contexts (this adds some dissonance which might be what you want sometimes, e.g. in a chord that is supposed to resolve to a more consonant chord).

Samples

(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.)

(18edo)

(31edo)

(21edo for comparison)

‎(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.

Chords and extended harmony

Chords are given in oneirotonic MOS interval notation. For example, M5 means major mosfifth (squashed fifth).

"Rising" means that a triad uses the perfect mos6th (major 5th); "falling" means that a triad uses a major mos5th (minor 5th)

  • R-Mo3-Mo5: Falling Major Triad; Squashed Major Triad
  • R-mo3-Mo5: Falling Minor Triad; Squashed Minor Triad
  • R-mo3-mo5: Squashed Dim Triad
  • R-Mo3-Ao5: Squashed Aug Triad
  • R-Mo3-Mo5-Ao6: Falling Major Triad Add6
  • R-mo3-Mo5-Ao6: Falling Minor Triad Add6
  • R-Mo3-Mo5-Mo7: Falling Major Tetrad
  • R-mo3-Mo5-Mo7: Falling Minor Tetrad
  • R-mo3-mo5-Mo7: Oneiro Half-Diminished Tetrad
  • R-mo3-mo5-mo7: Orwell Tetrad, Oneiro Diminished Tetrad
  • R-Mo3-Ao6: Squashed 1st Inversion Minor Triad; Sephiroth Triad (approximates 8:10:13 in 13edo and 31edo)
  • R-Mo3-Ao6-Mo8: Sephiroth Triad Add7
  • R-Mo3-Ao6-(Mo2)-(Po4): Sephiroth Tetrad Add9
  • R-Mo3-Ao6-(mo2)-(Po4): Sephiroth Tetrad Addm9
  • R-Mo3-Ao6-(Po4): Sephiroth Tetrad
  • R-mo3-Po6: Rising Minor Triad; Squashed 1st Inversion Major Triad
  • R-Mo3-Po6: Rising Major Triad
  • R-mo3-Mo7: Minor add6 no5
  • R-mo3-mo7: Minor addm6 no5
  • R-mo5-Mo7: Falling no3 add6
  • R-mo5-mo7: Falling no3 add6
  • R-Mo3-Mo8: Major 7th no5
  • R-mo3-Mo8: Minor Major 7th no5
  • R-Mo3-Mo5-Mo8: Falling Major Seventh Tetrad
  • R-mo3-Mo5-Mo8: Falling Minor Major Seventh Tetrad
  • R-Mo3-Mo7-Mo8: no5 Major Seventh Add6
  • R-mo3-Mo7-Mo8: no5 Minor Major Seventh Add6
  • R-Mo3-Po6-Mo8: Rising Major Seventh
  • R-mo3-Po6-Mo8: Rising Oneiro Minor Major Seventh
  • R-Mo3-(Mo2): Oneiro Major Add9
  • R-mo3-(Mo2): Oneiro Minor Add9
  • R-Mo3-Mo5-(Mo2): Falling Major Triad Add9
  • R-mo3-Mo5-(Mo2): Falling Minor Triad Add9
  • R-Mo3-(Mo2)-(Po4): no5 Major Add9 Sub11
  • R-mo3-(Mo2)-(Po4): no5 Minor Add9 sub11
  • R-Mo2-Po4: Sus24 No5
  • R-Mo2-Mo5: Falling Sus2 Triad
  • R-Po4-Mo5: Falling Sus4 Triad
  • R-Mo2-Po4-Mo5: Falling Sus24
  • R-Po4-Mo7: Oneiro Quartal Triad
  • R-Po4-Mo7-(Mo2): Oneiro Quartal Tetrad, Core Tetrad
  • R-Po4-Mo7-(Mo2)-(Mo5): Oneiro Quartal Pentad, Core Pentad
  • R-Po4-Mo7-(Mo2)-(Mo5)-(Mo8): Oneiro Quartal Hexad
  • R-Po4-Mo7-Mo8: Oneiro Quartal Seventh Tetrad
  • R-Po4-mo8: Expanding Quartal Triad
  • R-Mo2-Po4-mo8: Expanding Quartal Triad add2
  • R-mo3-Po4-mo8: Expanding Quartal Triad Addm3
  • R-mo5-mo8: Contracting Quartal Triad
  • R-mo5-mo7-mo8: Contracting Quartal Triad Addm7
  • R-Mo3-Mo5-mo8: Falling Major Triad addm7

Hyposoft oneiro theory

21edo has the soft oneirotonic (5L 3s) MOS with generator 8\21; in addition to the naiadics (457.14¢) and extremely sharp fifths (742.85¢) that generate it, it has neutral thirds (instead of major thirds as in 13edo oneirotonic), neogothic minor thirds, and meantone-like diatonic semitones. The oneirofifths (4-step intervals) are more tritone-like than fifth-like, unlike in 13edo, although they do have a consonant, even JI-like quality to them. In terms of JI, it mainly approximates 5:9:11:13 (R-mo8-Mo2-Po4) and 16:23:30 (R-Mo5-Mo8). Importantly, the sharp fifth is now harmonically much more fifth-like than the flat fifth, unlike in 13edo and harder tunings. Rather than squashed tertian triads, it may be preferable to use triads with sharp fifths, quartal harmony, stacks of seconds and thirds, third+sixth and third+seventh chords, and using the JI approximations (subsets of 5:9:11:13 (R-mo8-Mo2-Po4), 9:10:11:13 (R-Mo2-Mo3-Mo5), and 8:15:23 (R-Mo7-Mo5)).

34edo (semisoft) oneirotonic is broadly similar, except the small steps are more 12edo-like and less meantone-like, and it is a bit more optimized for the 5:9:11:13 approximation.

Primodal theory

Main article: 5L 3s/Primodal theory

Temperaments

Main article: 5L 3s/Temperaments

Samples

WT13C Prelude II (J Oneirominor) (score) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.

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

(13edo, L Illarnekian)

(by Igliashon Jones, 13edo, J Celephaïsian)

See also

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
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
7\18 3 1 3 466.667 933.333 200.000 666.667 L/s = 3
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 Boundary of propriety (generators smaller than this are proper)
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 Optimum rank range (L/s=3/2) oneirotonic
11\29 4 3 4 455.172 910.345 165.517 620.690
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
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