Xenharmonic Wiki

5L 3s

Revision as of 05:23, 26 March 2021 by Inthar (talk | contribs) (→Notation)
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 ↘ 
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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) 5L3s 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. When viewed as a chord (with undetermined voicing) we call it the Oneiro Core Pentad. (Note: 3L 5s scales also have 3L 2s subsets.)

In terms of regular temperaments, there are at least two melodically viable ways to interpret oneirotonic (see also Tuning ranges):

  1. When the generator is between 457.14¢ (8\21) and 461.54¢ (5\13): Petrtri (13&21, a 2.5.9.11.13.17 temperament that mainly approximates the harmonic series chord 5:9:11:13)
  2. 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:

  • 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.
  • 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.

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.

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

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
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
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

A-Team (13&18)

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

A short definition of A-Team is "meantone oneirotonic". This is because A-Team tunings share the following features with meantone diatonic tunings:

  • 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 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.

  • 13edo has characteristically small major mosseconds of about 185c. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best 11/8 out of all A-team tunings.
  • 18edo can be used for a large 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 is very close to the 2.9.5.21 POTE tuning, and can be used to make the major mos3rd a near-just 5/4.
  • 44edo (generator 17\44 = 463.64¢), 57edo (generator 22\57 = 463.16¢), and 70edo (generator 27\70 = 462.857¢) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.

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

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.

Petrtri (13&21)

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,

  • 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.

  • 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.

13edo 21edo 34edo Optimal (2.5.9.11.13.17 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

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 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.

21edo 29edo 37edo Optimal (POTE) tuning JI intervals represented (2.3.7/5.11/5.13/5 subgroup)
generator (g) 8\21, 457.14 11\29, 455.17 14\37, 454.05 455.22 13/10
L (3g - octave) 3\21, 171.43 4\29, 165.52 5\37, 162.16 165.65 11/10
s (-5g + 2 octaves) 2\21, 114.29 3\29, 124.14 4\37, 129.73 123.91 14/13, 15/14

Buzzard (48&53)

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

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 oneirofourth (aka minor fourth, falling fourth) -1 O perfect oneirosixth (aka major fifth, rising fifth)
2 P major oneiroseventh -2 L minor oneirothird
3 K major oneirosecond -3 Q minor oneiroeighth
4 N major oneirofifth (aka minor fifth, falling fifth) -4 N@ minor oneirofifth (aka major fourth, rising fourth)
5 Q& major oneiroeighth -5 K@ minor oneirosecond
6 L& major oneirothird -6 P@ minor oneiroseventh
7 O& augmented oneirosixth -7 M@ diminished oneirofourth
The chromatic 13-note MOS also has the following intervals (from some root):
8 J& augmented unison -8 J@ diminished octave
9 M& augmented oneirofourth -9 O@ diminished oneirosixth
10 P& augmented oneiroseventh -10 L@ diminished oneirothird
11 K& augmented oneirosecond -11 Q@ diminished oneiroeighth
12 N& augmented oneirofifth -12 N@@ diminished oneirofifth

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 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:

  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 2nd = L + chroma)
  • the distorted Freygish SASLSLLS
  • Celephaïsian &4 &6 LsAsLsLs

A-Team 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".

Pentatonic subsets

The Oneiro Falling Suspended Pentatonic, i.e. P1-M2-P4-M5-M7 (on J, J-K-M-N-P), is also an important subset in ana modes: it roughly implies the "least" tonality (In particular, it only implies ana-ness, not major or minor tonality), and it sounds floaty, and suspended, much like suspended and quartal chords do in diatonic contexts. The Oneiro Rising Suspended Pentatonic P1-M2-P4-P6-M7 (J-K-M-O-P) can be used for similar effect.

Modes of the oneiro-pentatonic MOS:

  1. P1-M2-P4-M5-M7 Oneiro Falling Suspended Pentatonic
  2. P1-M2-P4-P6-M7 Oneiro Rising Suspended Pentatonic
  3. P1-m3-P4-P6-M7 Oneiro Symmetrical Pentatonic
  4. P1-m3-P4-P6-m8 Oneiro Expanding Quartal Pentatonic
  5. P1-m3-m5-P6-m8 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.

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 for comparison)

(31edo 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

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-M3-M5: Falling Major Triad; Squashed Major Triad
  • R-m3-M5: Falling Minor Triad; Squashed Minor Triad
  • R-m3-m5: Squashed Dim Triad
  • R-M3-A5: Squashed Aug Triad
  • R-M3-M5-A6: Falling Major Triad Add6
  • R-m3-M5-A6: Falling Minor Triad Add6
  • R-M3-M5-M7: Falling Major Tetrad
  • R-m3-M5-M7: Falling Minor Tetrad
  • R-m3-m5-M7: Half-Diminished Tetrad
  • R-m3-m5-m7: Orwell Tetrad, Diminished Tetrad
  • R-M3-A6: Squashed 1st Inversion Minor Triad; Sephiroth Triad (approximates 8:10:13 in 13edo and 31edo)
  • R-M3-A6-M8: Sephiroth Triad Add7
  • R-M3-A6-(M2)-(P4): Sephiroth Triad Add9 Sub11
  • R-M3-A6-(m2)-(P4): Sephiroth Triad Addm9 Sub11
  • R-M3-A6-(P4): Sephiroth Triad Sub11
  • R-m3-P6: Rising Minor Triad; Squashed 1st Inversion Major Triad
  • R-M3-P6: Rising Major Triad
  • R-m3-M7: Minor add6 no5
  • R-m3-m7: Minor addm6 no5
  • R-m5-M7: Falling no3 add6
  • R-m5-m7: Falling no3 add6
  • R-M3-M8: Major 7th no5
  • R-m3-M8: Minor Major 7th no5
  • R-M3-M5-M8: Falling Major Seventh Tetrad
  • R-m3-M5-M8: Falling Minor Major Seventh Tetrad
  • R-M3-M7-M8: no5 Major Seventh Add6
  • R-m3-M7-M8: no5 Minor Major Seventh Add6
  • R-M3-P6-M8: Rising Major Seventh
  • R-m3-P6-M8: Rising Oneiro Minor Major Seventh
  • R-M3-(M2): Oneiro Major Add9
  • R-m3-(M2): Oneiro Minor Add9
  • R-M3-M5-(M2): Falling Major Triad Add9
  • R-m3-M5-(M2): Falling Minor Triad Add9
  • R-M3-(M2)-(P4): no5 Major Add9 Sub11
  • R-m3-(M2)-(P4): no5 Minor Add9 Sub11
  • R-m3-P6-M7-(M2)-(P4)-(A6)-(M8)
  • R-M2-P4: Sus24 No5
  • R-M2-M5: Falling Sus2 Triad
  • R-P4-M5: Falling Sus4 Triad
  • R-M2-P4-M5: Falling 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
  • R-M3-M5-m8: Falling Major Triad addm7

Primodal theory

18edo may be a better basis for a style of oneirotonic primodality using comma sharp and comma flat fifths than 13edo (in particular diesis sharp and diesis flat fifths; diesis is a category with a central region of 32 to 40¢). In 18edo both the major fifth (+31.4¢) and the minor fifth (-35.3¢) are about a diesis off from a just perfect fifth. In 13edo only the major fifth is a diesis sharp, and it is +36.5¢ off from just; so there's less wiggle room for a neji if you want every major fifth to be at most a diesis sharp).

31nejis and 34nejis (though 34edo is not an A-Team tuning) also provide opportunities to use dieses directly, since 1\31 (38.71¢) and 1\34 (35.29¢) are both dieses.

Primodal chords

Some relatively low-complexity oneirotonic-inspired primodal chords. They are grouped by prime family.

/11

  • 22:25:26:29:32:34:38:42:44 Undecimal Celephaïsian
  • 22:25:26:29:32:34:38:40:44 Undecimal Ultharian

/13

  • 13:15:19 Tridecimal Falling Ultraminor Triad
  • 13:16:19 Tridecimal Falling Submajor Triad
  • 13:16:21 Tridecimal Squashed 1st Inversion Minor Triad
  • 13:17:19 Tridecimal Naiadic Maj2; Tridecimal Falling Sus4
  • 13:17:20 Tridecimal Rising Sus4
  • 13:17:21 Tridecimal Squashed 2nd Inversion Major Triad
  • 13:16:19:22 Tridecimal Falling Major Tetrad
  • 26:29:38 Tridecimal Falling Sus2 Triad
  • 26:31:38 Tridecimal Falling Bright Minor Triad
  • 26:33:38 Tridecimal Falling Bright Major Triad
  • 26:29:34:38 Tridecimal Falling Sus24

/17

  • 17:20:25 Septendecimal Falling Minor Triad
  • 17:21:25 Septen Falling Major Triad
  • 17:20:26 Septen Rising Minor Triad
  • 17:20:25:29 Septen Falling Minor Tetrad
  • 17:21:25:29 Septen Falling Major Tetrad
  • 17:20:26:29 Septen Rising Minor Triad addM6
  • 34:41:50 Septen Falling Bright Minor Triad
  • 34:43:50 Septen Falling Octodecous Major Triad (octodecous means '18edo-like')
  • 34:40:47:55 Septen Orwell Tetrad
  • 34:40:52:58:76:89:102:129 (Celephaïsian + P5; R-min3-r5-M6-M9-sub11-P12(fc)-M14)
  • 34:40:52:58:76:89:102:110:129 (Celephaïsian + P5; R-min3-r5-M6-M9-sub11-P12(fc)-supmin13-M14)
  • 34:40:50:58:89:102:129 (R-min3-f5-M6-M9-sub11-P12(rc)-M14)
  • 34:40:50:58:89:102:110:129 (R-min3-f5-M6-M9-sub11-P12(rc)-supmin13-M14)
  • 34:40:50:58:76:89:110:129 (R-m3-f5-M6-M9-sub11-supm13-M7)
  • 34:40:50:58:76:89:102:110:129:208 (R-m3-f5-M6-M9-sub11-P12(rc)-supm13-M14-r19(rc^2))
  • 34:38:40:44:49:52:58:64:68 Septen Celephaïsian

/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 Falling Bright Major Triad
  • 19:23:28 Novem Falling Supraminor Triad
  • 19:22:28 Novem Falling Ultraminor Triad
  • 19:24:29 Novem Rising Major Triad
  • 19:24:30 Novem Augmented Triad
  • 19:24:43 Novem Major no5 add9
  • 19:24:43:50 Novem Major no5 add9sub11
  • 19:24:28:43:50 Novem Falling Major Triad add9 sub11
  • 19:24:29:43:50 Novem Rising Major Triad add9 sub11
  • 19:25:34 Novem Expanding Quartal
  • 19:26:34 Novem Contracting Quartal
  • 38:43:56 Novem Falling Minor Triad
  • 38:45:56 Novemdecimal Falling Dark Major Triad
  • 38:48:56:65 Novem Falling Major Tetrad
  • 38:48:73 Novem Major Seventh no5
  • 38:48:63 Novem Falling Major Triad
  • 38:50:65 Novem Oneiro Quartal Triad
  • 38:50:65:73 Novem Oneiro Quartal Seventh Tetrad
  • 38:50:65:86 Novem Oneiro Core Tetrad
  • 38:50:65:86:112 Novem Oneiro Core Pentad
  • 38:50:65:86:112:146 Novem Oneiro Core Hexad
  • 38:50:63 Novem Squashed First Inversion Neutral Triad
  • 38:43:45:50:56:58:65:72:76 Novem Bright Celephaïsian
  • 38:42:44:49:55:58:65:72:76 Novem Dark Celephaïsian

/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 Falling Min4 no5
  • 23:27:30:35:44 Vice Falling Min4 addM5,M7
  • 23:27:37 Vice Orwell Tetrad no4
  • 23:29:34 Vice Octodecous Falling Major Triad
  • 46:54:68 Vice Octodecous Falling Minor Triad
  • 46:54:60:67 Vice Falling 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 Tridecous Falling Minor Triad (tridecous means '13edo-like')
  • 46:57:67 Vice Tridecous Falling Major Triad
  • 46:54:67:78 Vice Tridecous Falling Minor Tetrad
  • 46:57:67:78 Vice Tridecous Falling Major Tetrad
  • 46:54:60:67:78 Vice Falling Minor Tetrad Add Min4
  • 46:60:67 Vice Falling Sus4
  • 46:54:60:67 Vice Falling Min3 Sus4
  • 46:52:58:60:68:76:79:89:92 Vice Bright Dylathian
  • 46:51:57:60:67:75:78:88:92 Vice Dark Dylathian
  • 46:52:58:60:68:71:79:89:92 Vice Bright Illarnekian
  • 46:51:57:60:67:70:78:88:92 Vice Dark Illarnekian
  • 46:52:54:60:68:71:79:89:92 Vice Bright Celephaïsian
  • 46:51:54:60:67:70:78:88:92 Vice Dark Celephaïsian
  • 46:52:54:60:68:71:79:83:92 Vice Bright Ultharian
  • 46:51:54:60:67:70:78:82:92 Vice Dark Ultharian
  • 46:52:54:60:64:71:79:83:92 Vice Bright Mnarian
  • 46:51:54:60:63:70:78:82:92 Vice Dark Mnarian
  • 46:49:54:60:64:71:79:83:92 Vice Bright Kadathian
  • 46:48:54:60:63:70:78:82:92 Vice Dark Kadathian
  • 46:49:54:60:64:71:75:83:92 Vice Bright Hlanithian
  • 46:48:54:60:63:70:74:82:92 Vice Dark Hlanithian
  • 46:49:54:58:64:71:75:83:92 Vice Bright Sarn
  • 46:48:54:57:63:70:74:82:92 Vice Dark Sarn

/29

  • 29:34:38 Vicesimononal Falling Sus4
  • 29:34:42 Vicenon Falling Minor Triad
  • 29:36:42 Vicenon Falling Major Triad
  • 29:34:40:47 Vicenon Orwell Tetrad
  • 29:38:52 Vicenon Expanding Quartal Triad
  • 29:40:52 Vicenon Contracting Quartal Triad
  • 29:38:65:84:99 Vicenon Oneiro Core Pentad
  • 29:38:65:84:99:110 Vicenon Oneiro Core Hexad
  • 58:65:76:84:99:116 Vicenon Oneiro Falling Suspended Pentatonic
  • 58:65:76:89:99:116 Vicenon Oneiro Rising Suspended Pentatonic
  • 58:72:76:89:99:116 Vicenon Oneiro Symmetrical Pentatonic
  • 58:72:76:89:104:116 Vicenon Oneiro Expanding Quartal Pentatonic
  • 58:72:80:89:104:116 Vicenon Oneiro Diminished Pentatonic
  • 58:65:72:80:84:94:99:110:116 Vicenon Dylydian
  • 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 Illarmixian
  • 58:65:68:76:84:94:99:110:116 Vicenon Celdorian
  • 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 Mnionian
  • 58:65:68:76:80:89:94:104:116 Vicenon Mnaeolian
  • 58:61:68:76:80:89:99:104:116 Vicenon Kadathian
  • 58:61:68:76:84:89:99:104:116 Vicenon Ulphrygian
  • 58:61:68:76:80:89:94:104:116 Vicenon Hlanithian
  • 58:61:68:72:80:89:99:104:116 Vicenon Sardorian
  • 58:61:68:72:80:89:94:104:116 Vicenon Sarnathian
  • 58:61:68:72:80:84:94:104:116 Vicenon Sarlocrian

/47

  • 47:52:55:61:68:72:80:89:94 Quadseptimal Celephaïsian

/61

  • 61:68:72:80:89:93:104:116:122 Sessantunesimal Celephaïsian

Over small prime multiples

Some oneirotonic nejis

The reader is encouraged to tweak these nejis and add more nejis that they like.

13nejis

  1. 58:61:65:68:72:76:80:84:89:94:99:104:110:116 - A low-complexity 13neji; has /13, /17, /19, and /29 prime modes
    • For lower complexity, can use 64 instead of 65 or 100 instead of 99
  2. 92:97:102:108:114:120:127:134:141:149:157:165:174:184 - Vice 13neji

18nejis

  1. 92:96:100:104:108:112:116:120:125:130:136:141:146:152:158:164:170:177:184 - 18neji with /13, /17, /19, /23, and /29 prime modes

21nejis

  1. 128:132:137:141:146:151:156:161:166:172:178:184:190:197:204:210:217:224:232:240:248:256

31nejis

  1. 92:94:96:98:101:103:105:108:110:113:115:118:120:123:126:129:132:135:138:141:144:147:150:154:157:161:165:168:172:176:180:184

34nejis

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

Technical data

Comma list: 196/195, 847/845, 1001/1000

Mapping (for 2, 3, 7/5, 11/5, 13/5): [⟨1 5 2 0 1], ⟨0 -9 -4 3 1]]

Mapping generators: ~2, ~13/10

Optimal ET sequence: 21, 29, 37

Intervals

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

Degree Size in 21edo Size in 29edo Size in 37edo Size in POTE tuning Note name on Q Approximate ratios #Gens up
1 0\21, 0.00 0\29, 0.00 0\37, 0.00 0.00 Q 1/1 0
2 3\21, 171.43 4\29, 165.52 5\37, 163.16 165.65 J 11/10, 10/9 +3
3 6\21, 342.86 8\29, 331.03 10\37, 324.32 331.31 K 11/9, 6/5 +6
4 8\21, 457.14 11\29, 455.17 14\37, 454.05 455.17 L 13/10, 9/7 +1
5 11\21, 628.57 15\29, 620.69 19\37, 616.22 620.87 M 13/9, 10/7 +4
6 14\21, 800.00 19\29, 786.21 23\37, 778.38 786.52 N 11/7 +7
7 16\21, 914.29 22\29, 910.34 28\37, 908.11 910.44 O 22/13 +2
8 19\21, 1085.71 26\29, 1075.86 33\37, 1070.27 1076.09 P 13/7, 28/15 +5

Petrtri

Subgroup: 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 Q Approximate ratios #Gens up
1 0\13, 0.00 0\21, 0.00 0\34, 0.00 0.00 Q 1/1 0
2 2\13, 184.62 3\21, 171.43 5\34, 176.47 177.45 J 10/9, 11/10 +3
3 4\13, 369.23 6\21, 342.86 10\34, 352.94 354.90 K 11/9, 16/13 +6
4 5\13, 461.54 8\21, 457.14 13\34, 458.82 459.15 L 13/10, 17/13, 22/17 +1
5 7\13, 646.15 11\21, 628.57 18\34, 635.294 636.60 M 13/9, 16/11, 23/16 (esp. 21edo) +4
6 9\13, 830.77 14\21, 800.00 23\34, 811.77 814.05 N 8/5 +7
7 10\13, 923.08 16\21, 914.29 26\34, 917.65 918.30 O 17/10 +2
8 12\13, 1107.69 19\21, 1085.71 31\34, 1094.12 1095.75 P 17/9, 32/17, 15/8 +5

A-Team

Subgroup: 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 Q Approximate ratios[1] #Gens up
1 0\13, 0.00 0\18, 0.00 0\31, 0.00 Q 1/1 0
2 2\13, 184.62 3\18, 200.00 5\31, 193.55 J 9/8, 10/9 +3
3 4\13, 369.23 6\18, 400.00 10\31, 387.10 K 5/4 +6
4 5\13, 461.54 7\18, 466.67 12\31, 464.52 L 21/16, 13/10 +1
5 7\13, 646.15 10\18, 666.66 17\31, 658.06 M 13/9, 16/11 +4
6 9\13, 830.77 13\18, 866.66 22\31, 851.61 N 13/8, 18/11 +7
7 10\13, 923.08 14\18, 933.33 24\31, 929.03 O 12/7 +2
8 12\13, 1107.69 17\18, 1133.33 29\31, 1122.58 P +5
  1. ↑ The ratio interpretations that are not valid for 18edo are italicized.

Buzzard

Subgroup: 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 Q Approximate ratios #Gens up
1 0\38, 0.00 0\53, 0.00 0\63, 0.00 0.00 Q 1/1 0
2 7\38, 221.05 10\53, 226.42 12\63, 228.57 227.07 J 8/7 +3
3 14\38, 442.10 20\53, 452.83 24\63, 457.14 453.81 K 13/10, 9/7 +6
4 15\38, 473.68 21\53, 475.47 25\63, 476.19 475.63 L 21/16 +1
5 22\38, 694.73 31\53, 701.89 37\63, 704.76 702.54 M 3/2 +4
6 29\38, 915.78 41\53, 928.30 49\63, 933.33 929.45 N 12/7, 22/13 +7
7 30\38, 947.36 42\53, 950.94 50\63, 952.38 951.27 O 26/15 +2
8 37\38, 1168.42 52\53, 1177.36 62\63, 1180.95 1178.18 P 108/55, 160/81 +5

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

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