5L 3s: Difference between revisions

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TAMNAMS consistency correction
Line 40: Line 40:


== Intervals ==
== 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 large oneirothird is reached by six subfourth generators, [[18edo]]'s large oneirothird is 6*466.67 mod 1200 = 2800 mod 1200 = 400¢, same as the [[12edo]] major third.
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.
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! # generators up
! # generators up
! Notation (1/1 = J)
! Notation (1/1 = J)
! Octatonic interval category name
! [[TAMNAMS]] name
! Abbrev.
! Abbrev.
! # generators up
! # generators up
! Notation of 2/1 inverse  
! Notation of 2/1 inverse  
! Octatonic interval category name
! [[TAMNAMS]] name
! Abbrev.
! Abbrev.
|-
|-
Line 66: Line 66:
| M
| M
| perfect oneirofourth (aka minor fourth, falling fourth)
| perfect oneirofourth (aka minor fourth, falling fourth)
| Po4
| Pon4
| -1
| -1
| O
| O
| perfect oneirosixth (aka major fifth, rising fifth)
| perfect oneirosixth (aka major fifth, rising fifth)
| Po6
| Pon6
|-
|-
| 2
| 2
| P
| P
| large oneiroseventh
| major oneiroseventh
| Lo7
| Lon7
| -2
| -2
| L
| L
| small oneirothird
| minor oneirothird
| so3
| son3
|-
|-
| 3
| 3
| K
| K
| large oneirosecond
| major oneirosecond
| Lo2
| Lon2
| -3
| -3
| Q
| Q
| small oneiroeighth
| minor oneiroeighth
| so8
| son8
|-
|-
| 4
| 4
| N
| N
| large oneirofifth (aka minor fifth, falling fifth)
| major oneirofifth (aka minor fifth, falling fifth)
| Lo5
| Lon5
| -4
| -4
| N@
| N@
| small oneirofifth (aka major fourth, rising fourth)
| minor oneirofifth (aka major fourth, rising fourth)
| so4
| son4
|-
|-
| 5
| 5
| Q&
| Q&
| large oneiroeighth
| major oneiroeighth
| Lo8
| Lon8
| -5
| -5
| K@
| K@
| small oneirosecond
| minor oneirosecond
| so2
| son2
|-
|-
| 6
| 6
| L&
| L&
| large oneirothird
| major oneirothird
| Lo3
| Lon3
| -6
| -6
| P@
| P@
| small oneiroseventh
| minor oneiroseventh
| so7
| son7
|-
|-
| 7
| 7
| O&
| O&
| augmented oneirosixth
| augmented oneirosixth
| Ao6
| Aon6
| -7
| -7
| M@
| M@
| diminished oneirofourth
| diminished oneirofourth
| d-o4
| d-on4
|-
|-
| colspan="8" style="text-align:center" | The chromatic 13-note MOS (either [[5L 8s]], [[8L 5s]], or [[13edo]]) also has the following intervals (from some root):
| colspan="8" style="text-align:center" | The chromatic 13-note MOS (either [[5L 8s]], [[8L 5s]], or [[13edo]]) also has the following intervals (from some root):
Line 140: Line 140:
| M&
| M&
| augmented oneirofourth
| augmented oneirofourth
| Ao4
| Aon4
| -9
| -9
| O@
| O@
| diminished oneirosixth
| diminished oneirosixth
| d-o6
| d-on6
|-
|-
| 10
| 10
| P&
| P&
| augmented oneiroseventh
| augmented oneiroseventh
| Ao7
| Aon7
| -10
| -10
| L@
| L@
| diminished oneirothird
| diminished oneirothird
| d-o3
| d-on3
|-
|-
| 11
| 11
| K&
| K&
| augmented oneirosecond
| augmented oneirosecond
| Ao2
| Aon2
| -11
| -11
| Q@
| Q@
| diminished oneiroeighth
| diminished oneiroeighth
| d-o8
| d-on8
|-
|-
| 12
| 12
| N&
| N&
| augmented oneirofifth
| augmented oneirofifth
| Ao5
| Aon5
| -12
| -12
| N@@
| N@@
| diminished oneirofifth
| diminished oneirofifth
| d-o5
| d-on5
|}
|}


Line 233: Line 233:
| 0
| 0
|-
|-
| small o2nd
| minor on2nd
| 1\13, 92.31
| 1\13, 92.31
| 1\18, 66.67
| 1\18, 66.67
Line 241: Line 241:
| -5
| -5
|-
|-
| large o2nd
| major on2nd
| 2\13, 184.62
| 2\13, 184.62
| 3\18, 200.00
| 3\18, 200.00
Line 249: Line 249:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| small o3rd
| minor on3rd
| 3\13, 276.92
| 3\13, 276.92
| 4\18, 266.67
| 4\18, 266.67
Line 257: Line 257:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| large o3rd
| major on3rd
| 4\13, 369.23
| 4\13, 369.23
| 6\18, 400.00
| 6\18, 400.00
Line 265: Line 265:
| +6
| +6
|-
|-
| dim. o4th
| dim. on4th
| 4\13, 369.23
| 4\13, 369.23
| 5\18, 333.33
| 5\18, 333.33
Line 273: Line 273:
| -7
| -7
|-
|-
| perf. o4th
| perf. on4th
| 5\13, 461.54  
| 5\13, 461.54  
| 7\18, 466.67
| 7\18, 466.67
Line 281: Line 281:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| small o5th
| minor on5th
| 6\13, 553.85
| 6\13, 553.85
| 8\18, 533.33
| 8\18, 533.33
Line 289: Line 289:
| -4
| -4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| large o5th
| major on5th
| 7\13, 646.15
| 7\13, 646.15
| 10\18, 666.66
| 10\18, 666.66
Line 297: Line 297:
| +4
| +4
|-
|-
| perf. o6th
| perf. on6th
| 8\13, 738.46
| 8\13, 738.46
| 11\18, 733.33
| 11\18, 733.33
Line 305: Line 305:
| -1
| -1
|-
|-
| aug. o6th
| aug. on6th
| 9\13, 830.77
| 9\13, 830.77
| 13\18, 866.66
| 13\18, 866.66
Line 313: Line 313:
| +7
| +7
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| small o7th
| minor on7th
| 9\13, 830.77
| 9\13, 830.77
| 12\18, 800.00
| 12\18, 800.00
Line 321: Line 321:
| -6
| -6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| large o7th
| major on7th
| 10\13, 923.08
| 10\13, 923.08
| 14\18, 933.33
| 14\18, 933.33
Line 329: Line 329:
| +2
| +2
|-
|-
| small o8th
| minor on8th
| 11\13, 1015.39
| 11\13, 1015.39
| 15\18, 1000.00
| 15\18, 1000.00
Line 337: Line 337:
| -3
| -3
|-
|-
| large o8th
| major on8th
| 12\13, 1107.69
| 12\13, 1107.69
| 17\18, 1133.33
| 17\18, 1133.33
Line 351: Line 351:
* 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¢).
* 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-Lo2-Lo3-Lo5 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¢).
* [[21edo]]'s P1-Lon2-Lon3-Lon5 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.
* [[34edo]]'s 9:10:11:13 is even better.


Line 393: Line 393:
| 0
| 0
|-
|-
| small o2nd
| minor on2nd
| 2\21, 114.29
| 2\21, 114.29
| 3\34, 105.88
| 3\34, 105.88
Line 400: Line 400:
| -5
| -5
|-
|-
| large o2nd
| major on2nd
| 3\21, 171.43
| 3\21, 171.43
| 5\34, 176.47
| 5\34, 176.47
Line 407: Line 407:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| small o3rd
| minor on3rd
| 5\21, 285.71
| 5\21, 285.71
| 8\34, 282.35
| 8\34, 282.35
Line 414: Line 414:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| large o3rd
| major on3rd
| 6\21, 342.86
| 6\21, 342.86
| 10\34, 352.94
| 10\34, 352.94
Line 421: Line 421:
| +6
| +6
|-
|-
| dim. o4th
| dim. on4th
| 7\21, 400.00
| 7\21, 400.00
| 11\34, 388.24
| 11\34, 388.24
Line 428: Line 428:
| -7
| -7
|-
|-
| perf. o4th
| perf. on4th
| 7\18, 457.14
| 7\18, 457.14
| 12\31, 458.82
| 12\31, 458.82
Line 435: Line 435:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| small o5th
| minor on5th
| 10\21, 571.43
| 10\21, 571.43
| 16\34, 564.72
| 16\34, 564.72
Line 442: Line 442:
| -4
| -4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| large o5th
| major on5th
| 11\21, 628.57
| 11\21, 628.57
| 18\34, 635.29
| 18\34, 635.29
Line 449: Line 449:
| +4
| +4
|-
|-
| perf. o6th
| perf. on6th
| 13\21, 742.86
| 13\21, 742.86
| 21\34, 741.18
| 21\34, 741.18
Line 456: Line 456:
| -1
| -1
|-
|-
| aug. o6th
| aug. on6th
| 14\21, 800.00
| 14\21, 800.00
| 23\34, 811.77
| 23\34, 811.77
Line 463: Line 463:
| +7
| +7
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| small o7th
| minor on7th
| 15\21, 857.14
| 15\21, 857.14
| 24\34, 847.06
| 24\34, 847.06
Line 470: Line 470:
| -6
| -6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| large o7th
| major on7th
| 16\21, 914.29
| 16\21, 914.29
| 26\34, 917.65
| 26\34, 917.65
Line 477: Line 477:
| +2
| +2
|-
|-
| small o8th
| minor on8th
| 18\21, 1028.57
| 18\21, 1028.57
| 29\34, 1023.53
| 29\34, 1023.53
Line 484: Line 484:
| -3
| -3
|-
|-
| large o8th
| major on8th
| 19\21, 1085.71
| 19\21, 1085.71
| 31\34, 1094.12
| 31\34, 1094.12
Line 536: Line 536:
| +8
| +8
|-
|-
| dim. o2nd
| dim. on2nd
| 2\29, 82.8
| 2\29, 82.8
| K@@
| K@@
Line 542: Line 542:
| -13
| -13
|-
|-
| small o2nd
| minor on2nd
| 3\29, 124.1
| 3\29, 124.1
| K@
| K@
Line 548: Line 548:
| -5
| -5
|-
|-
| large o2nd
| major on2nd
| 4\29, 165.5
| 4\29, 165.5
| K
| K
Line 554: Line 554:
| +3
| +3
|-
|-
| aug. o2nd
| aug. on2nd
| 5\29, 206.9
| 5\29, 206.9
| K&
| K&
Line 560: Line 560:
| +11
| +11
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. o3rd
| dim. on3rd
| 6\29, 248.3
| 6\29, 248.3
| L@
| L@
Line 566: Line 566:
| -10
| -10
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| small o3rd
| minor on3rd
| 7\29, 289.7
| 7\29, 289.7
| L
| L
Line 572: Line 572:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| large o3rd
| major on3rd
| 8\29, 331.0
| 8\29, 331.0
| L&
| L&
Line 578: Line 578:
| +6
| +6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. o3rd
| aug. on3rd
| 9\29, 372.4
| 9\29, 372.4
| L&&
| L&&
Line 584: Line 584:
| +14
| +14
|-
|-
| doubly dim. o4th
| doubly dim. on4th
| 9\29, 372.4
| 9\29, 372.4
| M@@
| M@@
Line 590: Line 590:
| -15
| -15
|-
|-
| dim. o4th
| dim. on4th
| 10\29, 413.8
| 10\29, 413.8
| M@
| M@
Line 596: Line 596:
| -7
| -7
|-
|-
| perf. o4th
| perf. on4th
| 11\29, 455.2
| 11\29, 455.2
| M
| M
Line 602: Line 602:
| +1
| +1
|-
|-
| aug. o4th
| aug. on4th
| 12\29, 496.6
| 12\29, 496.6
| M&
| M&
Line 608: Line 608:
| +9
| +9
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. o5th
| dim. on5th
| 13\29, 537.9
| 13\29, 537.9
| N@@
| N@@
Line 614: Line 614:
| -12
| -12
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| small o5th
| minor on5th
| 14\29, 579.3
| 14\29, 579.3
| N@
| N@
Line 620: Line 620:
| -4
| -4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| large o5th
| major on5th
| 15\29 620.7
| 15\29 620.7
| N
| N
Line 626: Line 626:
| +4
| +4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. o5th
| aug. on5th
| 16\29 662.1
| 16\29 662.1
| N&
| N&
Line 632: Line 632:
| +12
| +12
|-
|-
| dim. o6th
| dim. on6th
| 17\29, 703.4
| 17\29, 703.4
| O@
| O@
Line 638: Line 638:
| -9
| -9
|-
|-
| perf. o6th
| perf. on6th
| 18\29, 755.2
| 18\29, 755.2
| O
| O
Line 644: Line 644:
| -1
| -1
|-
|-
| aug. o6th
| aug. on6th
| 19\29, 786.2
| 19\29, 786.2
| O&
| O&
Line 650: Line 650:
| +7
| +7
|-
|-
| doubly aug. o6th
| doubly aug. on6th
| 20\29 827.6
| 20\29 827.6
| O&&
| O&&
Line 656: Line 656:
| +15
| +15
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. o7th
| dim. on7th
| 20\29 827.6
| 20\29 827.6
| P@@
| P@@
Line 662: Line 662:
| -14
| -14
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| small o7th
| minor on7th
| 21\29 869.0
| 21\29 869.0
| P@
| P@
Line 668: Line 668:
| -6
| -6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| large o7th
| major on7th
| 22\29, 910.3
| 22\29, 910.3
| P
| P
Line 674: Line 674:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. o7th
| aug. on7th
| 23\29, 951.7
| 23\29, 951.7
| P&
| P&
Line 680: Line 680:
| +10
| +10
|-
|-
| dim. o8th
| dim. on8th
| 24\29, 993.1
| 24\29, 993.1
| Q@
| Q@
Line 686: Line 686:
| -11
| -11
|-
|-
| small o8th
| minor on8th
| 25\29, 1034.5
| 25\29, 1034.5
| Q
| Q
Line 692: Line 692:
| -3
| -3
|-
|-
| large o8th
| major on8th
| 26\29, 1075.9
| 26\29, 1075.9
| Q&
| Q&
Line 698: Line 698:
| +5
| +5
|-
|-
| aug. o8th
| aug. on8th
| 27\29, 1117.2
| 27\29, 1117.2
| Q&&
| Q&&

Revision as of 23:36, 17 April 2021

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 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).

5L 3s 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 mode LLsLLsLs.

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

5L 3s has a pentatonic MOS subset 3L 2s (SLSLL). (Note: 3L 5s scales also have 3L 2s subsets.)

Names

The TAMNAMS system, used by this article, uses the name oneirotonic (/oʊnaɪrəˈtɒnɪk/ oh-ny-rə-TON-ik or /ənaɪrə-/ ə-ny-rə-) or 'oneiro' for short. The name oneirotonic (from Greek oneiros 'dream') is coined after the Dreamlands in H.P. Lovecraft's Dream Cycle mythos.

'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as father is technically an abstract regular temperament, not a generator range. A more correct way to say it would be 'father[8]' or 'father octatonic'. "Father" is also vague because optimal generators for it also generate 3L 2s.

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) TAMNAMS name Abbrev. # generators up Notation of 2/1 inverse TAMNAMS 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) Pon4 -1 O perfect oneirosixth (aka major fifth, rising fifth) Pon6
2 P major oneiroseventh Lon7 -2 L minor oneirothird son3
3 K major oneirosecond Lon2 -3 Q minor oneiroeighth son8
4 N major oneirofifth (aka minor fifth, falling fifth) Lon5 -4 N@ minor oneirofifth (aka major fourth, rising fourth) son4
5 Q& major oneiroeighth Lon8 -5 K@ minor oneirosecond son2
6 L& major oneirothird Lon3 -6 P@ minor oneiroseventh son7
7 O& augmented oneirosixth Aon6 -7 M@ diminished oneirofourth d-on4
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 Aon4 -9 O@ diminished oneirosixth d-on6
10 P& augmented oneiroseventh Aon7 -10 L@ diminished oneirothird d-on3
11 K& augmented oneirosecond Aon2 -11 Q@ diminished oneiroeighth d-on8
12 N& augmented oneirofifth Aon5 -12 N@@ diminished oneirofifth d-on5

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
minor on2nd 1\13, 92.31 1\18, 66.67 2\31, 77.42 K@ 21/20, 22/21 -5
major on2nd 2\13, 184.62 3\18, 200.00 5\31, 193.55 K 9/8, 10/9 +3
minor on3rd 3\13, 276.92 4\18, 266.67 7\31, 270.97 L 7/6 -2
major on3rd 4\13, 369.23 6\18, 400.00 10\31, 387.10 L& 5/4 +6
dim. on4th 4\13, 369.23 5\18, 333.33 9\31, 348.39 M@ 16/13, 11/9 -7
perf. on4th 5\13, 461.54 7\18, 466.67 12\31, 464.52 M 21/16, 13/10, 17/13 +1
minor on5th 6\13, 553.85 8\18, 533.33 14\31, 541.94 N@ 11/8 -4
major on5th 7\13, 646.15 10\18, 666.66 17\31, 658.06 N 13/9, 16/11 +4
perf. on6th 8\13, 738.46 11\18, 733.33 19\31, 735.48 O 26/17 -1
aug. on6th 9\13, 830.77 13\18, 866.66 22\31, 851.61 O& 13/8, 18/11 +7
minor on7th 9\13, 830.77 12\18, 800.00 21\31, 812.90 P@ 8/5 -6
major on7th 10\13, 923.08 14\18, 933.33 24\31, 929.03 P 12/7 +2
minor on8th 11\13, 1015.39 15\18, 1000.00 26\31, 1006.45 Q 9/5, 16/9 -3
major on8th 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-Lon2-Lon3-Lon5 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
minor on2nd 2\21, 114.29 3\34, 105.88 K@ 16/15 -5
major on2nd 3\21, 171.43 5\34, 176.47 K 10/9, 11/10 +3
minor on3rd 5\21, 285.71 8\34, 282.35 L 13/11, 20/17 -2
major on3rd 6\21, 342.86 10\34, 352.94 L& 11/9 +6
dim. on4th 7\21, 400.00 11\34, 388.24 M@ 5/4 -7
perf. on4th 7\18, 457.14 12\31, 458.82 M 13/10 +1
minor on5th 10\21, 571.43 16\34, 564.72 N@ 18/13, 32/23 -4
major on5th 11\21, 628.57 18\34, 635.29 N 13/9, 23/16 +4
perf. on6th 13\21, 742.86 21\34, 741.18 O 20/13 -1
aug. on6th 14\21, 800.00 23\34, 811.77 O& 8/5 +7
minor on7th 15\21, 857.14 24\34, 847.06 P@ 18/11 -6
major on7th 16\21, 914.29 26\34, 917.65 P 22/13, 17/10 +2
minor on8th 18\21, 1028.57 29\34, 1023.53 Q 9/5 -3
major on8th 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. on2nd 2\29, 82.8 K@@ -13
minor on2nd 3\29, 124.1 K@ 14/13 -5
major on2nd 4\29, 165.5 K 11/10 +3
aug. on2nd 5\29, 206.9 K& 9/8 +11
dim. on3rd 6\29, 248.3 L@ 15/13 -10
minor on3rd 7\29, 289.7 L 13/11 -2
major on3rd 8\29, 331.0 L& +6
aug. on3rd 9\29, 372.4 L&& +14
doubly dim. on4th 9\29, 372.4 M@@ -15
dim. on4th 10\29, 413.8 M@ 14/11 -7
perf. on4th 11\29, 455.2 M 13/10 +1
aug. on4th 12\29, 496.6 M& 4/3 +9
dim. on5th 13\29, 537.9 N@@ 15/11 -12
minor on5th 14\29, 579.3 N@ 7/5 -4
major on5th 15\29 620.7 N 10/7 +4
aug. on5th 16\29 662.1 N& 22/15 +12
dim. on6th 17\29, 703.4 O@ 3/2 -9
perf. on6th 18\29, 755.2 O 20/13 -1
aug. on6th 19\29, 786.2 O& 11/7 +7
doubly aug. on6th 20\29 827.6 O&& +15
dim. on7th 20\29 827.6 P@@ -14
minor on7th 21\29 869.0 P@ -6
major on7th 22\29, 910.3 P 22/13 +2
aug. on7th 23\29, 951.7 P& 26/15 +10
dim. on8th 24\29, 993.1 Q@ 16/9 -11
minor on8th 25\29, 1034.5 Q 20/11 -3
major on8th 26\29, 1075.9 Q& 13/7 +5
aug. on8th 27\29, 1117.2 Q&& +13
dim. o9th 28\29, 1158.6 J@ -8

Approaches

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 Cents L s L/s Comments
3\8 450.000 1 1 1.000
17\45 453.333 6 5 1.200
14\37 454.054 5 4 1.250
34\59 454.545 9 7 1.286
11\29 455.172 4 3 1.333
30\79 455.696 11 8 1.375
19\50 456.000 7 5 1.400
27\71 456.338 10 7 1.429
8\21 457.143 3 2 1.500 L/s = 3/2
29\76 457.895 11 7 1.571
21\55 458.182 8 5 1.600
34\89 458.427 13 8 1.625 Golden oneirotonic
13\34 458.824 5 3 1.667
31\81 459.259 12 7 1.714
18\47 459.574 7 4 1.750
23\60 460.000 9 5 1.800
5\13 461.538 2 1 2.000 Basic oneirotonic
(generators smaller than this are proper)
22\57 463.158 9 4 2.250
17\44 463.636 7 3 2.333
29\75 464.000 12 5 2.400
12\31 464.516 5 2 2.500
31\80 465.000 13 5 2.600
19\49 465.306 8 3 2.667
26\67 465.672 11 4 2.750
7\18 466.667 3 1 3.000 L/s = 3/1
23\59 467.797 10 3 3.333
16\41 468.293 7 2 3.500
25\64 468.750 11 3 3.667
9\23 469.565 4 1 4.000
20\51 470.588 9 2 4.500
11\28 471.429 5 1 5.000
13\33 472.727 6 1 6.000
2\5 480.000 1 0 → inf
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