5L 3s: Difference between revisions

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remove key sig section; standardize to "mos-" instead of "oneiro-"
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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 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& ...
The chain of perfect mosfourths becomes: ... P@ K@ N@ Q L O J M P K N Q& L& O& ...


Thus the [[13edo]] gamut is as follows:  
Thus the [[13edo]] gamut is as follows:  
Line 43: Line 43:


== 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 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.
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 mosthird is reached by six subfourth generators, [[18edo]]'s major mosthird is 6*466.67 mod 1200 = 2800 mod 1200 = 400¢, same as the [[12edo]] major third.
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
Line 68: Line 68:
| 1
| 1
| M
| M
| perfect oneirofourth (aka minor fourth, falling fourth)
| perfect mosfourth (aka minor fourth, falling fourth)
| Pon4
| Pmos4
| -1
| -1
| O
| O
| perfect oneirosixth (aka major fifth, rising fifth)
| perfect mossixth (aka major fifth, rising fifth)
| Pon6
| Pmos6
|-
|-
| 2
| 2
| P
| P
| major oneiroseventh
| major mosseventh
| Lon7
| Lmos7
| -2
| -2
| L
| L
| minor oneirothird
| minor mosthird
| son3
| smos3
|-
|-
| 3
| 3
| K
| K
| major oneirosecond
| major mossecond
| Lon2
| Lmos2
| -3
| -3
| Q
| Q
| minor oneiroeighth
| minor moseighth
| son8
| smos8
|-
|-
| 4
| 4
| N
| N
| major oneirofifth (aka minor fifth, falling fifth)
| major mosfifth (aka minor fifth, falling fifth)
| Lon5
| Lmos5
| -4
| -4
| N@
| N@
| minor oneirofifth (aka major fourth, rising fourth)
| minor mosfifth (aka major fourth, rising fourth)
| son4
| smos4
|-
|-
| 5
| 5
| Q&
| Q&
| major oneiroeighth
| major moseighth
| Lon8
| Lmos8
| -5
| -5
| K@
| K@
| minor oneirosecond
| minor mossecond
| son2
| smos2
|-
|-
| 6
| 6
| L&
| L&
| major oneirothird
| major mosthird
| Lon3
| Lmos3
| -6
| -6
| P@
| P@
| minor oneiroseventh
| minor mosseventh
| son7
| smos7
|-
|-
| 7
| 7
| O&
| O&
| augmented oneirosixth
| augmented mossixth
| Aon6
| Amos6
| -7
| -7
| M@
| M@
| diminished oneirofourth
| diminished mosfourth
| d-on4
| d-mos4
|-
|-
| 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 133: Line 133:
| 8
| 8
| J&
| J&
| augmented oneirounison (aka oneirochroma)
| augmented mosunison (aka oneirochroma)
| Ao1
| Ao1
| -8
| -8
| J@
| J@
| diminished oneirooctave (aka diminished oneironinth)
| diminished mosoctave (aka diminished mosninth)
| d-o9
| d-o9
|-
|-
| 9
| 9
| M&
| M&
| augmented oneirofourth
| augmented mosfourth
| Aon4
| Amos4
| -9
| -9
| O@
| O@
| diminished oneirosixth
| diminished mossixth
| d-on6
| d-mos6
|-
|-
| 10
| 10
| P&
| P&
| augmented oneiroseventh
| augmented mosseventh
| Aon7
| Amos7
| -10
| -10
| L@
| L@
| diminished oneirothird
| diminished mosthird
| d-on3
| d-mos3
|-
|-
| 11
| 11
| K&
| K&
| augmented oneirosecond
| augmented mossecond
| Aon2
| Amos2
| -11
| -11
| Q@
| Q@
| diminished oneiroeighth
| diminished moseighth
| d-on8
| d-mos8
|-
|-
| 12
| 12
| N&
| N&
| augmented oneirofifth
| augmented mosfifth
| Aon5
| Amos5
| -12
| -12
| N@@
| N@@
| diminished oneirofifth
| diminished mosfifth
| d-on5
| d-mos5
|}
|}


Line 240: Line 240:
| 0
| 0
|-
|-
| minor on2nd
| minor mos2nd
| 1\13, 92.31
| 1\13, 92.31
| 1\18, 66.67
| 1\18, 66.67
Line 248: Line 248:
| -5
| -5
|-
|-
| major on2nd
| major mos2nd
| 2\13, 184.62
| 2\13, 184.62
| 3\18, 200.00
| 3\18, 200.00
Line 256: Line 256:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor on3rd
| minor mos3rd
| 3\13, 276.92
| 3\13, 276.92
| 4\18, 266.67
| 4\18, 266.67
Line 264: Line 264:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major on3rd
| major mos3rd
| 4\13, 369.23
| 4\13, 369.23
| 6\18, 400.00
| 6\18, 400.00
Line 272: Line 272:
| +6
| +6
|-
|-
| dim. on4th
| dim. mos4th
| 4\13, 369.23
| 4\13, 369.23
| 5\18, 333.33
| 5\18, 333.33
Line 280: Line 280:
| -7
| -7
|-
|-
| perf. on4th
| perf. mos4th
| 5\13, 461.54  
| 5\13, 461.54  
| 7\18, 466.67
| 7\18, 466.67
Line 288: Line 288:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor on5th
| minor mos5th
| 6\13, 553.85
| 6\13, 553.85
| 8\18, 533.33
| 8\18, 533.33
Line 296: Line 296:
| -4
| -4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major on5th
| major mos5th
| 7\13, 646.15
| 7\13, 646.15
| 10\18, 666.66
| 10\18, 666.66
Line 304: Line 304:
| +4
| +4
|-
|-
| perf. on6th
| perf. mos6th
| 8\13, 738.46
| 8\13, 738.46
| 11\18, 733.33
| 11\18, 733.33
Line 312: Line 312:
| -1
| -1
|-
|-
| aug. on6th
| aug. mos6th
| 9\13, 830.77
| 9\13, 830.77
| 13\18, 866.66
| 13\18, 866.66
Line 320: Line 320:
| +7
| +7
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor on7th
| minor mos7th
| 9\13, 830.77
| 9\13, 830.77
| 12\18, 800.00
| 12\18, 800.00
Line 328: Line 328:
| -6
| -6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major on7th
| major mos7th
| 10\13, 923.08
| 10\13, 923.08
| 14\18, 933.33
| 14\18, 933.33
Line 336: Line 336:
| +2
| +2
|-
|-
| minor on8th
| minor mos8th
| 11\13, 1015.39
| 11\13, 1015.39
| 15\18, 1000.00
| 15\18, 1000.00
Line 344: Line 344:
| -3
| -3
|-
|-
| major on8th
| major mos8th
| 12\13, 1107.69
| 12\13, 1107.69
| 17\18, 1133.33
| 17\18, 1133.33
Line 359: Line 359:
* 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-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¢).
* [[21edo]]'s P1-Lmos2-Lmos3-Lmos5 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 403: Line 403:
| 0
| 0
|-
|-
| minor on2nd
| minor mos2nd
| 2\21, 114.29
| 2\21, 114.29
| 3\34, 105.88
| 3\34, 105.88
Line 410: Line 410:
| -5
| -5
|-
|-
| major on2nd
| major mos2nd
| 3\21, 171.43
| 3\21, 171.43
| 5\34, 176.47
| 5\34, 176.47
Line 417: Line 417:
| +3
| +3
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor on3rd
| minor mos3rd
| 5\21, 285.71
| 5\21, 285.71
| 8\34, 282.35
| 8\34, 282.35
Line 424: Line 424:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major on3rd
| major mos3rd
| 6\21, 342.86
| 6\21, 342.86
| 10\34, 352.94
| 10\34, 352.94
Line 431: Line 431:
| +6
| +6
|-
|-
| dim. on4th
| dim. mos4th
| 7\21, 400.00
| 7\21, 400.00
| 11\34, 388.24
| 11\34, 388.24
Line 438: Line 438:
| -7
| -7
|-
|-
| perf. on4th
| perf. mos4th
| 7\18, 457.14
| 7\18, 457.14
| 12\31, 458.82
| 12\31, 458.82
Line 445: Line 445:
| +1
| +1
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor on5th
| minor mos5th
| 10\21, 571.43
| 10\21, 571.43
| 16\34, 564.72
| 16\34, 564.72
Line 452: Line 452:
| -4
| -4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major on5th
| major mos5th
| 11\21, 628.57
| 11\21, 628.57
| 18\34, 635.29
| 18\34, 635.29
Line 459: Line 459:
| +4
| +4
|-
|-
| perf. on6th
| perf. mos6th
| 13\21, 742.86
| 13\21, 742.86
| 21\34, 741.18
| 21\34, 741.18
Line 466: Line 466:
| -1
| -1
|-
|-
| aug. on6th
| aug. mos6th
| 14\21, 800.00
| 14\21, 800.00
| 23\34, 811.77
| 23\34, 811.77
Line 473: Line 473:
| +7
| +7
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor on7th
| minor mos7th
| 15\21, 857.14
| 15\21, 857.14
| 24\34, 847.06
| 24\34, 847.06
Line 480: Line 480:
| -6
| -6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major on7th
| major mos7th
| 16\21, 914.29
| 16\21, 914.29
| 26\34, 917.65
| 26\34, 917.65
Line 487: Line 487:
| +2
| +2
|-
|-
| minor on8th
| minor mos8th
| 18\21, 1028.57
| 18\21, 1028.57
| 29\34, 1023.53
| 29\34, 1023.53
Line 494: Line 494:
| -3
| -3
|-
|-
| major on8th
| major mos8th
| 19\21, 1085.71
| 19\21, 1085.71
| 31\34, 1094.12
| 31\34, 1094.12
Line 546: Line 546:
| +8
| +8
|-
|-
| dim. on2nd
| dim. mos2nd
| 2\29, 82.8
| 2\29, 82.8
| K@@
| K@@
Line 552: Line 552:
| -13
| -13
|-
|-
| minor on2nd
| minor mos2nd
| 3\29, 124.1
| 3\29, 124.1
| K@
| K@
Line 558: Line 558:
| -5
| -5
|-
|-
| major on2nd
| major mos2nd
| 4\29, 165.5
| 4\29, 165.5
| K
| K
Line 564: Line 564:
| +3
| +3
|-
|-
| aug. on2nd
| aug. mos2nd
| 5\29, 206.9
| 5\29, 206.9
| K&
| K&
Line 570: Line 570:
| +11
| +11
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. on3rd
| dim. mos3rd
| 6\29, 248.3
| 6\29, 248.3
| L@
| L@
Line 576: Line 576:
| -10
| -10
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor on3rd
| minor mos3rd
| 7\29, 289.7
| 7\29, 289.7
| L
| L
Line 582: Line 582:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major on3rd
| major mos3rd
| 8\29, 331.0
| 8\29, 331.0
| L&
| L&
Line 588: Line 588:
| +6
| +6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. on3rd
| aug. mos3rd
| 9\29, 372.4
| 9\29, 372.4
| L&&
| L&&
Line 594: Line 594:
| +14
| +14
|-
|-
| doubly dim. on4th
| doubly dim. mos4th
| 9\29, 372.4
| 9\29, 372.4
| M@@
| M@@
Line 600: Line 600:
| -15
| -15
|-
|-
| dim. on4th
| dim. mos4th
| 10\29, 413.8
| 10\29, 413.8
| M@
| M@
Line 606: Line 606:
| -7
| -7
|-
|-
| perf. on4th
| perf. mos4th
| 11\29, 455.2
| 11\29, 455.2
| M
| M
Line 612: Line 612:
| +1
| +1
|-
|-
| aug. on4th
| aug. mos4th
| 12\29, 496.6
| 12\29, 496.6
| M&
| M&
Line 618: Line 618:
| +9
| +9
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. on5th
| dim. mos5th
| 13\29, 537.9
| 13\29, 537.9
| N@@
| N@@
Line 624: Line 624:
| -12
| -12
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor on5th
| minor mos5th
| 14\29, 579.3
| 14\29, 579.3
| N@
| N@
Line 630: Line 630:
| -4
| -4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major on5th
| major mos5th
| 15\29 620.7
| 15\29 620.7
| N
| N
Line 636: Line 636:
| +4
| +4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. on5th
| aug. mos5th
| 16\29 662.1
| 16\29 662.1
| N&
| N&
Line 642: Line 642:
| +12
| +12
|-
|-
| dim. on6th
| dim. mos6th
| 17\29, 703.4
| 17\29, 703.4
| O@
| O@
Line 648: Line 648:
| -9
| -9
|-
|-
| perf. on6th
| perf. mos6th
| 18\29, 755.2
| 18\29, 755.2
| O
| O
Line 654: Line 654:
| -1
| -1
|-
|-
| aug. on6th
| aug. mos6th
| 19\29, 786.2
| 19\29, 786.2
| O&
| O&
Line 660: Line 660:
| +7
| +7
|-
|-
| doubly aug. on6th
| doubly aug. mos6th
| 20\29 827.6
| 20\29 827.6
| O&&
| O&&
Line 666: Line 666:
| +15
| +15
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. on7th
| dim. mos7th
| 20\29 827.6
| 20\29 827.6
| P@@
| P@@
Line 672: Line 672:
| -14
| -14
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor on7th
| minor mos7th
| 21\29 869.0
| 21\29 869.0
| P@
| P@
Line 678: Line 678:
| -6
| -6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major on7th
| major mos7th
| 22\29, 910.3
| 22\29, 910.3
| P
| P
Line 684: Line 684:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. on7th
| aug. mos7th
| 23\29, 951.7
| 23\29, 951.7
| P&
| P&
Line 690: Line 690:
| +10
| +10
|-
|-
| dim. on8th
| dim. mos8th
| 24\29, 993.1
| 24\29, 993.1
| Q@
| Q@
Line 696: Line 696:
| -11
| -11
|-
|-
| minor on8th
| minor mos8th
| 25\29, 1034.5
| 25\29, 1034.5
| Q
| Q
Line 702: Line 702:
| -3
| -3
|-
|-
| major on8th
| major mos8th
| 26\29, 1075.9
| 26\29, 1075.9
| Q&
| Q&
Line 708: Line 708:
| +5
| +5
|-
|-
| aug. on8th
| aug. mos8th
| 27\29, 1117.2
| 27\29, 1117.2
| Q&&
| Q&&
Line 845: Line 845:
| 108/55, 160/81
| 108/55, 160/81
| +5
| +5
|}
|}-->
 
== 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 ==
== Modes ==
Oneirotonic modes are named after cities in the Dreamlands.
Oneirotonic modes are named after cities in the Dreamlands.

Revision as of 00:12, 18 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.)

Standing assumptions

The TAMNAMS system is used in this article to name 5L 3s intervals and step size ratios and step ratio ranges.

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 perfect mosfourths 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

Names

The TAMNAMS system suggests 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.

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 mosthird is reached by six subfourth generators, 18edo's major mosthird 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 mosfourth (aka minor fourth, falling fourth) Pmos4 -1 O perfect mossixth (aka major fifth, rising fifth) Pmos6
2 P major mosseventh Lmos7 -2 L minor mosthird smos3
3 K major mossecond Lmos2 -3 Q minor moseighth smos8
4 N major mosfifth (aka minor fifth, falling fifth) Lmos5 -4 N@ minor mosfifth (aka major fourth, rising fourth) smos4
5 Q& major moseighth Lmos8 -5 K@ minor mossecond smos2
6 L& major mosthird Lmos3 -6 P@ minor mosseventh smos7
7 O& augmented mossixth Amos6 -7 M@ diminished mosfourth d-mos4
The chromatic 13-note MOS (either 5L 8s, 8L 5s, or 13edo) also has the following intervals (from some root):
8 J& augmented mosunison (aka oneirochroma) Ao1 -8 J@ diminished mosoctave (aka diminished mosninth) d-o9
9 M& augmented mosfourth Amos4 -9 O@ diminished mossixth d-mos6
10 P& augmented mosseventh Amos7 -10 L@ diminished mosthird d-mos3
11 K& augmented mossecond Amos2 -11 Q@ diminished moseighth d-mos8
12 N& augmented mosfifth Amos5 -12 N@@ diminished mosfifth d-mos5

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.

Also, in 18edo and 31edo, the minor mosthird is close to 7/6.

The set of identifications above is associated with A-Team temperament.

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 mos2nd 1\13, 92.31 1\18, 66.67 2\31, 77.42 K@ 21/20, 22/21 -5
major mos2nd 2\13, 184.62 3\18, 200.00 5\31, 193.55 K 9/8, 10/9 +3
minor mos3rd 3\13, 276.92 4\18, 266.67 7\31, 270.97 L 7/6 -2
major mos3rd 4\13, 369.23 6\18, 400.00 10\31, 387.10 L& 5/4 +6
dim. mos4th 4\13, 369.23 5\18, 333.33 9\31, 348.39 M@ 16/13, 11/9 -7
perf. mos4th 5\13, 461.54 7\18, 466.67 12\31, 464.52 M 21/16, 13/10, 17/13 +1
minor mos5th 6\13, 553.85 8\18, 533.33 14\31, 541.94 N@ 11/8 -4
major mos5th 7\13, 646.15 10\18, 666.66 17\31, 658.06 N 13/9, 16/11 +4
perf. mos6th 8\13, 738.46 11\18, 733.33 19\31, 735.48 O 26/17 -1
aug. mos6th 9\13, 830.77 13\18, 866.66 22\31, 851.61 O& 13/8, 18/11 +7
minor mos7th 9\13, 830.77 12\18, 800.00 21\31, 812.90 P@ 8/5 -6
major mos7th 10\13, 923.08 14\18, 933.33 24\31, 929.03 P 12/7 +2
minor mos8th 11\13, 1015.39 15\18, 1000.00 26\31, 1006.45 Q 9/5, 16/9 -3
major mos8th 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-Lmos2-Lmos3-Lmos5 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.

This set of JI identifications is associated with petrtri temperament. (P1-Mmos2-Pmos4 could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" petrtri temperament is.)

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 mos2nd 2\21, 114.29 3\34, 105.88 K@ 16/15 -5
major mos2nd 3\21, 171.43 5\34, 176.47 K 10/9, 11/10 +3
minor mos3rd 5\21, 285.71 8\34, 282.35 L 13/11, 20/17 -2
major mos3rd 6\21, 342.86 10\34, 352.94 L& 11/9 +6
dim. mos4th 7\21, 400.00 11\34, 388.24 M@ 5/4 -7
perf. mos4th 7\18, 457.14 12\31, 458.82 M 13/10 +1
minor mos5th 10\21, 571.43 16\34, 564.72 N@ 18/13, 32/23 -4
major mos5th 11\21, 628.57 18\34, 635.29 N 13/9, 23/16 +4
perf. mos6th 13\21, 742.86 21\34, 741.18 O 20/13 -1
aug. mos6th 14\21, 800.00 23\34, 811.77 O& 8/5 +7
minor mos7th 15\21, 857.14 24\34, 847.06 P@ 18/11 -6
major mos7th 16\21, 914.29 26\34, 917.65 P 22/13, 17/10 +2
minor mos8th 18\21, 1028.57 29\34, 1023.53 Q 9/5 -3
major mos8th 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. mos2nd 2\29, 82.8 K@@ -13
minor mos2nd 3\29, 124.1 K@ 14/13 -5
major mos2nd 4\29, 165.5 K 11/10 +3
aug. mos2nd 5\29, 206.9 K& 9/8 +11
dim. mos3rd 6\29, 248.3 L@ 15/13 -10
minor mos3rd 7\29, 289.7 L 13/11 -2
major mos3rd 8\29, 331.0 L& +6
aug. mos3rd 9\29, 372.4 L&& +14
doubly dim. mos4th 9\29, 372.4 M@@ -15
dim. mos4th 10\29, 413.8 M@ 14/11 -7
perf. mos4th 11\29, 455.2 M 13/10 +1
aug. mos4th 12\29, 496.6 M& 4/3 +9
dim. mos5th 13\29, 537.9 N@@ 15/11 -12
minor mos5th 14\29, 579.3 N@ 7/5 -4
major mos5th 15\29 620.7 N 10/7 +4
aug. mos5th 16\29 662.1 N& 22/15 +12
dim. mos6th 17\29, 703.4 O@ 3/2 -9
perf. mos6th 18\29, 755.2 O 20/13 -1
aug. mos6th 19\29, 786.2 O& 11/7 +7
doubly aug. mos6th 20\29 827.6 O&& +15
dim. mos7th 20\29 827.6 P@@ -14
minor mos7th 21\29 869.0 P@ -6
major mos7th 22\29, 910.3 P 22/13 +2
aug. mos7th 23\29, 951.7 P& 26/15 +10
dim. mos8th 24\29, 993.1 Q@ 16/9 -11
minor mos8th 25\29, 1034.5 Q 20/11 -3
major mos8th 26\29, 1075.9 Q& 13/7 +5
aug. mos8th 27\29, 1117.2 Q&& +13
dim. o9th 28\29, 1158.6 J@ -8

Modes

Oneirotonic modes are named after cities in the Dreamlands.

Mode UDP Name
LLsLLsLs 7|0 Dylathian (də-LA(H)TH-iən)
LLsLsLLs 6|1 Illarnekian (ill-ar-NEK-iən)
LsLLsLLs 5|2 Celephaïsian (kel-ə-FAY-zhən)
LsLLsLsL 4|3 Ultharian (ul-THA(I)R-iən)
LsLsLLsL 3|4 Mnarian (mə-NA(I)R-iən)
sLLsLLsL 2|5 Kadathian (kə-DA(H)TH-iən)
sLLsLsLL 1|6 Hlanithian (lə-NITH-iən)
sLsLLsLL 0|7 Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"

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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