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switching to clearer 0-based "mosstep" names
Line 23: Line 23:
The notation used in this article is J Ultharian (LsLLsLsL) = JKLMNOPQJ, 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".)
The notation used in this article is J Ultharian (LsLLsLsL) = JKLMNOPQJ, 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".)


The chain of perfect mosfourths becomes: ... P@ K@ N@ Q L O J M P K N Q& L& O& ...
The chain of perfect 3-mossteps 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 ''k'' of generators it takes to reach the interval and reducing mod 1200 if necessary (so you can use "''k''*''g'' % 1200" for search engines, for plugged-in values of ''k'' and ''g''). 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.
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 ''k'' of generators it takes to reach the interval and reducing mod 1200 if necessary (so you can use "''k''*''g'' % 1200" for search engines, for plugged-in values of ''k'' and ''g''). For example, since the major 2-stepis reached by six subfourth generators, [[18edo]]'s major 2-step 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 mosfourth
| perfect 3-step
| 2L + 1s
| 2L + 1s
| -1
| -1
| O
| O
| perfect mossixth
| perfect 5-step
| 3L + 2s
| 3L + 2s
|-
|-
| 2
| 2
| P
| P
| major mosseventh
| major 6-step
| 4L + 2s
| 4L + 2s
| -2
| -2
| L
| L
| minor mosthird
| minor 2-step
| 1L + 1s
| 1L + 1s
|-
|-
| 3
| 3
| K
| K
| major mossecond
| major 1-step
| 1L + 0s
| 1L + 0s
| -3
| -3
| Q
| Q
| minor moseighth
| minor 7-step
| 4L + 3s
| 4L + 3s
|-
|-
| 4
| 4
| N
| N
| major mosfifth
| major 4-step
| 3L + 1s
| 3L + 1s
| -4
| -4
| N@
| N@
| minor mosfifth
| minor 4-step
| 2L + 2s
| 2L + 2s
|-
|-
| 5
| 5
| Q&
| Q&
| major moseighth
| major 7-step
| 5L + 2s
| 5L + 2s
| -5
| -5
| K@
| K@
| minor mossecond
| minor 1-step
| 0L + 1s
| 0L + 1s
|-
|-
| 6
| 6
| L&
| L&
| major mosthird
| major 2-step
| 2L + 0s
| 2L + 0s
| -6
| -6
| P@
| P@
| minor mosseventh
| minor 6-step
| 3L + 3s
| 3L + 3s
|-
|-
| 7
| 7
| O&
| O&
| augmented mossixth
| augmented 5-step
| 4L + 1s
| 4L + 1s
| -7
| -7
| M@
| M@
| diminished mosfourth
| diminished 3-step
| 1L + 2s
| 1L + 2s
|-
|-
Line 133: Line 133:
| 8
| 8
| J&
| J&
| augmented mosunison (aka moschroma)
| augmented 0-step (aka moschroma)
| 1L - 1s
| 1L - 1s
| -8
| -8
| J@
| J@
| diminished mosoctave (aka diminished mosninth)
| diminished 8-step (aka diminished mosninth)
| 4L + 4s
| 4L + 4s
|-
|-
| 9
| 9
| M&
| M&
| augmented mosfourth
| augmented 3-step
| 3L + 0s
| 3L + 0s
| -9
| -9
| O@
| O@
| diminished mossixth
| diminished 5-step
| 2L + 3s
| 2L + 3s
|-
|-
| 10
| 10
| P&
| P&
| augmented mosseventh
| augmented 6-step
| 5L + 1s
| 5L + 1s
| -10
| -10
| L@
| L@
| diminished mosthird
| diminished 2-step
| 0L + 2s
| 0L + 2s
|-
|-
| 11
| 11
| K&
| K&
| augmented mossecond
| augmented 1-step
| 2L - 1s
| 2L - 1s
| -11
| -11
| Q@
| Q@
| diminished moseighth
| diminished 7-step
| 3L + 4s
| 3L + 4s
|-
|-
| 12
| 12
| N&
| N&
| augmented mosfifth
| augmented 4-step
| 4L + 0s
| 4L + 0s
| -12
| -12
| N@@
| N@@
| diminished mosfifth
| diminished 4-step
| 1L + 3s
| 1L + 3s
|}
|}
Line 183: Line 183:
Hypohard oneirotonic can be considered "meantone oneirotonic". This is because these tunings share the following features with [[meantone]] diatonic tunings:  
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 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.
* The major 2-mosstep (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]].
Also, in [[18edo]] and [[31edo]], the minor 2-mosstep is close to [[7/6]].


The set of identifications above is associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.
The set of identifications above is associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.


EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]].
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.
* 13edo has characteristically small 1-mossteps 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.
* 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 5-mosstep) and falling fifths (666.7c, a major 4-mosstep) 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.
* 31edo  can be used to make the major 2-mosstep 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.
* [[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.


Line 240: Line 240:
| 0
| 0
|-
|-
| minor mos2nd
| minor 1-step
| 1\13, 92.31
| 1\13, 92.31
| 1\18, 66.67
| 1\18, 66.67
Line 248: Line 248:
| -5
| -5
|-
|-
| major mos2nd
| major 1-step
| 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 mos3rd
| minor 2-step
| 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 mos3rd
| major 2-step
| 4\13, 369.23
| 4\13, 369.23
| 6\18, 400.00
| 6\18, 400.00
Line 272: Line 272:
| +6
| +6
|-
|-
| dim. mos4th
| dim. 3-step
| 4\13, 369.23
| 4\13, 369.23
| 5\18, 333.33
| 5\18, 333.33
Line 280: Line 280:
| -7
| -7
|-
|-
| perf. mos4th
| perf. 3-step
| 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 mos5th
| minor 4-step
| 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 mos5th
| major 4-step
| 7\13, 646.15
| 7\13, 646.15
| 10\18, 666.66
| 10\18, 666.66
Line 304: Line 304:
| +4
| +4
|-
|-
| perf. mos6th
| perf. 5-step
| 8\13, 738.46
| 8\13, 738.46
| 11\18, 733.33
| 11\18, 733.33
Line 312: Line 312:
| -1
| -1
|-
|-
| aug. mos6th
| aug. 5-step
| 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 mos7th
| minor 6-step
| 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 mos7th
| major 6-step
| 10\13, 923.08
| 10\13, 923.08
| 14\18, 933.33
| 14\18, 933.33
Line 336: Line 336:
| +2
| +2
|-
|-
| minor mos8th
| minor 7-step
| 11\13, 1015.39
| 11\13, 1015.39
| 15\18, 1000.00
| 15\18, 1000.00
Line 344: Line 344:
| -3
| -3
|-
|-
| major mos8th
| major 7-step
| 12\13, 1107.69
| 12\13, 1107.69
| 17\18, 1133.33
| 17\18, 1133.33
Line 357: Line 357:
[[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,  
[[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 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¢).
* The major 2-mosstep (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¢).
* [[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¢).
Line 403: Line 403:
| 0
| 0
|-
|-
| minor mos2nd
| minor 1-step
| 2\21, 114.29
| 2\21, 114.29
| 3\34, 105.88
| 3\34, 105.88
Line 410: Line 410:
| -5
| -5
|-
|-
| major mos2nd
| major 1-step
| 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 mos3rd
| minor 2-step
| 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 mos3rd
| major 2-step
| 6\21, 342.86
| 6\21, 342.86
| 10\34, 352.94
| 10\34, 352.94
Line 431: Line 431:
| +6
| +6
|-
|-
| dim. mos4th
| dim. 3-step
| 7\21, 400.00
| 7\21, 400.00
| 11\34, 388.24
| 11\34, 388.24
Line 438: Line 438:
| -7
| -7
|-
|-
| perf. mos4th
| perf. 3-step
| 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 mos5th
| minor 4-step
| 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 mos5th
| major 4-step
| 11\21, 628.57
| 11\21, 628.57
| 18\34, 635.29
| 18\34, 635.29
Line 459: Line 459:
| +4
| +4
|-
|-
| perf. mos6th
| perf. 5-step
| 13\21, 742.86
| 13\21, 742.86
| 21\34, 741.18
| 21\34, 741.18
Line 466: Line 466:
| -1
| -1
|-
|-
| aug. mos6th
| aug. 5-step
| 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 mos7th
| minor 6-step
| 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 mos7th
| major 6-step
| 16\21, 914.29
| 16\21, 914.29
| 26\34, 917.65
| 26\34, 917.65
Line 487: Line 487:
| +2
| +2
|-
|-
| minor mos8th
| minor 7-step
| 18\21, 1028.57
| 18\21, 1028.57
| 29\34, 1023.53
| 29\34, 1023.53
Line 494: Line 494:
| -3
| -3
|-
|-
| major mos8th
| major 7-step
| 19\21, 1085.71
| 19\21, 1085.71
| 31\34, 1094.12
| 31\34, 1094.12
Line 546: Line 546:
| +8
| +8
|-
|-
| dim. mos2nd
| dim. 1-step
| 2\29, 82.8
| 2\29, 82.8
| K@@
| K@@
Line 552: Line 552:
| -13
| -13
|-
|-
| minor mos2nd
| minor 1-step
| 3\29, 124.1
| 3\29, 124.1
| K@
| K@
Line 558: Line 558:
| -5
| -5
|-
|-
| major mos2nd
| major 1-step
| 4\29, 165.5
| 4\29, 165.5
| K
| K
Line 564: Line 564:
| +3
| +3
|-
|-
| aug. mos2nd
| aug. 1-step
| 5\29, 206.9
| 5\29, 206.9
| K&
| K&
Line 570: Line 570:
| +11
| +11
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. mos3rd
| dim. 2-step
| 6\29, 248.3
| 6\29, 248.3
| L@
| L@
Line 576: Line 576:
| -10
| -10
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos3rd
| minor 2-step
| 7\29, 289.7
| 7\29, 289.7
| L
| L
Line 582: Line 582:
| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos3rd
| major 2-step
| 8\29, 331.0
| 8\29, 331.0
| L&
| L&
Line 588: Line 588:
| +6
| +6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos3rd
| aug. 2-step
| 9\29, 372.4
| 9\29, 372.4
| L&&
| L&&
Line 594: Line 594:
| +14
| +14
|-
|-
| doubly dim. mos4th
| doubly dim. 3-step
| 9\29, 372.4
| 9\29, 372.4
| M@@
| M@@
Line 600: Line 600:
| -15
| -15
|-
|-
| dim. mos4th
| dim. 3-step
| 10\29, 413.8
| 10\29, 413.8
| M@
| M@
Line 606: Line 606:
| -7
| -7
|-
|-
| perf. mos4th
| perf. 3-step
| 11\29, 455.2
| 11\29, 455.2
| M
| M
Line 612: Line 612:
| +1
| +1
|-
|-
| aug. mos4th
| aug. 3-step
| 12\29, 496.6
| 12\29, 496.6
| M&
| M&
Line 618: Line 618:
| +9
| +9
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. mos5th
| dim. 4-step
| 13\29, 537.9
| 13\29, 537.9
| N@@
| N@@
Line 624: Line 624:
| -12
| -12
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos5th
| minor 4-step
| 14\29, 579.3
| 14\29, 579.3
| N@
| N@
Line 630: Line 630:
| -4
| -4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos5th
| major 4-step
| 15\29 620.7
| 15\29 620.7
| N
| N
Line 636: Line 636:
| +4
| +4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos5th
| aug. 4-step
| 16\29 662.1
| 16\29 662.1
| N&
| N&
Line 642: Line 642:
| +12
| +12
|-
|-
| dim. mos6th
| dim. 5-step
| 17\29, 703.4
| 17\29, 703.4
| O@
| O@
Line 648: Line 648:
| -9
| -9
|-
|-
| perf. mos6th
| perf. 5-step
| 18\29, 755.2
| 18\29, 755.2
| O
| O
Line 654: Line 654:
| -1
| -1
|-
|-
| aug. mos6th
| aug. 5-step
| 19\29, 786.2
| 19\29, 786.2
| O&
| O&
Line 660: Line 660:
| +7
| +7
|-
|-
| doubly aug. mos6th
| doubly aug. 5-step
| 20\29 827.6
| 20\29 827.6
| O&&
| O&&
Line 666: Line 666:
| +15
| +15
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. mos7th
| dim. 6-step
| 20\29 827.6
| 20\29 827.6
| P@@
| P@@
Line 672: Line 672:
| -14
| -14
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos7th
| minor 6-step
| 21\29 869.0
| 21\29 869.0
| P@
| P@
Line 678: Line 678:
| -6
| -6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos7th
| major 6-step
| 22\29, 910.3
| 22\29, 910.3
| P
| P
Line 684: Line 684:
| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos7th
| aug. 6-step
| 23\29, 951.7
| 23\29, 951.7
| P&
| P&
Line 690: Line 690:
| +10
| +10
|-
|-
| dim. mos8th
| dim. 7-step
| 24\29, 993.1
| 24\29, 993.1
| Q@
| Q@
Line 696: Line 696:
| -11
| -11
|-
|-
| minor mos8th
| minor 7-step
| 25\29, 1034.5
| 25\29, 1034.5
| Q
| Q
Line 702: Line 702:
| -3
| -3
|-
|-
| major mos8th
| major 7-step
| 26\29, 1075.9
| 26\29, 1075.9
| Q&
| Q&
Line 708: Line 708:
| +5
| +5
|-
|-
| aug. mos8th
| aug. 7-step
| 27\29, 1117.2
| 27\29, 1117.2
| Q&&
| Q&&
Line 745: Line 745:
| +8
| +8
|-
|-
| minor mos2nd
| minor 1-step
| 1\23, 52.2
| 1\23, 52.2
| K@
| K@
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| -5
| -5
|-
|-
| major mos2nd
| major 1-step
| 4\23, 208.7
| 4\23, 208.7
| K
| K
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| +3
| +3
|-
|-
| aug. mos2nd
| aug. 1-step
| 7\23, 365.2
| 7\23, 365.2
| K&
| K&
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| +11
| +11
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. mos3rd
| dim. 2-step
| 2\23, 104.3
| 2\23, 104.3
| L@
| L@
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| -10
| -10
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos3rd
| minor 2-step
| 5\23, 260.9
| 5\23, 260.9
| L
| L
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| -2
| -2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos3rd
| major 2-step
| 8\23, 417.4
| 8\23, 417.4
| L&
| L&
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| +6
| +6
|-
|-
| dim. mos4th
| dim. 3-step
| 6\23, 313.0
| 6\23, 313.0
| M@
| M@
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| -7
| -7
|-
|-
| perf. mos4th
| perf. 3-step
| 9\23, 469.6
| 9\23, 469.6
| M
| M
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| +1
| +1
|-
|-
| aug. mos4th
| aug. 3-step
| 12\23, 626.1
| 12\23, 626.1
| M&
| M&
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| +9
| +9
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| dim. mos5th
| dim. 4-step
| 7\23, 365.2
| 7\23, 365.2
| N@@
| N@@
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| -12
| -12
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos5th
| minor 4-step
| 10\23, 521.7
| 10\23, 521.7
| N@
| N@
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| -4
| -4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos5th
| major 4-step
| 13\23, 678.3
| 13\23, 678.3
| N
| N
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| +4
| +4
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos5th
| aug. 4-step
| 16\23, 834.8
| 16\23, 834.8
| N&
| N&
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| +12
| +12
|-
|-
| dim. mos6th
| dim. 5-step
| 11\23, 573.9
| 11\23, 573.9
| O@
| O@
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| -9
| -9
|-
|-
| perf. mos6th
| perf. 5-step
| 14\23, 730.4
| 14\23, 730.4
| O
| O
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| -1
| -1
|-
|-
| aug. mos6th
| aug. 5-step
| 17\23, 887.0
| 17\23, 887.0
| O&
| O&
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|-
|-
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| minor mos7th
| minor 6-step
| 15\23 782.6
| 15\23 782.6
| P@
| P@
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| -6
| -6
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| major mos7th
| major 6-step
| 18\23, 939.1
| 18\23, 939.1
| P
| P
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| +2
| +2
|-bgcolor="#eaeaff"
|-bgcolor="#eaeaff"
| aug. mos7th
| aug. 6-step
| 21\23, 1095.7
| 21\23, 1095.7
| P&
| P&
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| +10
| +10
|-
|-
| dim. mos8th
| dim. 7-step
| 16\23, 834.8
| 16\23, 834.8
| Q@
| Q@
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| -11
| -11
|-
|-
| minor mos8th
| minor 7-step
| 19\23, 991.3
| 19\23, 991.3
| Q
| Q
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| -3
| -3
|-
|-
| major mos8th
| major 7-step
| 22\23, 1147.8
| 22\23, 1147.8
| Q&
| Q&

Revision as of 00:35, 4 June 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 ¢)
View • Talk • Edit

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 warped diatonic scale, 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, 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".)

The chain of perfect 3-mossteps 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 (although a very inaccurate one), not a generator range. A more correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, 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 k of generators it takes to reach the interval and reducing mod 1200 if necessary (so you can use "k*g % 1200" for search engines, for plugged-in values of k and g). For example, since the major 2-stepis reached by six subfourth generators, 18edo's major 2-step is 6*466.67 mod 1200 = 2800 mod 1200 = 400¢, same as the 12edo major third.

# generators up Notation (1/1 = J) TAMNAMS name In L's and s's # generators up Notation of 2/1 inverse TAMNAMS name In L's and s's
The 8-note MOS has the following intervals (from some root):
0 J perfect unison 0L + 0s 0 J octave 5L + 3s
1 M perfect 3-step 2L + 1s -1 O perfect 5-step 3L + 2s
2 P major 6-step 4L + 2s -2 L minor 2-step 1L + 1s
3 K major 1-step 1L + 0s -3 Q minor 7-step 4L + 3s
4 N major 4-step 3L + 1s -4 N@ minor 4-step 2L + 2s
5 Q& major 7-step 5L + 2s -5 K@ minor 1-step 0L + 1s
6 L& major 2-step 2L + 0s -6 P@ minor 6-step 3L + 3s
7 O& augmented 5-step 4L + 1s -7 M@ diminished 3-step 1L + 2s
The chromatic 13-note MOS (either 5L 8s, 8L 5s, or 13edo) also has the following intervals (from some root):
8 J& augmented 0-step (aka moschroma) 1L - 1s -8 J@ diminished 8-step (aka diminished mosninth) 4L + 4s
9 M& augmented 3-step 3L + 0s -9 O@ diminished 5-step 2L + 3s
10 P& augmented 6-step 5L + 1s -10 L@ diminished 2-step 0L + 2s
11 K& augmented 1-step 2L - 1s -11 Q@ diminished 7-step 3L + 4s
12 N& augmented 4-step 4L + 0s -12 N@@ diminished 4-step 1L + 3s

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 2-mosstep (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 2-mosstep 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 1-mossteps 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 5-mosstep) and falling fifths (666.7c, a major 4-mosstep) 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 2-mosstep 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 1-step 1\13, 92.31 1\18, 66.67 2\31, 77.42 K@ 21/20, 22/21 -5
major 1-step 2\13, 184.62 3\18, 200.00 5\31, 193.55 K 9/8, 10/9 +3
minor 2-step 3\13, 276.92 4\18, 266.67 7\31, 270.97 L 7/6 -2
major 2-step 4\13, 369.23 6\18, 400.00 10\31, 387.10 L& 5/4 +6
dim. 3-step 4\13, 369.23 5\18, 333.33 9\31, 348.39 M@ 16/13, 11/9 -7
perf. 3-step 5\13, 461.54 7\18, 466.67 12\31, 464.52 M 21/16, 13/10, 17/13 +1
minor 4-step 6\13, 553.85 8\18, 533.33 14\31, 541.94 N@ 11/8 -4
major 4-step 7\13, 646.15 10\18, 666.66 17\31, 658.06 N 13/9, 16/11 +4
perf. 5-step 8\13, 738.46 11\18, 733.33 19\31, 735.48 O 26/17 -1
aug. 5-step 9\13, 830.77 13\18, 866.66 22\31, 851.61 O& 13/8, 18/11 +7
minor 6-step 9\13, 830.77 12\18, 800.00 21\31, 812.90 P@ 8/5 -6
major 6-step 10\13, 923.08 14\18, 933.33 24\31, 929.03 P 12/7 +2
minor 7-step 11\13, 1015.39 15\18, 1000.00 26\31, 1006.45 Q 9/5, 16/9 -3
major 7-step 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 2-mosstep (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 1-step 2\21, 114.29 3\34, 105.88 K@ 16/15 -5
major 1-step 3\21, 171.43 5\34, 176.47 K 10/9, 11/10 +3
minor 2-step 5\21, 285.71 8\34, 282.35 L 13/11, 20/17 -2
major 2-step 6\21, 342.86 10\34, 352.94 L& 11/9 +6
dim. 3-step 7\21, 400.00 11\34, 388.24 M@ 5/4 -7
perf. 3-step 7\18, 457.14 12\31, 458.82 M 13/10 +1
minor 4-step 10\21, 571.43 16\34, 564.72 N@ 18/13, 32/23 -4
major 4-step 11\21, 628.57 18\34, 635.29 N 13/9, 23/16 +4
perf. 5-step 13\21, 742.86 21\34, 741.18 O 20/13 -1
aug. 5-step 14\21, 800.00 23\34, 811.77 O& 8/5 +7
minor 6-step 15\21, 857.14 24\34, 847.06 P@ 18/11 -6
major 6-step 16\21, 914.29 26\34, 917.65 P 22/13, 17/10 +2
minor 7-step 18\21, 1028.57 29\34, 1023.53 Q 9/5 -3
major 7-step 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 chord 10:11:13 is very well approximated in 29edo.

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. 1-step 2\29, 82.8 K@@ -13
minor 1-step 3\29, 124.1 K@ 14/13 -5
major 1-step 4\29, 165.5 K 11/10 +3
aug. 1-step 5\29, 206.9 K& 9/8 +11
dim. 2-step 6\29, 248.3 L@ 15/13 -10
minor 2-step 7\29, 289.7 L 13/11 -2
major 2-step 8\29, 331.0 L& +6
aug. 2-step 9\29, 372.4 L&& +14
doubly dim. 3-step 9\29, 372.4 M@@ -15
dim. 3-step 10\29, 413.8 M@ 14/11 -7
perf. 3-step 11\29, 455.2 M 13/10 +1
aug. 3-step 12\29, 496.6 M& 4/3 +9
dim. 4-step 13\29, 537.9 N@@ 15/11 -12
minor 4-step 14\29, 579.3 N@ 7/5 -4
major 4-step 15\29 620.7 N 10/7 +4
aug. 4-step 16\29 662.1 N& 22/15 +12
dim. 5-step 17\29, 703.4 O@ 3/2 -9
perf. 5-step 18\29, 755.2 O 20/13 -1
aug. 5-step 19\29, 786.2 O& 11/7 +7
doubly aug. 5-step 20\29 827.6 O&& +15
dim. 6-step 20\29 827.6 P@@ -14
minor 6-step 21\29 869.0 P@ -6
major 6-step 22\29, 910.3 P 22/13 +2
aug. 6-step 23\29, 951.7 P& 26/15 +10
dim. 7-step 24\29, 993.1 Q@ 16/9 -11
minor 7-step 25\29, 1034.5 Q 20/11 -3
major 7-step 26\29, 1075.9 Q& 13/7 +5
aug. 7-step 27\29, 1117.2 Q&& +13
dim. mos9th 28\29, 1158.6 J@ -8

Parahard

23edo oneiro combines the sound of neogothic tunings like 46edo and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as 46edo's neogothic major second, and is both a warped 22edo superpyth diatonic and a warped 24edo semaphore semiquartal (and both nearby scales are superhard MOSes).

Intervals

The intervals of the extended generator chain (-12 to +12 generators) are as follows in various oneirotonic tunings close to 23edo.

Degree Size in 23edo (superhard) Note name on J Approximate ratios (23edo) #Gens up
unison 0\23, 0.0 J 1/1 0
oneirochroma 3\23, 156.5 J& +8
minor 1-step 1\23, 52.2 K@ -5
major 1-step 4\23, 208.7 K +3
aug. 1-step 7\23, 365.2 K& 21/17, inverse φ +11
dim. 2-step 2\23, 104.3 L@ 17/16 -10
minor 2-step 5\23, 260.9 L -2
major 2-step 8\23, 417.4 L& 14/11 +6
dim. 3-step 6\23, 313.0 M@ 6/5 -7
perf. 3-step 9\23, 469.6 M 21/16 +1
aug. 3-step 12\23, 626.1 M& +9
dim. 4-step 7\23, 365.2 N@@ 21/17, inverse φ -12
minor 4-step 10\23, 521.7 N@ -4
major 4-step 13\23, 678.3 N +4
aug. 4-step 16\23, 834.8 N& 34/21, φ +12
dim. 5-step 11\23, 573.9 O@ -9
perf. 5-step 14\23, 730.4 O 32/21 -1
aug. 5-step 17\23, 887.0 O& 5/3 +7
minor 6-step 15\23 782.6 P@ 11/7 -6
major 6-step 18\23, 939.1 P +2
aug. 6-step 21\23, 1095.7 P& 32/17 +10
dim. 7-step 16\23, 834.8 Q@ 34/21, φ -11
minor 7-step 19\23, 991.3 Q -3
major 7-step 22\23, 1147.8 Q& +5
dim. mos9th 20\23, 1043.5 J@ -8

Ultrahard

Buzzard is an oneirotonic rank-2 temperament in the pseudopaucitonic range. It represents the only harmonic entropy minimum of the oneirotonic spectrum.

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 18edo 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) Buzzard 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

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

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

The Angels' Library by Inthar in the Sarnathian (23233233) mode of 21edo oneirotonic (score)

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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