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Line 20: Line 20:
Thus the [[13edo]] gamut is as follows:  
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'''
'''J''' J&/K@ '''K'''/L@ '''L'''/K& L&/M@ '''M''' M&/N@ '''N'''/O@ '''O'''/N& O&/P@ '''P'''/Q@ '''Q'''/P& Q&/J@ '''J'''


The [[18edo]] gamut is notated as follows:
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'''
'''J''' K@ J&/L@ '''K''' '''L''' K&/M@ L& '''M''' N@ M&/O@ '''N''' '''O''' N&/P@ O&/Q@ '''P''' '''Q''' P&/J@ Q& '''J'''


The [[21edo]] gamut:
The [[21edo]] gamut:
Line 52: Line 52:
| colspan="8" style="text-align:center" |The 8-note MOS has the following intervals (from some root):
| colspan="8" style="text-align:center" |The 8-note MOS has the following intervals (from some root):
|-
|-
|0
| 0
|J
| J
|perfect unison
| perfect unison
|0L + 0s
| 0L + 0s
|0
| 0
|J
| J
| octave
| octave
|5L + 3s
| 5L + 3s
|-
|-
|1
| 1
|M
| M
|perfect 3-step
| perfect 3-step
|2L + 1s
| 2L + 1s
| -1
| -1
| O
| O
|perfect 5-step
| perfect 5-step
|3L + 2s
| 3L + 2s
|-
|-
|2
| 2
|P
| P
|major 6-step
| major 6-step
| 4L + 2s
| 4L + 2s
| -2
| -2
| L
| L
|minor 2-step
| minor 2-step
| 1L + 1s
| 1L + 1s
|-
|-
|3
| 3
|K
| K
|major (1-)step
| major (1-)step
|1L + 0s
| 1L + 0s
| -3
| -3
| Q
| Q
|minor 7-step
| minor 7-step
| 4L + 3s
| 4L + 3s
|-
|-
|4
| 4
|N
| N
|major 4-step
| major 4-step
| 3L + 1s
| 3L + 1s
| -4
| -4
Line 97: Line 97:
| 2L + 2s
| 2L + 2s
|-
|-
|5
| 5
|Q&
| Q&
|major 7-step
| major 7-step
| 5L + 2s
| 5L + 2s
| -5
| -5
| K@
| K@
| minor (1-)step
| minor (1-)step
|0L + 1s
| 0L + 1s
|-
|-
|6
| 6
|L&
| L&
|major 2-step
| major 2-step
| 2L + 0s
| 2L + 0s
| -6
| -6
Line 115: Line 115:
| 3L + 3s
| 3L + 3s
|-
|-
|7
| 7
|O&
| O&
|augmented 5-step
| augmented 5-step
|4L + 1s
| 4L + 1s
| -7
| -7
| M@
| M@
| diminished 3-step
| diminished 3-step
|1L + 2s
| 1L + 2s
|-
|-
| 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):
|-
|-
|8
| 8
|J&
| J&
|augmented 0-step (aka moschroma)
| augmented 0-step (aka moschroma)
|1L - 1s
| 1L - 1s
| -8
| -8
| J@
| J@
| diminished 8-step (aka diminished mosoctave)
| diminished 8-step (aka diminished mosoctave)
|4L + 4s
| 4L + 4s
|-
|-
|9
| 9
|M&
| M&
|augmented 3-step
| augmented 3-step
|3L + 0s
| 3L + 0s
| -9
| -9
| O@
| O@
| diminished 5-step
| diminished 5-step
|2L + 3s
| 2L + 3s
|-
|-
|10
| 10
| P&
| P&
|augmented 6-step
| augmented 6-step
|5L + 1s
| 5L + 1s
| -10
| -10
|L@
| L@
|diminished 2-step
| diminished 2-step
|0L + 2s
| 0L + 2s
|-
|-
|11
| 11
| K&
| K&
|augmented 1-step
| augmented 1-step
|2L - 1s
| 2L - 1s
| -11
| -11
|Q@
| Q@
|diminished 7-step
| diminished 7-step
|3L + 4s
| 3L + 4s
|-
|-
|12
| 12
| N&
| N&
|augmented 4-step
| augmented 4-step
|4L + 0s
| 4L + 0s
| -12
| -12
|N@@
| N@@
|diminished 4-step
| diminished 4-step
|1L + 3s
| 1L + 3s
|}
|}


Line 178: Line 178:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 13edo (basic)
! Size in 13edo (basic)
!Size in 18edo (hard)
! Size in 18edo (hard)
! Size in 21edo (soft)
! Size in 21edo (soft)
! class="unsortable" |Note name on J
! class="unsortable" |Note name on J
!#Gens up
! #Gens up
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|unison
| unison
|0\13, 0.00
| 0\13, 0.00
|0\18, 0.00
| 0\18, 0.00
|0\21, 0.00
| 0\21, 0.00
|J
| J
|0
| 0
|-
|-
|minor step
| minor step
|1\13, 92.31
| 1\13, 92.31
|1\18, 66.67
| 1\18, 66.67
|2\21, 114.29
| 2\21, 114.29
| K@
| K@
| -5
| -5
|-
|-
|major step
| major step
|2\13, 184.62
| 2\13, 184.62
| 3\18, 200.00
| 3\18, 200.00
| 3\21, 171.43
| 3\21, 171.43
Line 205: Line 205:
| +3
| +3
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 2-step
| minor 2-step
| 3\13, 276.92
| 3\13, 276.92
| 4\18, 266.67
| 4\18, 266.67
Line 212: Line 212:
| -2
| -2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 2-step
| major 2-step
| 4\13, 369.23
| 4\13, 369.23
| 6\18, 400.00
| 6\18, 400.00
Line 219: Line 219:
| +6
| +6
|-
|-
|dim. 3-step
| dim. 3-step
|4\13, 369.23
| 4\13, 369.23
| 5\18, 333.33
| 5\18, 333.33
| 7\21, 400.00
| 7\21, 400.00
|M@
| M@
| -7
| -7
|-
|-
|perf. 3-step
| perf. 3-step
| 5\13, 461.54
| 5\13, 461.54
|7\18, 466.67
| 7\18, 466.67
| 8\21, 457.14
| 8\21, 457.14
| M
| M
| +1
| +1
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 4-step
| minor 4-step
| 6\13, 553.85
| 6\13, 553.85
| 8\18, 533.33
| 8\18, 533.33
| 10\21, 571.43
| 10\21, 571.43
|N@
| N@
| -4
| -4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 4-step
| major 4-step
| 7\13, 646.15
| 7\13, 646.15
| 10\18, 666.66
| 10\18, 666.66
|11\31, 628.57
| 11\31, 628.57
|N
| N
| +4
| +4
|-
|-
|perf. 5-step
| perf. 5-step
| 8\13, 738.46
| 8\13, 738.46
| 11\18, 733.33
| 11\18, 733.33
|13\21, 742.86
| 13\21, 742.86
|O
| O
| -1
| -1
|-
|-
|aug. 5-step
| aug. 5-step
|9\13, 830.77
| 9\13, 830.77
| 13\18, 866.66
| 13\18, 866.66
|14\21, 800.00
| 14\21, 800.00
|O&
| O&
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 6-step
| minor 6-step
| 9\13, 830.77
| 9\13, 830.77
| 12\18, 800.00
| 12\18, 800.00
|15\21, 857.14
| 15\21, 857.14
|P@
| P@
|  -6
|  -6
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6-step
| major 6-step
| 10\13, 923.08
| 10\13, 923.08
|14\18, 933.33
| 14\18, 933.33
|16\21, 914.29
| 16\21, 914.29
|P
| P
| +2
| +2
|-
|-
|minor 7-step
| minor 7-step
| 11\13, 1015.39
| 11\13, 1015.39
|15\18, 1000.00
| 15\18, 1000.00
|18\21, 1028.57
| 18\21, 1028.57
|Q
| Q
| -3
| -3
|-
|-
| major 7-step
| major 7-step
| 12\13, 1107.69
| 12\13, 1107.69
|17\18, 1133.33
| 17\18, 1133.33
|19\21, 1085.71
| 19\21, 1085.71
|Q&
| Q&
| +5
| +5
|}
|}


Line 315: Line 315:
![[31edo]] (semihard)
![[31edo]] (semihard)
|-
|-
|generator (g)
| generator (g)
|5\13, 461.54
| 5\13, 461.54
| 7\18, 466.67
| 7\18, 466.67
| 12\31, 464.52
| 12\31, 464.52
|-
|-
|L (3g - octave)
| L (3g - octave)
|2\13, 184.62
| 2\13, 184.62
| 3\18, 200.00
| 3\18, 200.00
| 5\31, 193.55
| 5\31, 193.55
|-
|-
|s (-5g + 2 octaves)
| s (-5g + 2 octaves)
| 1\13, 92.31
| 1\13, 92.31
|1\18, 66.67
| 1\18, 66.67
|2\31, 77.42
| 2\31, 77.42
|}
|}


Line 337: Line 337:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 13edo (basic)
! Size in 13edo (basic)
!Size in 18edo (hard)
! Size in 18edo (hard)
! Size in 31edo (semihard)
! Size in 31edo (semihard)
! class="unsortable" |Note name on J
! class="unsortable" |Note name on J
! class="unsortable" |Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>
! class="unsortable" |Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>
!#Gens up
! #Gens up
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|unison
| unison
|0\13, 0.00
| 0\13, 0.00
|0\18, 0.00
| 0\18, 0.00
|0\31, 0.00
| 0\31, 0.00
|J
| J
|1/1
| 1/1
|0
| 0
|-
|-
|minor step
| minor step
|1\13, 92.31
| 1\13, 92.31
|1\18, 66.67
| 1\18, 66.67
|2\31, 77.42
| 2\31, 77.42
|K@
| K@
|21/20, ''22/21''
| 21/20, ''22/21''
| -5
| -5
|-
|-
|major step
| major step
|2\13, 184.62
| 2\13, 184.62
| 3\18, 200.00
| 3\18, 200.00
| 5\31, 193.55
| 5\31, 193.55
| K
| K
|9/8, 10/9
| 9/8, 10/9
| +3
| +3
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 2-step
| minor 2-step
| 3\13, 276.92
| 3\13, 276.92
| 4\18, 266.67
| 4\18, 266.67
| 7\31, 270.97
| 7\31, 270.97
| L
| L
|7/6
| 7/6
| -2
| -2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 2-step
| major 2-step
| 4\13, 369.23
| 4\13, 369.23
| 6\18, 400.00
| 6\18, 400.00
| 10\31, 387.10
| 10\31, 387.10
|L&
| L&
|5/4
| 5/4
| +6
| +6
|-
|-
|dim. 3-step
| dim. 3-step
|4\13, 369.23
| 4\13, 369.23
| 5\18, 333.33
| 5\18, 333.33
| 9\31, 348.39
| 9\31, 348.39
| M@
| M@
|''16/13, 11/9''
| ''16/13, 11/9''
| -7
| -7
|-
|-
|perf. 3-step
| perf. 3-step
| 5\13, 461.54
| 5\13, 461.54
|7\18, 466.67
| 7\18, 466.67
| 12\31, 464.52
| 12\31, 464.52
|M
| M
|21/16, ''13/10'', 17/13
| 21/16, ''13/10'', 17/13
| +1
| +1
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 4-step
| minor 4-step
| 6\13, 553.85
| 6\13, 553.85
| 8\18, 533.33
| 8\18, 533.33
| 14\31, 541.94
| 14\31, 541.94
|N@
| N@
|''11/8''
| ''11/8''
| -4
| -4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 4-step
| major 4-step
| 7\13, 646.15
| 7\13, 646.15
| 10\18, 666.66
| 10\18, 666.66
|17\31, 658.06
| 17\31, 658.06
|N
| N
|''13/9'', ''16/11''
| ''13/9'', ''16/11''
| +4
| +4
|-
|-
|perf. 5-step
| perf. 5-step
| 8\13, 738.46
| 8\13, 738.46
| 11\18, 733.33
| 11\18, 733.33
|19\31, 735.48
| 19\31, 735.48
|O
| O
|26/17
| 26/17
| -1
| -1
|-
|-
|aug. 5-step
| aug. 5-step
|9\13, 830.77
| 9\13, 830.77
| 13\18, 866.66
| 13\18, 866.66
|22\31, 851.61
| 22\31, 851.61
|O&
| O&
|''13/8'', ''18/11''
| ''13/8'', ''18/11''
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 6-step
| minor 6-step
| 9\13, 830.77
| 9\13, 830.77
| 12\18, 800.00
| 12\18, 800.00
|21\31, 812.90
| 21\31, 812.90
|P@
| P@
|8/5
| 8/5
| -6
| -6
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6-step
| major 6-step
| 10\13, 923.08
| 10\13, 923.08
|14\18, 933.33
| 14\18, 933.33
|24\31, 929.03
| 24\31, 929.03
|P
| P
|12/7
| 12/7
| +2
| +2
|-
|-
| minor 7-step
| minor 7-step
| 11\13, 1015.39
| 11\13, 1015.39
|15\18, 1000.00
| 15\18, 1000.00
|26\31, 1006.45
| 26\31, 1006.45
|Q
| Q
| 9/5, 16/9
| 9/5, 16/9
| -3
| -3
Line 458: Line 458:
| major 7-step
| major 7-step
| 12\13, 1107.69
| 12\13, 1107.69
|17\18, 1133.33
| 17\18, 1133.33
|29\31, 1122.58
| 29\31, 1122.58
|Q&
| Q&
|
|
| +5
| +5
Line 468: Line 468:
===Hyposoft===
===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,  
[[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 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¢).
* 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-L1ms-L2ms-L4ms 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-L1ms-L2ms-L4ms 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.


This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri|petrtri]] temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" [[petrtri]] temperament is.)
This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri|petrtri]] temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" [[petrtri]] temperament is.)
Line 480: Line 480:
|-
|-
!
!
![[21edo]] (soft)
! [[21edo]] (soft)
![[34edo]] (semisoft)
! [[34edo]] (semisoft)
|-
|-
|generator (g)
| generator (g)
|8\21, 457.14
| 8\21, 457.14
| 13\34, 458.82
| 13\34, 458.82
|-
|-
|L (3g - octave)
| L (3g - octave)
|3\21, 171.43
| 3\21, 171.43
| 5\34, 176.47
| 5\34, 176.47
|-
|-
|s (-5g + 2 octaves)
| s (-5g + 2 octaves)
| 2\21, 114.29
| 2\21, 114.29
| 3\34, 105.88
| 3\34, 105.88
Line 502: Line 502:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 21edo (soft)
! Size in 21edo (soft)
! Size in 34edo (semisoft)
! Size in 34edo (semisoft)
! class="unsortable" |Note name on J
! class="unsortable" |Note name on J
! class="unsortable" |Approximate ratios
! class="unsortable" |Approximate ratios
!#Gens up
! #Gens up
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|unison
| unison
|0\21, 0.00
| 0\21, 0.00
|0\34, 0.00
| 0\34, 0.00
|J
| J
|1/1
| 1/1
|0
| 0
|-
|-
|minor step
| minor step
|2\21, 114.29
| 2\21, 114.29
| 3\34, 105.88
| 3\34, 105.88
| K@
| K@
|16/15
| 16/15
| -5
| -5
|-
|-
|major step
| major step
|3\21, 171.43
| 3\21, 171.43
| 5\34, 176.47
| 5\34, 176.47
| K
| K
|10/9, 11/10
| 10/9, 11/10
| +3
| +3
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 2-step
| minor 2-step
| 5\21, 285.71
| 5\21, 285.71
| 8\34, 282.35
| 8\34, 282.35
| L
| L
|13/11, 20/17
| 13/11, 20/17
| -2
| -2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 2-step
| major 2-step
| 6\21, 342.86
| 6\21, 342.86
| 10\34, 352.94
| 10\34, 352.94
|L&
| L&
|11/9
| 11/9
| +6
| +6
|-
|-
| dim. 3-step
| dim. 3-step
|7\21, 400.00
| 7\21, 400.00
| 11\34, 388.24
| 11\34, 388.24
|M@
| M@
|5/4
| 5/4
| -7
| -7
|-
|-
|perf. 3-step
| perf. 3-step
| 8\21, 457.14
| 8\21, 457.14
| 12\31, 458.82
| 12\31, 458.82
|M
| M
|13/10
| 13/10
| +1
| +1
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 4-step
| minor 4-step
| 10\21, 571.43
| 10\21, 571.43
|16\34, 564.72
| 16\34, 564.72
|N@
| N@
|18/13, 32/23
| 18/13, 32/23
| -4
| -4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 4-step
| major 4-step
| 11\21, 628.57
| 11\21, 628.57
|18\34, 635.29
| 18\34, 635.29
|N
| N
|13/9, 23/16
| 13/9, 23/16
| +4
| +4
|-
|-
|perf. 5-step
| perf. 5-step
| 13\21, 742.86
| 13\21, 742.86
|21\34, 741.18
| 21\34, 741.18
|O
| O
|20/13
| 20/13
| -1
| -1
|-
|-
|aug. 5-step
| aug. 5-step
|14\21, 800.00
| 14\21, 800.00
|23\34, 811.77
| 23\34, 811.77
|O&
| O&
|8/5
| 8/5
| +7
| +7
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 6-step
| minor 6-step
| 15\21, 857.14
| 15\21, 857.14
|24\34, 847.06
| 24\34, 847.06
|P@
| P@
|18/11
| 18/11
| -6
| -6
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6-step
| major 6-step
| 16\21, 914.29
| 16\21, 914.29
|26\34, 917.65
| 26\34, 917.65
|P
| P
|22/13, 17/10
| 22/13, 17/10
| +2
| +2
|-
|-
|minor 7-step
| minor 7-step
| 18\21, 1028.57
| 18\21, 1028.57
|29\34, 1023.53
| 29\34, 1023.53
|Q
| Q
|9/5
| 9/5
| -3
| -3
|-
|-
|major 7-step
| major 7-step
| 19\21, 1085.71
| 19\21, 1085.71
|31\34, 1094.12
| 31\34, 1094.12
|Q&
| Q&
| 15/8
| 15/8
| +5
| +5
Line 621: Line 621:
|-
|-
!
!
![[29edo]] (supersoft)
! [[29edo]] (supersoft)
![[37edo]]
! [[37edo]]
|-
|-
| generator (g)
| generator (g)
|11\29, 455.17
| 11\29, 455.17
|14\37, 454.05
| 14\37, 454.05
|-
|-
|L (3g - octave)
| L (3g - octave)
|4\29, 165.52
| 4\29, 165.52
| 5\37, 162.16
| 5\37, 162.16
|-
|-
|s (-5g + 2 octaves)
| s (-5g + 2 octaves)
| 3\29, 124.14
| 3\29, 124.14
| 4\37, 129.73
| 4\37, 129.73
Line 641: Line 641:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 29edo (supersoft)
! Size in 29edo (supersoft)
! class="unsortable" |Note name on J
! class="unsortable" |Note name on J
! class="unsortable" |Approximate ratios (29edo)
! class="unsortable" |Approximate ratios (29edo)
!#Gens up
!#Gens up
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|unison
| unison
|0\29, 0.00
| 0\29, 0.00
|J
| J
|1/1
| 1/1
|0
| 0
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|oneirochroma
| oneirochroma
| 1\29, 41.4
| 1\29, 41.4
|J&
| J&
|
|
| +8
| +8
|-
|-
|dim. step
| dim. step
|2\29, 82.8
| 2\29, 82.8
|K@@
| K@@
|
|
| -13
| -13
|-
|-
|minor step
| minor step
|3\29, 124.1
| 3\29, 124.1
|K@
| K@
|14/13
| 14/13
| -5
| -5
|-
|-
|major step
| major step
|4\29, 165.5
| 4\29, 165.5
|K
| K
|11/10
| 11/10
| +3
| +3
|-
|-
|aug. step
| aug. step
|5\29, 206.9
| 5\29, 206.9
|K&
| K&
|9/8
| 9/8
| +11
| +11
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 2-step
| dim. 2-step
|6\29, 248.3
| 6\29, 248.3
|L@
| L@
|15/13
| 15/13
| -10
| -10
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 2-step
| minor 2-step
| 7\29, 289.7
| 7\29, 289.7
|L
| L
|13/11
| 13/11
| -2
| -2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 2-step
| major 2-step
| 8\29, 331.0
| 8\29, 331.0
|L&
| L&
|
|
| +6
| +6
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 2-step
| aug. 2-step
|9\29, 372.4
| 9\29, 372.4
|L&&
| L&&
|
|
| +14
| +14
|-
|-
|doubly dim. 3-step
| doubly dim. 3-step
|9\29, 372.4
| 9\29, 372.4
|M@@
| M@@
|
|
| -15
| -15
|-
|-
|dim. 3-step
| dim. 3-step
|10\29, 413.8
| 10\29, 413.8
| M@
| M@
|14/11
| 14/11
| -7
| -7
|-
|-
|perf. 3-step
| perf. 3-step
| 11\29, 455.2
| 11\29, 455.2
| M
| M
|13/10
| 13/10
| +1
| +1
|-
|-
|aug. 3-step
| aug. 3-step
|12\29, 496.6
| 12\29, 496.6
| M&
| M&
|4/3
| 4/3
| +9
| +9
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 4-step
| dim. 4-step
|13\29, 537.9
| 13\29, 537.9
| N@@
| N@@
|15/11
| 15/11
| -12
| -12
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 4-step
| minor 4-step
| 14\29, 579.3
| 14\29, 579.3
| N@
| N@
|7/5
| 7/5
| -4
| -4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 4-step
| major 4-step
| 15\29 620.7
| 15\29 620.7
|N
| N
|10/7
| 10/7
| +4
| +4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 4-step
| aug. 4-step
|16\29 662.1
| 16\29 662.1
|N&
| N&
|22/15
| 22/15
| +12
| +12
|-
|-
|dim. 5-step
| dim. 5-step
|17\29, 703.4
| 17\29, 703.4
| O@
| O@
|3/2
| 3/2
| -9
| -9
|-
|-
|perf. 5-step
| perf. 5-step
| 18\29, 755.2
| 18\29, 755.2
| O
| O
|20/13
| 20/13
| -1
| -1
|-
|-
|aug. 5-step
| aug. 5-step
|19\29, 786.2
| 19\29, 786.2
| O&
| O&
|11/7
| 11/7
| +7
| +7
|-
|-
| doubly aug. 5-step
| doubly aug. 5-step
|20\29 827.6
| 20\29 827.6
|O&&
| O&&
|
|
| +15
| +15
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 6-step
| dim. 6-step
|20\29 827.6
| 20\29 827.6
|P@@
| P@@
|
|
| -14
| -14
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 6-step
| minor 6-step
| 21\29 869.0
| 21\29 869.0
|P@
| P@
|
|
| -6
| -6
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6-step
| major 6-step
| 22\29, 910.3
| 22\29, 910.3
| P
| P
|22/13
| 22/13
| +2
| +2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 6-step
| aug. 6-step
|23\29, 951.7
| 23\29, 951.7
| P&
| P&
|26/15
| 26/15
| +10
| +10
|-
|-
|dim. 7-step
| dim. 7-step
|24\29, 993.1
| 24\29, 993.1
| Q@
| Q@
|16/9
| 16/9
| -11
| -11
|-
|-
|minor 7-step
| minor 7-step
| 25\29, 1034.5
| 25\29, 1034.5
|Q
| Q
|20/11
| 20/11
| -3
| -3
|-
|-
|major 7-step
| major 7-step
| 26\29, 1075.9
| 26\29, 1075.9
|Q&
| Q&
|13/7
| 13/7
| +5
| +5
|-
|-
| aug. 7-step
| aug. 7-step
|27\29, 1117.2
| 27\29, 1117.2
|Q&&
| Q&&
|  
|  
| +13
| +13
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 8-step
| dim. 8-step
|28\29, 1158.6
| 28\29, 1158.6
|J@
| J@
|
|
| -8
| -8
Line 840: Line 840:
|-
|-
! class="unsortable" |Degree
! class="unsortable" |Degree
!Size in 23edo (superhard)
! Size in 23edo (superhard)
! class="unsortable" |Note name on J
! class="unsortable" |Note name on J
! class="unsortable" |Approximate ratios (23edo)
! class="unsortable" |Approximate ratios (23edo)
!#Gens up
! #Gens up
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|unison
| unison
|0\23, 0.0
| 0\23, 0.0
|J
| J
|1/1
| 1/1
|0
| 0
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|oneirochroma
| oneirochroma
| 3\23, 156.5
| 3\23, 156.5
|J&
| J&
|
|
| +8
| +8
|-
|-
|minor step
| minor step
|1\23, 52.2
| 1\23, 52.2
|K@
| K@
|
|
| -5
| -5
|-
|-
|major step
| major step
|4\23, 208.7
| 4\23, 208.7
|K
| K
|
|
| +3
| +3
|-
|-
| aug. step
| aug. step
|7\23, 365.2
| 7\23, 365.2
|K&
| K&
|21/17, inverse φ
| 21/17, inverse φ
| +11
| +11
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 2-step
| dim. 2-step
|2\23, 104.3
| 2\23, 104.3
|L@
| L@
|17/16
| 17/16
| -10
| -10
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 2-step
| minor 2-step
| 5\23, 260.9
| 5\23, 260.9
|L
| L
|
|
| -2
| -2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 2-step
| major 2-step
| 8\23, 417.4
| 8\23, 417.4
|L&
| L&
|14/11
| 14/11
| +6
| +6
|-
|-
|dim. 3-step
| dim. 3-step
|6\23, 313.0
| 6\23, 313.0
|M@
| M@
|6/5
| 6/5
| -7
| -7
|-
|-
|perf. 3-step
| perf. 3-step
| 9\23, 469.6
| 9\23, 469.6
|M
| M
|21/16
| 21/16
| +1
| +1
|-
|-
|aug. 3-step
| aug. 3-step
|12\23, 626.1
| 12\23, 626.1
| M&
| M&
|
|
| +9
| +9
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 4-step
| dim. 4-step
|7\23, 365.2
| 7\23, 365.2
|N@@
| N@@
|21/17, inverse φ
| 21/17, inverse φ
| -12
| -12
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 4-step
| minor 4-step
| 10\23, 521.7
| 10\23, 521.7
| N@
| N@
Line 923: Line 923:
| -4
| -4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 4-step
| major 4-step
| 13\23, 678.3
| 13\23, 678.3
| N
| N
|
|
| +4
| +4
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 4-step
| aug. 4-step
|16\23, 834.8
| 16\23, 834.8
| N&
| N&
|34/21, φ
| 34/21, φ
| +12
| +12
|-
|-
|dim. 5-step
| dim. 5-step
|11\23, 573.9
| 11\23, 573.9
| O@
| O@
|
|
| -9
| -9
|-
|-
|perf. 5-step
| perf. 5-step
| 14\23, 730.4
| 14\23, 730.4
| O
| O
|32/21
| 32/21
| -1
| -1
|-
|-
|aug. 5-step
| aug. 5-step
|17\23, 887.0
| 17\23, 887.0
| O&
| O&
|5/3
| 5/3
| +7
| +7
|-
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|minor 6-step
| minor 6-step
| 15\23 782.6
| 15\23 782.6
|P@
| P@
|11/7
| 11/7
| -6
| -6
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|major 6-step
| major 6-step
| 18\23, 939.1
| 18\23, 939.1
| P
| P
|
|
| +2
| +2
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|aug. 6-step
| aug. 6-step
|21\23, 1095.7
| 21\23, 1095.7
|P&
| P&
|32/17
| 32/17
| +10
| +10
|-
|-
|dim. 7-step
| dim. 7-step
|16\23, 834.8
| 16\23, 834.8
| Q@
| Q@
|34/21, φ
| 34/21, φ
| -11
| -11
|-
|-
|minor 7-step
| minor 7-step
| 19\23, 991.3
| 19\23, 991.3
| Q
| Q
|
|
| -3
| -3
|-
|-
| major 7-step
| major 7-step
| 22\23, 1147.8
| 22\23, 1147.8
|Q&
| Q&
|
|
| +5
| +5
|-
|-
|- bgcolor="#eaeaff"
|- bgcolor="#eaeaff"
|dim. 8-step
| dim. 8-step
|20\23, 1043.5
| 20\23, 1043.5
|J@
| J@
|
|
| -8
| -8
Line 1,009: Line 1,008:
|-
|-
!
!
![[38edo]]
! [[38edo]]
! [[53edo]]
! [[53edo]]
![[63edo]]
! [[63edo]]
!Optimal ([[POTE]]) Buzzard tuning
! Optimal ([[POTE]]) Buzzard tuning
!JI intervals represented (2.3.5.7.13 subgroup)
! JI intervals represented (2.3.5.7.13 subgroup)
|-
|-
| generator (g)
| generator (g)
|15\38, 473.68
| 15\38, 473.68
|21\53, 475.47
| 21\53, 475.47
|25\63, 476.19
| 25\63, 476.19
|475.69
| 475.69
|4/3 21/16
| 4/3 21/16
|-
|-
| L (3g - octave)
| L (3g - octave)
|7/38, 221.04
| 7/38, 221.04
| 10/53, 226.41
| 10/53, 226.41
|12/63, 228.57
| 12/63, 228.57
|227.07
| 227.07
|8/7
| 8/7
|-
|-
|s (-5g + 2 octaves)
| s (-5g + 2 octaves)
| 1/38, 31.57
| 1/38, 31.57
|1/53 22.64
| 1/53 22.64
|1/63 19.05
| 1/63 19.05
|21.55
| 21.55
|50/49 81/80 91/90
| 50/49 81/80 91/90
|}
|}


Line 1,043: Line 1,042:
|-
|-
! Degree
! Degree
!Size in 38edo
! Size in 38edo
!Size in 53edo
! Size in 53edo
!Size in 63edo
! Size in 63edo
!Size in POTE tuning
! Size in POTE tuning
!Note name on Q
! Note name on Q
! class="unsortable" |Approximate ratios
! class="unsortable" |Approximate ratios
!#Gens up
! #Gens up
|-
|-
|1
| 1
|0\38, 0.00
| 0\38, 0.00
|0\53, 0.00
| 0\53, 0.00
|0\63, 0.00
| 0\63, 0.00
|0.00
| 0.00
|Q
| Q
|1/1
| 1/1
|0
| 0
|-
|-
|2
| 2
|7\38, 221.05
| 7\38, 221.05
| 10\53, 226.42
| 10\53, 226.42
|12\63, 228.57
| 12\63, 228.57
|227.07
| 227.07
|J
| J
|8/7
| 8/7
| +3
| +3
|-
|-
|3
| 3
|14\38, 442.10
| 14\38, 442.10
|20\53, 452.83
| 20\53, 452.83
|24\63, 457.14
| 24\63, 457.14
|453.81
| 453.81
|K
| K
|13/10, 9/7
| 13/10, 9/7
| +6
| +6
|-
|-
|4
| 4
|15\38, 473.68
| 15\38, 473.68
|21\53, 475.47
| 21\53, 475.47
|25\63, 476.19
| 25\63, 476.19
|475.63
| 475.63
|L
| L
|21/16
| 21/16
| +1
| +1
|-
|-
|5
| 5
|22\38, 694.73
| 22\38, 694.73
|31\53, 701.89
| 31\53, 701.89
|37\63, 704.76
| 37\63, 704.76
|702.54
| 702.54
|M
| M
|3/2
| 3/2
| +4
| +4
|-
|-
|6
| 6
|29\38, 915.78
| 29\38, 915.78
|41\53, 928.30
| 41\53, 928.30
|49\63, 933.33
| 49\63, 933.33
|929.45
| 929.45
|N
| N
|12/7, 22/13
| 12/7, 22/13
| +7
| +7
|-
|-
|7
| 7
|30\38, 947.36
| 30\38, 947.36
|42\53, 950.94
| 42\53, 950.94
|50\63, 952.38
| 50\63, 952.38
|951.27
| 951.27
|O
| O
|26/15
| 26/15
| +2
| +2
|-
|-
|8
| 8
|37\38, 1168.42
| 37\38, 1168.42
|52\53, 1177.36
| 52\53, 1177.36
|62\63, 1180.95
| 62\63, 1180.95
|1178.18
| 1178.18
|P
| P
|98/50, 160/81
| 98/50, 160/81
| +5
| +5
|}
|}


==Modes==
== Modes ==
Oneirotonic modes are named after cities in the Dreamlands.
Oneirotonic modes are named after cities in the Dreamlands.
{| class="wikitable"
{| class="wikitable"
|-
|-
| style="text-align:center;" |'''Mode'''
! style="text-align:center;" |'''Mode'''
| style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
! style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
| style="text-align:center;" |'''Name'''
! style="text-align:center;" |'''Name'''
|'''Other names'''
|'''Other names'''
|-
|-
Line 1,175: Line 1,174:


==Approaches==
==Approaches==
*[[5L 3s/Inthar's approach]]
* [[5L 3s/Inthar's approach]]
*[[5L 3s/Temperaments]]
* [[5L 3s/Temperaments]]


==Samples==
==Samples==
Line 1,208: Line 1,207:


==See also==
==See also==
*[[Well-Tempered 13-Tone Clavier]] (collab project to create 13edo oneirotonic keyboard pieces in a variety of keys and modes)
* [[Well-Tempered 13-Tone Clavier]] (collab project to create 13edo oneirotonic keyboard pieces in a variety of keys and modes)
==Scale tree==
==Scale tree==
Generator ranges:
* Bright generator: 450 cents (3\8) to 480 cents (2\5)
* Dark generator: 720 cents (3\5) to 750 cents (5\8)
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="6" |Generator
! colspan="6" |Bright generator
!Cents
!Cents
!L
!L
Line 1,218: Line 1,221:
!Comments
!Comments
|-
|-
|3\8|| || || || ||  || 450.000||1||1||1.000||
|3\8|| || || || ||  || 450.000 ||1||1||1.000||
|-
|-
| || || || || || 17\45 || 453.333||6||5||1.200||
| || || || || || 17\45 || 453.333 ||6||5||1.200||
|-
|-
| || || || ||14\37 ||  || 454.054||5||4||1.250||
| || || || ||14\37 ||  || 454.054 ||5||4||1.250||
|-
|-
| || || || || || 34\59 || 454.545||9||7||1.286||
| || || || || || 34\59 || 454.545 ||9||7||1.286||
|-
|-
| || || ||11\29 ||  || || 455.172||4||3||1.333||
| || || ||11\29 ||  || || 455.172 ||4||3||1.333||
|-
|-
| || || || || || 30\79 || 455.696||11||8||1.375||
| || || || || || 30\79 || 455.696 ||11||8||1.375||
|-
|-
| || || || ||19\50 ||  || 456.000||7||5||1.400||
| || || || ||19\50 ||  || 456.000 ||7||5||1.400||
|-
|-
| || || || || || 27\71 || 456.338||10||7||1.429||
| || || || || || 27\71 || 456.338 ||10||7||1.429||
|-
|-
| || ||8\21|| || || ||457.143||3||2||1.500||L/s = 3/2
| || ||8\21|| || || ||457.143 ||3||2||1.500||
|-
|-
|  || ||  || || || 29\76 || 457.895||11||7||1.571||
|  || ||  || || || 29\76 || 457.895 ||11||7||1.571||
|-
|-
| || || || ||21\55 ||  || 458.182||8||5||1.600||
| || || || ||21\55 ||  || 458.182 ||8||5||1.600||
|-
|-
| || || || || || 34\89 || 458.427||13||8||1.625||Golden oneirotonic
| || || || || || 34\89 || 458.427 ||13||8||1.625||Golden oneirotonic
|-
|-
| || || ||13\34 ||  || || 458.824||5||3||1.667||<!--Petrtri is in this region-->
| || || ||13\34 ||  || || 458.824 ||5||3||1.667||<!--Petrtri is in this region-->
|-
|-
|  || || || || || 31\81 || 459.259||12||7||1.714||
|  || || || || || 31\81 || 459.259 ||12||7||1.714||
|-
|-
| || || || ||18\47 ||  || 459.574||7||4||1.750||
| || || || ||18\47 ||  || 459.574 ||7||4||1.750||
|-
|-
| || || || || || 23\60 || 460.000||9||5||1.800||
| || || || || || 23\60 || 460.000 ||9||5||1.800||
|-
|-
| ||5\13|| || || || || 461.538||2||1||2.000||Basic oneirotonic<br>(generators smaller than this are proper)
| ||5\13|| || || || || 461.538 ||2||1||2.000||Basic oneirotonic<br>(generators smaller than this are proper)
|-
|-
| || || || || || 22\57 || 463.158||9||4||2.250||
| || || || || || 22\57 || 463.158 ||9||4||2.250||
|-
|-
| || || || ||17\44 ||  || 463.636||7||3||2.333||
| || || || ||17\44 ||  || 463.636 ||7||3||2.333||
|-
|-
| || || || || || 29\75 || 464.000||12||5||2.400||
| || || || || || 29\75 || 464.000 ||12||5||2.400||
|-
|-
| || || ||12\31 ||  || || 464.516||5||2||2.500||<!--A-Team is in this region-->
| || || ||12\31 ||  || || 464.516 ||5||2||2.500||<!--A-Team is in this region-->
|-
|-
|  || || || || || 31\80 || 465.000||13||5||2.600||
|  || || || || || 31\80 || 465.000 ||13||5||2.600||
|-
|-
| || || || ||19\49 ||  || 465.306||8||3||2.667||
| || || || ||19\49 ||  || 465.306 ||8||3||2.667||
|-
|-
| || || || || || 26\67 || 465.672||11||4||2.750||
| || || || || || 26\67 || 465.672 ||11||4||2.750||
|-
|-
| || ||7\18|| || || ||466.667||3||1||3.000||L/s = 3/1
| || ||7\18|| || || || 466.667 ||3||1||3.000||
|-
|-
|  || ||  || || || 23\59 || 467.797||10||3||3.333||
|  || ||  || || || 23\59 || 467.797 ||10||3||3.333||
|-
|-
| || || || ||16\41 ||  || 468.293||7||2||3.500||
| || || || ||16\41 ||  || 468.293 ||7||2||3.500||
|-
|-
| || || || || || 25\64 || 468.750||11||3||3.667||
| || || || || || 25\64 || 468.750 ||11||3||3.667||
|-
|-
| || || ||9\23|| || ||469.565||4||1||4.000||
| || || ||9\23|| || || 469.565 ||4||1||4.000||
|-
|-
|  || ||  ||  || || 20\51 || 470.588||9||2||4.500||
|  || ||  ||  || || 20\51 || 470.588 ||9||2||4.500||
|-
|-
| || || || ||11\28 ||  || 471.429||5||1||5.000||
| || || || ||11\28 ||  || 471.429 ||5||1||5.000||
|-
|-
| || || || || || 13\33 || 472.727||6||1||6.000||
| || || || || || 13\33 || 472.727 ||6||1||6.000||
|-
|-
|2\5||  || || || ||  || 480.000||1||0||→ inf||
|2\5||  || || || ||  || 480.000 ||1||0||→ inf||
|}
|}


Retrieved from "https://en.xen.wiki/w/5L_3s"