5L 3s: Difference between revisions
mNo edit summary |
No edit summary |
||
| Line 48: | Line 48: | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
! | ! | ||
! Notation (1/1 = J) | !Notation (1/1 = J) | ||
![[TAMNAMS]] name | |||
!In L's and s's | |||
!# generators up | |||
!Notation of 2/1 inverse | |||
! [[TAMNAMS]] name | ! [[TAMNAMS]] name | ||
!In L's and s's | |||
! In L's and s's | |||
|- | |- | ||
| 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 104: | Line 104: | ||
| 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 122: | Line 122: | ||
| 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 | ||
|} | |} | ||
== Tuning ranges == | == Tuning ranges== | ||
=== Simple tunings === | ===Simple tunings === | ||
Table of intervals in the simplest oneirotonic tunings: | Table of intervals in the simplest oneirotonic tunings: | ||
{| class="wikitable right-2 right-3 right-4 sortable " | {| class="wikitable right-2 right-3 right-4 sortable " | ||
|- | |- | ||
! 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 | ||
| K | | K | ||
| +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 218: | Line 218: | ||
| L | | L | ||
| -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 226: | Line 226: | ||
| +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 | ||
|} | |} | ||
=== Hypohard === | ===Hypohard=== | ||
[[Hypohard]] oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1. | [[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: | 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 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. | *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]]. | Also, in [[18edo]] and [[31edo]], the minor 2-mosstep is close to [[7/6]]. | ||
| Line 309: | Line 309: | ||
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 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. | *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. | *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 | *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. | ||
The sizes of the generator, large step and small step of oneirotonic are as follows in various hypohard oneiro tunings. | The sizes of the generator, large step and small step of oneirotonic are as follows in various hypohard oneiro tunings. | ||
| Line 318: | Line 318: | ||
|- | |- | ||
! | ! | ||
! [[13edo]] (basic) | ![[13edo]] (basic) | ||
! [[18edo]] (hard) | ![[18edo]] (hard) | ||
! [[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 | ||
|} | |} | ||
==== Intervals ==== | ==== Intervals==== | ||
Sortable table of major and minor intervals in hypohard oneiro tunings: | Sortable table of major and minor intervals in hypohard oneiro tunings: | ||
{| class="wikitable right-2 right-3 right-4 sortable " | {| class="wikitable right-2 right-3 right-4 sortable " | ||
|- | |- | ||
! 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 465: | Line 465: | ||
| 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 | ||
|} | |} | ||
<references/> | <references /> | ||
=== 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 486: | Line 486: | ||
{| class="wikitable right-2 right-3 right-4 right-5" | {| class="wikitable right-2 right-3 right-4 right-5" | ||
|- | |- | ||
! | ! | ||
! [[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 | ||
|} | |} | ||
==== Intervals ==== | ====Intervals==== | ||
Sortable table of major and minor intervals in hyposoft tunings (13edo not shown): | Sortable table of major and minor intervals in hyposoft tunings (13edo not shown): | ||
{| class="wikitable right-2 right-3 sortable " | {| class="wikitable right-2 right-3 sortable " | ||
|- | |- | ||
! 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 | ||
| 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 | ||
|} | |} | ||
=== Parasoft to ultrasoft tunings === | ===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 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. | ||
| Line 627: | Line 627: | ||
{| class="wikitable right-2 right-3 right-4 right-5" | {| class="wikitable right-2 right-3 right-4 right-5" | ||
|- | |- | ||
! | ! | ||
! [[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 | ||
|} | |} | ||
==== Intervals ==== | ==== Intervals==== | ||
The intervals of the extended generator chain (-15 to +15 generators) are as follows in various softer-than-soft oneirotonic tunings. | The intervals of the extended generator chain (-15 to +15 generators) are as follows in various softer-than-soft oneirotonic tunings. | ||
{| class="wikitable right-2 right-3 sortable " | {| class="wikitable right-2 right-3 sortable " | ||
|- | |- | ||
! 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. | | 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. mos9th | |dim. mos9th | ||
| 28\29, 1158.6 | |28\29, 1158.6 | ||
| J@ | |J@ | ||
| | | | ||
| -8 | | -8 | ||
|} | |} | ||
=== Parahard === | ===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). | 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 ==== | ====Intervals==== | ||
The intervals of the extended generator chain (-12 to +12 generators) are as follows in various oneirotonic tunings close to [[23edo]]. | The intervals of the extended generator chain (-12 to +12 generators) are as follows in various oneirotonic tunings close to [[23edo]]. | ||
{| class="wikitable right-2 right-3 sortable " | {| class="wikitable right-2 right-3 sortable " | ||
|- | |- | ||
! 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@ | ||
| | | | ||
| -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. mos9th | |dim. mos9th | ||
| 20\23, 1043.5 | |20\23, 1043.5 | ||
| J@ | |J@ | ||
| | | | ||
| -8 | | -8 | ||
|} | |} | ||
=== Ultrahard === | ===Ultrahard=== | ||
[[Buzzard]] is an oneirotonic rank-2 temperament in the [[Step ratio|pseudopaucitonic]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum. | [[Buzzard]] is an oneirotonic rank-2 temperament in the [[Step ratio|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 | 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 its own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[The_Archipelago|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. | 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. | ||
| Line 1,015: | Line 1,015: | ||
{| class="wikitable right-2 right-3 right-4 right-5" | {| class="wikitable right-2 right-3 right-4 right-5" | ||
|- | |- | ||
! | ! | ||
! [[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 | ||
|} | |} | ||
==== Intervals ==== | ==== Intervals==== | ||
Sortable table of intervals in the Dylathian mode and their Buzzard interpretations: | Sortable table of intervals in the Dylathian mode and their Buzzard interpretations: | ||
| Line 1,050: | Line 1,050: | ||
|- | |- | ||
! 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''' | ||
|- | |- | ||
| | LLsLLsLs | | |LLsLLsLs | ||
| style="text-align:center;" | 7|0 | | style="text-align:center;" |<nowiki>7|0</nowiki> | ||
| | Dylathian (də-LA(H)TH-iən) | | |Dylathian (də-LA(H)TH-iən) | ||
|- | |- | ||
| | LLsLsLLs | | |LLsLsLLs | ||
| style="text-align:center;" | 6|1 | | style="text-align:center;" |<nowiki>6|1</nowiki> | ||
| | Illarnekian (ill-ar-NEK-iən) | | |Illarnekian (ill-ar-NEK-iən) | ||
|- | |- | ||
| | LsLLsLLs | | |LsLLsLLs | ||
| style="text-align:center;" | 5|2 | | style="text-align:center;" |<nowiki>5|2</nowiki> | ||
| | Celephaïsian (kel-ə-FAY-zhən) | | |Celephaïsian (kel-ə-FAY-zhən) | ||
|- | |- | ||
| | LsLLsLsL | | |LsLLsLsL | ||
| style="text-align:center;" | 4|3 | | style="text-align:center;" |<nowiki>4|3</nowiki> | ||
| | Ultharian (ul-THA(I)R-iən) | | |Ultharian (ul-THA(I)R-iən) | ||
|- | |- | ||
| | LsLsLLsL | | |LsLsLLsL | ||
| style="text-align:center;" | 3|4 | | style="text-align:center;" |<nowiki>3|4</nowiki> | ||
| | Mnarian (mə-NA(I)R-iən) | | |Mnarian (mə-NA(I)R-iən) | ||
|- | |- | ||
| | sLLsLLsL | | |sLLsLLsL | ||
| style="text-align:center;" | 2|5 | | style="text-align:center;" |<nowiki>2|5</nowiki> | ||
| | Kadathian (kə-DA(H)TH-iən) | | |Kadathian (kə-DA(H)TH-iən) | ||
|- | |- | ||
| | sLLsLsLL | | |sLLsLsLL | ||
| style="text-align:center;" | 1|6 | | style="text-align:center;" |<nowiki>1|6</nowiki> | ||
| | Hlanithian (lə-NITH-iən) | | |Hlanithian (lə-NITH-iən) | ||
|- | |- | ||
| | sLsLLsLL | | |sLsLLsLL | ||
| style="text-align:center;" | 0|7 | | style="text-align:center;" |<nowiki>0|7</nowiki> | ||
| | Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn" | | |Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn" | ||
|} | |} | ||
== Approaches == | ==Approaches== | ||
* [[5L 3s/Inthar's approach]] | *[[5L 3s/Inthar's approach]] | ||
* [[5L 3s/Temperaments]] | *[[5L 3s/Temperaments]] | ||
== Samples == | ==Samples== | ||
[[File:The Angels' Library edo.mp3]] [[:File:The Angels' Library edo.mp3|The Angels' Library]] by [[Inthar]] in the Sarnathian (23233233) mode of 21edo oneirotonic ([[:File:The Angels' Library Score.pdf|score]]) | [[File:The Angels' Library edo.mp3]] [[:File:The Angels' Library edo.mp3|The Angels' Library]] by [[Inthar]] in the Sarnathian (23233233) mode of 21edo oneirotonic ([[:File:The Angels' Library Score.pdf|score]]) | ||
| Line 1,194: | Line 1,194: | ||
(by [[Igliashon Jones]], 13edo, J Celephaïsian) | (by [[Igliashon Jones]], 13edo, J Celephaïsian) | ||
== 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== | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! colspan="6" | Generator | ! colspan="6" |Generator | ||
! Cents | !Cents | ||
! L | !L | ||
! s | !s | ||
! L/s | !L/s | ||
! 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||L/s = 3/2 | ||
|- | |- | ||
| || || || || || 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||L/s = 3/1 | ||
|- | |- | ||
| || || || || || 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|| | ||
|} | |} | ||