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13edo scales: Difference between revisions

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Being a 7-note scale, the unison to octave interval categories remain the same as in the diatonic scale, except that we now have major fourths (6\13, approx. 11/8) and minor fourths (5\13, approx. 21/16), and their inversions minor and major fifths. An interesting feature is that you can switch whether you perceive an interval as minor or major by approaching it from opposite directions: for example, a minor sixth can be made to sound like a diatonic major sixth by walking up whole-whole-half-whole-whole steps from the tonic or like a diatonic minor sixth by walking down two whole steps.
Being a 7-note scale, the unison to octave interval categories remain the same as in the diatonic scale, except that we now have major fourths (6\13, approx. 11/8) and minor fourths (5\13, approx. 21/16), and their inversions minor and major fifths. An interesting feature is that you can switch whether you perceive an interval as minor or major by approaching it from opposite directions: for example, a minor sixth can be made to sound like a diatonic major sixth by walking up whole-whole-half-whole-whole steps from the tonic or like a diatonic minor sixth by walking down two whole steps.
===Scale===
===Scale===
The Ryonian mode:
Sortable table of intervals in the Ryonian mode (harmonics in bold):
{| class="wikitable sortable sorted"
{| class="wikitable sortable sorted"
|-
|-
Line 109: Line 109:
! | #Gens up
! | #Gens up
|-
|-
| | 1
| | 1, 8
| style="text-align:right;" | 0
| style="text-align:right;" | 0
| | J
| | J
| | 1/1
| | '''1/1'''
| | 0
| | 0
|-
|-
Line 118: Line 118:
| style="text-align:right;" | 184.615
| style="text-align:right;" | 184.615
| | K
| | K
| | 9/8, 10/9, 11/10, 19/17, 21/19  
| | '''9/8''', 10/9, 11/10, 19/17, 21/19  
| | +1
| | +1
|-
|-
Line 124: Line 124:
| style="text-align:right;" | 369.23
| style="text-align:right;" | 369.23
| | L
| | L
| | 5/4, 11/9, 16/13, 26/21
| | '''5/4''', 11/9, 16/13, 26/21
| | +2
| | +2
|-
|-
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| style="text-align:right;" | 553.85
| style="text-align:right;" | 553.85
| | M#
| | M#
| | 11/8, 18/13, 26/19   
| | '''11/8''', 18/13, 26/19   
| | +3
| | +3
|-
|-
Line 150: Line 150:
| | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| | +6
| | +6
|-
| | 8
| style="text-align:right;" | 1200
| | J
| | 2/1
| | 0
|}
|}


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The names I use for the oneirotonic interval classes are borrowed from diatonic interval categories: "second", "third", "fourth", "tritone" (4-step intervals), "fifth" (5-step intervals), "sixth" (6-step intervals), "seventh" (7-step intervals) and octave. You just have to remember that there's an extra category between fourths and fifths and that fourths and fifths are dissonant. Like in archeotonic you can change the perception of an interval by approaching it from different directions, but in oneirotonic it will change what diatonic interval class you hear it as: say, as both a third and a fourth, rather than both a major and a minor third.
The names I use for the oneirotonic interval classes are borrowed from diatonic interval categories: "second", "third", "fourth", "tritone" (4-step intervals), "fifth" (5-step intervals), "sixth" (6-step intervals), "seventh" (7-step intervals) and octave. You just have to remember that there's an extra category between fourths and fifths and that fourths and fifths are dissonant. Like in archeotonic you can change the perception of an interval by approaching it from different directions, but in oneirotonic it will change what diatonic interval class you hear it as: say, as both a third and a fourth, rather than both a major and a minor third.
===Scale===
===Scale===
The Dylathian mode is the most otonal mode, and is the basis for Kentaku note names JKLMNOPQJ (J is approx. 180 Hz, or an 11/8 above middle C)
The Dylathian mode is the most otonal mode, and is the basis for Kentaku note names JKLMNOPQJ (J is approx. 180 Hz, or an 11/8 above middle C). Sortable table of Dylathian with harmonics in bold:
{| class="wikitable sortable sorted"
{| class="wikitable sortable sorted"
|-
|-
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! | #Gens up
! | #Gens up
|-
|-
| | 1
| | 1, 9
| style="text-align:right;" | 0
| style="text-align:right;" | 0, 1200
| | J
| | J
| | 1/1
| | '''1/1'''
| | 0
| | 0
|-
|-
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| style="text-align:right;" | 184.615
| style="text-align:right;" | 184.615
| | K
| | K
| | 9/8, 10/9, 11/10, 19/17, 21/19  
| | '''9/8''', 10/9, 11/10, 19/17, 21/19  
| | +3
| | +3
|-
|-
Line 197: Line 191:
| style="text-align:right;" | 369.23
| style="text-align:right;" | 369.23
| | L
| | L
| | 5/4, 11/9, 16/13, 26/21
| | '''5/4''', 11/9, 16/13, 26/21
| | +6
| | +6
|-
|-
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| style="text-align:right;" | 461.54  
| style="text-align:right;" | 461.54  
| | M
| | M
| | 13/10, 17/13, 21/16, 22/17  
| | 13/10, 17/13, '''21/16''', 22/17  
| | +1
| | +1
|-
|-
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| style="text-align:right;" | 830.77
| style="text-align:right;" | 830.77
| | O
| | O
| | 8/5, 13/8, 18/11, 21/13  
| | 8/5, '''13/8''', 18/11, 21/13  
| | +7
| | +7
|-
|-
Line 229: Line 223:
| | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| | +5
| | +5
|-
| | 9
| style="text-align:right;" | 1200
| | J
| | 2/1
| | 0
|}
|}


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The brightest mode is LsLsLsLss or 0-2-3-5-6-8-9-11-12-13. The triad 4:11:13 occurs on degrees 1, 2, 3, 5, 7 and 9; these can be extended to either 4:10:11:13:17, 4:9:11:13:21, or 4:5:9:11:13 depending on what degree you're on. Since you get 21/16 as the minor version of 11/8, you also get two 8:13:17:21's with the same interval classes, on degrees 6 and 8. Degree 4 has a 4:5:11.
The brightest mode is LsLsLsLss or 0-2-3-5-6-8-9-11-12-13. The triad 4:11:13 occurs on degrees 1, 2, 3, 5, 7 and 9; these can be extended to either 4:10:11:13:17, 4:9:11:13:21, or 4:5:9:11:13 depending on what degree you're on. Since you get 21/16 as the minor version of 11/8, you also get two 8:13:17:21's with the same interval classes, on degrees 6 and 8. Degree 4 has a 4:5:11.


Sortable table with harmonics in bold:
{| class="wikitable sortable sorted"
{| class="wikitable sortable sorted"
|-
|-
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| style="text-align:right;" | 0.000
| style="text-align:right;" | 0.000
| | J
| | J
| | 1/1
| | '''1/1'''
| | 0
| | 0
|-
|-
Line 322: Line 311:
| style="text-align:right;" | 184.615
| style="text-align:right;" | 184.615
| | K
| | K
| | 9/8, 10/9, 11/10, 19/17, 21/19
| | '''9/8''', 10/9, 11/10, 19/17, 21/19
| | +5
| | +5
|-
|-
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| style="text-align:right;" | 276.92
| style="text-align:right;" | 276.92
| | Lb
| | Lb
| | 7/6, 13/11, 20/17, 19/16, 22/19  
| | 7/6, 13/11, 20/17, '''19/16''', 22/19  
| | +1
| | +1
|-
|-
Line 334: Line 323:
| style="text-align:right;" | 461.54
| style="text-align:right;" | 461.54
| | M
| | M
| | 13/10, 17/13, 21/16, 22/17  
| | 13/10, 17/13, '''21/16''', 22/17  
| | +6
| | +6
|-
|-
Line 340: Line 329:
| style="text-align:right;" | 553.85  
| style="text-align:right;" | 553.85  
| | M#/Nb
| | M#/Nb
| | 11/8, 18/13, 26/19  
| | '''11/8''', 18/13, 26/19  
| | +2
| | +2
|-
|-
Line 352: Line 341:
| style="text-align:right;" | 830.77
| style="text-align:right;" | 830.77
| | O
| | O
| | 8/5, 13/8, 18/11, 21/13  
| | 8/5, '''13/8''', 18/11, 21/13  
| | +3
| | +3
|-
|-
Line 366: Line 355:
| | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21  
| | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21  
| | +4
| | +4
|-
| | 10
| style="text-align:right;" | 1200
| | J
| | 2/1
| | 0
|}
|}
===Musical examples===
===Musical examples===
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The decatonic scale is excellent for 4:5:13 triads. It's generated by a major third, and two of them span a 4:5:13 triad, spanning degrees 1-4-8, and three of them span a 4:5:13:21 tetrad. This means that 8 of the 10 degrees have a 4:5:13 triad, and 7 of them in turn have a 4:5:13:21.
The decatonic scale is excellent for 4:5:13 triads. It's generated by a major third, and two of them span a 4:5:13 triad, spanning degrees 1-4-8, and three of them span a 4:5:13:21 tetrad. This means that 8 of the 10 degrees have a 4:5:13 triad, and 7 of them in turn have a 4:5:13:21.


The LsssLssLss mode is the only mode that has a 4:5:9:13:21 on the tonic. If you want an 11/8 instead of a 21/16 you can sharpen the 5th degree to get LssLsssLsss which is the only mode to have a 4:5:9:11:13 on the tonic.
The LsssLssLss mode is the only mode that has a 4:5:9:13:21 on the tonic. If you want an 11/8 instead of a 21/16 you can sharpen the 5th degree to get LssLsssLsss which is the only mode to have a 4:5:9:11:13 on the tonic. Sortable table of LsssLssLss with harmonics in bold:
{| class="wikitable sortable sorted"
{| class="wikitable sortable sorted"
|-
|-
Line 391: Line 374:
| style="text-align:right;" | 0
| style="text-align:right;" | 0
| | J
| | J
| | 1/1
| | '''1/1'''
| | 0
| | 0
|-
|-
Line 397: Line 380:
| style="text-align:right;" | 184.615
| style="text-align:right;" | 184.615
| | K
| | K
| | 9/8, 10/9, 11/10, 19/17, 21/19  
| | '''9/8''', 10/9, 11/10, 19/17, 21/19  
| | +7
| | +7
|-
|-
Line 403: Line 386:
| style="text-align:right;" | 276.92
| style="text-align:right;" | 276.92
| | Lb
| | Lb
| | 7/6, 13/11, 20/17, 19/16, 22/19
| | 7/6, 13/11, 20/17, '''19/16''', 22/19
| | +4
| | +4
|-
|-
Line 409: Line 392:
| style="text-align:right;" | 369.23
| style="text-align:right;" | 369.23
| | L
| | L
| | 5/4, 11/9, 16/13, 26/21  
| | '''5/4''', 11/9, 16/13, 26/21  
| | +1
| | +1
|-
|-
Line 415: Line 398:
| style="text-align:right;" | 461.54
| style="text-align:right;" | 461.54
| | M
| | M
| | 13/10, 17/13, 21/16, 22/17  
| | 13/10, 17/13, '''21/16''', 22/17  
| | -2
| | -2
|-
|-
Line 433: Line 416:
| style="text-align:right;" | 830.77
| style="text-align:right;" | 830.77
| | O
| | O
| | 8/5, 13/8, 18/11, 21/13  
| | 8/5, '''13/8''', 18/11, 21/13  
| | -1
| | -1
|-
|-
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