Xenharmonic Wiki

13edo scales: Difference between revisions

← Older editNewer edit →

Inthar (talk | contribs)

2 user groups
15,678 edits
VisualWikitext
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,678 edits
m better name/acronym
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,678 edits
m added more tables, corrected Kentaku notation, etc.
Line 1: Line 1:
Since 13 is prime, [[13edo]] has more MOS scale types than 12edo.
Since 13 is prime, [[13edo]] has more MOS scale types than 12edo; somewhat amazingly, all of them have a good number of consonant chords if you know where to look.


==Overview==
==Overview==
13edo intervals:
The intervals themselves are not very alien except near the middle of the octave, just a bit darker compared to their 12edo counterparts. They are familiar enough that they can be given pseudo-diatonic names:
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 99: Line 99:


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===
The Ryonian mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on J
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | J
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 184.615
| | K
| | 9/8, 10/9, 11/10, 19/17, 21/19
| | +1
|-
| | 3
| style="text-align:right;" | 369.23
| | L
| | 5/4, 11/9, 16/13, 26/21
| | +2
|-
| | 4
| style="text-align:right;" | 553.85
| | M#
| | 11/8, 18/13, 26/19 
| | +3
|-
| | 5
| style="text-align:right;" | 738.46
| | Ob
| | 17/11, 20/13, 26/17, 32/21
| | +4
|-
| | 6
| style="text-align:right;" | 923.08
| | P
| | 17/10, 12/7, 22/13, 19/11
| | +5
|-
| | 7
| style="text-align:right;" | 1107.69
| | Q
| | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| | +6
|-
| | 8
| style="text-align:right;" | 1200
| | J
| | 2/1
| | 0
|}


===Chords===
===Chords===
Line 114: Line 172:


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===
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)
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on J
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | J
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 184.615
| | K
| | 9/8, 10/9, 11/10, 19/17, 21/19
| | +3
|-
| | 3
| style="text-align:right;" | 369.23
| | L
| | 5/4, 11/9, 16/13, 26/21
| | +6
|-
| | 4
| style="text-align:right;" | 461.54
| | M
| | 13/10, 17/13, 21/16, 22/17
| | +1
|-
| | 5
| style="text-align:right;" | 646.15
| | N
| | 16/11, 13/9, 19/13
| | +4
|-
| | 6
| style="text-align:right;" | 830.77
| | O
| | 8/5, 13/8, 18/11, 21/13
| | +7
|-
| | 7
| style="text-align:right;" | 923.08
| | P
| | 17/10, 12/7, 22/13, 19/11
| | +2
|-
| | 8
| style="text-align:right;" | 1107.69
| | Q
| | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| | +5
|-
| | 9
| style="text-align:right;" | 1200
| | J
| | 2/1
| | 0
|}
===Chords===
===Chords===
Like in archaeotonic, seconds and thirds are similar in consonance to 12edo seconds and thirds, and similarly sixths and sevenths are similar to diatonic sixths and sevenths. Minor fourths (21/16) are dissonant, but they work a lot like diatonic perfect fourths do e.g. in "sus24" chords that resolve down to thirds, and can also be spread out to make convincing 4:9:21 chords which are common in oneirotonic.
Like in archaeotonic, seconds and thirds are similar in consonance to 12edo seconds and thirds, and similarly sixths and sevenths are similar to diatonic sixths and sevenths. Minor fourths (21/16) are dissonant, but they work a lot like diatonic perfect fourths do e.g. in "sus24" chords that resolve down to thirds, and can also be spread out to make convincing 4:9:21 chords which are common in oneirotonic.
Line 119: Line 242:
As in archeotonic harmony, root-third-ninth chords may be considered basic harmonic triads; oneirotonic scales have 5 such triads, 2 "major" and 3 "minor". J-L-K (4:5:9) and its minor counterpart J-Lb-K (4:19:9 if we're stretching it) work well with an added sixth or seventh, even when the resulting chord does not approximate an obvious JI chord.
As in archeotonic harmony, root-third-ninth chords may be considered basic harmonic triads; oneirotonic scales have 5 such triads, 2 "major" and 3 "minor". J-L-K (4:5:9) and its minor counterpart J-Lb-K (4:19:9 if we're stretching it) work well with an added sixth or seventh, even when the resulting chord does not approximate an obvious JI chord.


Oneirotonic provides two Orwell tetrads, made of three stacked minor thirds making one minor sixth; we get them by taking every second degree of the scale, JLNP or KMOQ. They sound like squashed diminished chords, but not quite. One could play an earbending trick where a movement up a major third and up 3 minor thirds will get you back to where you started unlike in 12edo. The two Orwell tetrads contain the two copies of 8:11:13 in oneirotonic, Q-M-O or L-P-J in J Ilarnekian.
Oneirotonic provides two Orwell tetrads, made of three stacked minor thirds making one minor sixth; we get them by taking every second degree of the scale, JLNP or KMOQ. They sound like squashed diminished chords, but not quite. One could play an earbending trick where a movement up a major third and up 3 minor thirds will get you back to where you started unlike in 12edo. The two Orwell tetrads contain the two copies of 8:11:13 in oneirotonic, Q-M-Ob or L-P-J in J Ilarnekian.


Minor tritones (approximating 11/8) work like tritones and they like to resolve inward to a third. Major tritones (16/11) are the opposite: they like to resolve outward to a sixth. Unlike in 12edo, fourths and tritones, and their octave inversions are very different in quality. Minor fourths and minor tritones are more consonant than their inversions major tritones and major fifths; they can also both be spread out to make them more consonant, whereas their inversions cannot.
Minor tritones (approximating 11/8) work like tritones and they like to resolve inward to a third. Major tritones (16/11) are the opposite: they like to resolve outward to a sixth. Unlike in 12edo, fourths and tritones, and their octave inversions are very different in quality. Minor fourths and minor tritones are more consonant than their inversions major tritones and major fifths; they can also both be spread out to make them more consonant, whereas their inversions cannot.


The diminished fourth can work either like the diatonic diminished fourth, or (uniquely in 13edo among all oneirotonic tunings) serve as an extra 5/4 in the scale and can be part of extra consonant chords (such as the aforementioned BEST that represents both 8:10:11:13 and 13:16:18:21, which occurs as O-J-K-M in J Ilarnekian, but it only represents 13:16:18:21 in other oneirotonic-supporting tunings such as [[31edo]]).
The diminished fourth can work either like the diatonic diminished fourth, or (uniquely in 13edo among all oneirotonic tunings) serve as an extra 5/4 in the scale and can be part of extra consonant chords (such as the aforementioned BEST that represents both 8:10:11:13 and 13:16:18:21, which occurs as Ob-J-K-M in J Ilarnekian, but it only represents 13:16:18:21 in other oneirotonic-supporting tunings such as [[31edo]]).


Basic chord progressions can move by minor fourths or major seconds: J major-M minor-P minor-O major-J major (in Ilarnekian) or J major-K major-O# major-M major-J major (in Dylathian)
Basic chord progressions can move by minor fourths or major seconds: J major-M minor-P minor-Ob major-J major (in Ilarnekian) or J major-K major-O major-M major-J major (in Dylathian)


===Modal harmony===
===Modal harmony===
Line 180: Line 303:
Generated by 3\13, the 276.9-cent minor third approximating [[13/11]], this scale sounds a little like the octatonic scale in 12edo with an extra small step inserted. Two of these make an 11/8 and three make a 13/8, making this scale very good for 4:11:13 triads. (In terms of regular temperament theory, this makes 13edo a tuning for the [[Color notation|bithotrilu]] temperament that tempers out the bithotrilu comma 1352/1331 = {{monzo|3 0 0 0 -3 2}}, aka "lovecraft temperament".) [[17edo]] also supports bithotrilu temperament and thus has a similar 4L 5s scale, generated by the 4\17 minor third. Similar scales also exist in 22edo and 31edo with flatter generators, but they use a [[Orwell|different temperament]] and won't approximate the 13th harmonic.
Generated by 3\13, the 276.9-cent minor third approximating [[13/11]], this scale sounds a little like the octatonic scale in 12edo with an extra small step inserted. Two of these make an 11/8 and three make a 13/8, making this scale very good for 4:11:13 triads. (In terms of regular temperament theory, this makes 13edo a tuning for the [[Color notation|bithotrilu]] temperament that tempers out the bithotrilu comma 1352/1331 = {{monzo|3 0 0 0 -3 2}}, aka "lovecraft temperament".) [[17edo]] also supports bithotrilu temperament and thus has a similar 4L 5s scale, generated by the 4\17 minor third. Similar scales also exist in 22edo and 31edo with flatter generators, but they use a [[Orwell|different temperament]] and won't approximate the 13th harmonic.
===Scale===
===Scale===
{| class="wikitable"
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.
 
{| class="wikitable sortable sorted"
|-
|-
! style="text-align:right" | Degree
! style="text-align:right" | Degree
! | Cents
! | Cents
! | Note name on J
! | Approximate ratios
! | Approximate ratios
! | #Gens
! | # generators up
|-
|-
| | 1
| | 1
| style="text-align:right;" | 0.000
| style="text-align:right;" | 0.000
| | J
| | 1/1
| | 1/1
| | 0
| | 0
Line 194: Line 321:
| | 2
| | 2
| style="text-align:right;" | 184.615
| style="text-align:right;" | 184.615
| | 9/8~10/9~11/10
| | K
| | 9/8, 10/9, 11/10, 19/17, 21/19
| | +5
| | +5
|-
|-
| | 3
| | 3
| style="text-align:right;" | 276.92
| style="text-align:right;" | 276.92
| | 13/11
| | Lb
| | 7/6, 13/11, 20/17, 19/16, 22/19
| | +1
| | +1
|-
|-
| | 4
| | 4
| style="text-align:right;" | 461.54
| style="text-align:right;" | 461.54
| | 21/16~13/10~17/13
| | M
| | 13/10, 17/13, 21/16, 22/17
| | +6
| | +6
|-
|-
| | 5
| | 5
| style="text-align:right;" | 553.85  
| style="text-align:right;" | 553.85  
| | 11/8
| | M#/Nb
| | 11/8, 18/13, 26/19
| | +2
| | +2
|-
|-
| | 6
| | 6
| style="text-align:right;" | 738.46
| style="text-align:right;" | 738.46
| | 20/13
| | N#/Ob
| | 17/11, 20/13, 26/17, 32/21
| | +7
| | +7
|-
|-
| | 7
| | 7
| style="text-align:right;" | 830.77
| style="text-align:right;" | 830.77
| | 13/8~8/5
| | O
| | 8/5, 13/8, 18/11, 21/13
| | +3
| | +3
|-
|-
| | 8
| | 8
| style="text-align:right;" | 1015.385
| style="text-align:right;" | 1015.385
| | 9/5
| | P#
| | 9/5, 16/9, 20/11, 34/19, 38/21
| | +8
| | +8
|-
|-
| | 9
| | 9
| style="text-align:right;" | 1107.69
| style="text-align:right;" | 1107.69
| |  
| | Q
| | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| | +4
| | +4
|-
|-
| | 10
| | 10
| style="text-align:right;" | 1200
| style="text-align:right;" | 1200
| | J
| | 2/1
| | 2/1
| | 0
| | 0
|}
|}
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.
===Musical examples===
===Musical examples===
[https://www.youtube.com/watch?v=x4Yesl8n6gc Brusselator Sprouts (by Xotla)] (The main riffs are in this scale, although key changes and notes outside the 9 note subset are used too.)
[https://www.youtube.com/watch?v=x4Yesl8n6gc Brusselator Sprouts (by Xotla)] (The main riffs are in this scale, although key changes and notes outside the 9 note subset are used too.)


==Decatonic (3L 7s)==
==Decatonic (3L 7s)==
The decatonic scale is optimized 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.
Line 251: Line 384:
! style="text-align:right" | Degree
! style="text-align:right" | Degree
! | Cents
! | Cents
! | Note name on J
! | Approximate ratios
! | Approximate ratios
! | #Gens
! | # generators up
|-
|-
| | 1
| | 1
| style="text-align:right;" | 0
| style="text-align:right;" | 0
| | J
| | 1/1
| | 1/1
| | 0
| | 0
|-
|-
| | 2
| | 2
| style="text-align:right;" | 185
| style="text-align:right;" | 184.615
| | 9/8~10/9~11/10
| | K
| | 9/8, 10/9, 11/10, 19/17, 21/19
| | +7
| | +7
|-
|-
| | 3
| | 3
| style="text-align:right;" | 277
| style="text-align:right;" | 276.92
| | 13/11
| | Lb
| | 7/6, 13/11, 20/17, 19/16, 22/19
| | +4
| | +4
|-
|-
| | 4
| | 4
| style="text-align:right;" | 369
| style="text-align:right;" | 369.23
| | 5/4~16/13
| | L
| | 5/4, 11/9, 16/13, 26/21
| | +1
| | +1
|-
|-
| | 5
| | 5
| style="text-align:right;" | 461
| style="text-align:right;" | 461.54
| | 21/16
| | M
| | 13/10, 17/13, 21/16, 22/17
| | -2
| | -2
|-
|-
| | 6
| | 6
| style="text-align:right;" | 646
| style="text-align:right;" | 646.15
| | 16/11
| | N
| | 16/11, 13/9, 19/13
| | +5
| | +5
|-
|-
| | 7
| | 7
| style="text-align:right;" | 731
| style="text-align:right;" | 738.46
| |  
| | Ob
| | 17/11, 20/13, 26/17, 32/21
| | +2
| | +2
|-
|-
| | 8
| | 8
| style="text-align:right;" | 831
| style="text-align:right;" | 830.77
| | 8/5~13/8~21/13
| | O
| | 8/5, 13/8, 18/11, 21/13  
| | -1
| | -1
|-
|-
| | 9
| | 9
| style="text-align:right;" | 1015
| style="text-align:right;" | 1015.385
| | 16/9~9/5~20/11
| | P#
| | 9/5, 16/9, 20/11, 34/19, 38/21
| | +6
| | +6
|-
|-
| | 10
| | 10
| style="text-align:right;" | 1108
| style="text-align:right;" | 1107.69
| |  
| | Q
| | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| | +3
| | +3
|-
|-
| | 11
| | 11
| style="text-align:right;" | 1200
| style="text-align:right;" | 1200
| | J
| | 2/1
| | 2/1
| | 0
| | 0
Retrieved from "https://en.xen.wiki/w/13edo_scales"