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13edo scales: Difference between revisions

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Line 86: Line 86:
| 4:5:9
| 4:5:9
| 4:5:9:11, 4:5:9:13
| 4:5:9:11, 4:5:9:13
|-
| Father pentatonic (32323)
| minor fourth (5\13)
|
|
|-
|-
| oneirotonic (21221221)
| oneirotonic (21221221)
Line 92: Line 97:
| Basic triads with added 6ths and 7ths
| Basic triads with added 6ths and 7ths
|-
|-
| nonatonic (212121211)
| Lovecraft nonatonic (212121211)
| minor third (3\13)
| minor third (3\13)
| 4:11:13
| 4:11:13
| 4:9:11:13
| 4:9:11:13
|-
|-
| decatonic (1313131)
| Sephiroth decatonic (1313131)
| major third (4\13)
| major third (4\13)
| 4:5:13
| 4:5:13
Line 124: Line 129:
| 0.00, 1200.00
| 0.00, 1200.00
| J
| J
| '''1/1''', '''2/1'''
| 1/1, 2/1
| 0
| 0
|-
|-
Line 130: Line 135:
| 184.62
| 184.62
| 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 136: Line 141:
| 369.23
| 369.23
| L
| L
| '''5/4''', 11/9, 16/13, 26/21
| 5/4, 11/9, 16/13, 26/21
| +2
| +2
|-
|-
Line 142: Line 147:
| 553.85
| 553.85
| M#
| M#
| '''11/8''', 18/13, 26/19   
| 11/8, 18/13, 26/19   
| +3
| +3
|-
|-
Line 154: Line 159:
| 923.08
| 923.08
| Pb
| Pb
| 8/5, '''13/8''', 18/11, 21/13
| 8/5, 13/8, 18/11, 21/13
| -2
| -2
|-
|-
Line 177: Line 182:


== Oneirotonic (5L 3s) ==
== Oneirotonic (5L 3s) ==
:''Main article: [[5L 3s]]''
[[File:Oneirotonic_Scale_-_Dylathian_in_L.png|alt=Oneirotonic Scale - Dylathian in L.png|800x135px|Oneirotonic Scale - Dylathian in L.png]]


The oneirotonic scale is the darker, damper, more "minory" cousin of archaeotonic. Only 2 out of 8 oneirotonic modes (Dylathian and Ilarnekian) are "major" in the sense of having a major third, and both sound pretty bittersweet.
[[:File:Oneirotonic_Scale_-_Dylathian_in_L.svg|Oneirotonic Scale - Dylathian in L.svg]]
 
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). Sortable table of Dylathian (Harmonics are in bold; this is useful for seeing a chord's complexity when you sort the intervals according to the generator chain):
 
{| class="wikitable right-1 right-2 sortable"
|-
! style="text-align:right" | Degree
! Cents
! Note name on J
! Approximate ratios
! #Gens up
|-
| 1, 9
| 0.00, 1200.00
| J
| '''1/1''', '''2/1'''
| 0
|-
| 2
| 184.62
| K
| '''9/8''', 10/9, 11/10, 19/17, 21/19
| +3
|-
| 3
| 369.23
| L
| '''5/4''', 11/9, 16/13, 26/21
| +6
|-
| 4
| 461.54
| M
| 13/10, 17/13, '''21/16''', 22/17
| +1
|-
| 5
| 646.15
| N
| 16/11, 13/9, 19/13
| +4
|-
| 6
| 830.77
| O
| 8/5, '''13/8''', 18/11, 21/13
| +7
|-
| 7
| 923.08
| P
| 17/10, 12/7, 22/13, 19/11
| +2
|-
| 8
| 1107.69
| Q
| 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| +5
|}
 
=== Chords ===
 
Despite being melodically familiar, oneirotonic may be the most harmonically complex of the 13edo scales; the most common consonant triad is a fairly complex 4:9:21. Hence oneirotonic especially benefits from either using inharmonic timbres in addition to harmonic ones, or using a well-tempered or [[primodal]] JI version of 13edo. The availability of primes also varies greatly by mode: for example, only Dylathian, Ilarnekian and Sarnathian have a 5/4 on the tonic, and only Mnarian, Kadathian, Hlanithian and Sarnathian have an 11/8 on the tonic.
 
=== Modal harmony ===
 
How I think about the 8 oneirotonic modes:
 
#Dylathian: 2 2 1 2 2 1 2 1 (major with hints of Mixolydian and "#5")
#Ilarnekian: 2 2 1 2 1 2 2 1 (major with hints of "b6")
#Celephaïsian: 2 1 2 2 1 2 2 1 (the oneirotonic melodic minor. Very classical-sounding; Easley Blackwood's 13-note etude uses this as its home mode.)
#Ultharian: 2 1 2 2 1 2 1 2
#Mnarian: 2 1 2 1 2 2 1 2
#Kadathian: 1 2 2 1 2 2 1 2
#Hlanithian: 1 2 2 1 2 1 2 2
#Sarnathian: 1 2 1 2 2 1 2 2
 
==== Modes with sharp tritone ====
 
The brighter modes can be viewed as providing a distorted version of diatonic functional harmony. For example, in the Dylathian mode, the 4:5:9 triad on the sixth degree can sound like both "V" and "III of iv" depending on context. Basic chord progressions can move by minor fourths, thirds, or major seconds: for example, 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).
 
==== Modes with flat tritone ====
 
The darker modes are radically different in character than the brighter modes. Because of the consonant 11/8 minor tritone and the 13/8 minor sixth, 11/8 sounds more like a stable scale function and the relatively dissonant 21/16 minor fourth wants to be a major third resolving up to the 11/8.
 
=== 8-note scales as modified 12edo scales ===
As stated many times, octatonic scales in 13edo are heptatonic scales in 12edo with one extra small step. Key: You have a choice over where to put the new small step!
 
You can also view oneirotonic modes as scales made of two shrunken 12edo tetrachords, each spanning a minor fourth, and one trichord spanning a minor third. This will let you build 13edo "tetrachordal" scales with a similar structure that is not one of the 8 modes, with tetrachord structures similar to 12edo ones. For example:
*[2 2 1] [2 1 2] [1 2] is a kind of Mixolydian
*[2 1 2] [2 2 1] [2 1] is a kind of Dorian. I personally think this sounds better than Ultharian.
*[2 1 2] [2 1] [1 3 1] is a kind of harmonic minor (also obtained by lowering the 7th degree of the Celephaïsian mode)
*[1 3 1] [2 1] [1 2 2] is a kind of Phrygian dominant scale (which also contains 1 3 1 2 2 2 2, a chromatic modification of the Zo-Kalarian mode of the archeotonic scale).
*[1 3 1] [2 1] [2 2 1] another Phrygian dominant


=== Samples ===
[[:File:13edo-fretboard-template.svg|13edo-fretboard-template.svg]]
 
[[File:Oneirotonic 3 part sample.mp3]]
 
‎(A rather classical-sounding 3-part harmonization of the ascending J Ilarnekian scale; tuning is 13edo)


== Switching between archeo- and oneirotonic ==
== Switching between archeo- and oneirotonic ==
Line 298: Line 203:
| Dylathian  2 2 1 2 2 1 2 1 || ↔ || Oukranian  2 2 1 2 2 2 2
| Dylathian  2 2 1 2 2 1 2 1 || ↔ || Oukranian  2 2 1 2 2 2 2
|-
|-
| Ilarnekian 2 2 1 2 1 2 2 1 || ↔ || Ryonian    2 2 2 2 2 2 1
| Illarnekian 2 2 1 2 1 2 2 1 || ↔ || Ryonian    2 2 2 2 2 2 1
|-
|-
| Ultharian  2 1 2 2 1 2 1 2 || ↔ || Tamashian  2 1 2 2 2 2 2
| Ultharian  2 1 2 2 1 2 1 2 || ↔ || Tamashian  2 1 2 2 2 2 2
Line 312: Line 217:
== Nonatonic (4L 5s) ==
== Nonatonic (4L 5s) ==


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 [[support]]s 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 ===
Line 331: Line 236:
| 0.00, 1200.00
| 0.00, 1200.00
| J
| J
| '''1/1''', '''2/1'''
| 1/1, 2/1
| 0
| 0
|-
|-
Line 337: Line 242:
| 184.62
| 184.62
| K
| K
| '''9/8''', 10/9, 11/10, 19/17, 21/19
| 9/8, 10/9, 11/10, 19/17, 21/19
| +5
| +5
|-
|-
Line 343: Line 248:
| 276.92
| 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 349: Line 254:
| 461.54
| 461.54
| M
| M
| 13/10, 17/13, '''21/16''', 22/17  
| 13/10, 17/13, 21/16, 22/17  
| +6
| +6
|-
|-
Line 355: Line 260:
| 553.85  
| 553.85  
| M#/Nb
| M#/Nb
| '''11/8''', 18/13, 26/19  
| 11/8, 18/13, 26/19  
| +2
| +2
|-
|-
Line 367: Line 272:
| 830.77
| 830.77
| O
| O
| 8/5, '''13/8''', 18/11, 21/13  
| 8/5, 13/8, 18/11, 21/13  
| +3
| +3
|-
|-
Line 394: Line 299:


== Sephiroth heptatonic (3L 4s) ==
== Sephiroth heptatonic (3L 4s) ==
The symmetric 1313131 mode:
{| class="wikitable right-1 right-2 sortable"
|-
! Degree
! Cents
! Note name on J
! Approximate ratios
! #Gens up
|-
| 1, 8
| 0.00, 1200.00
| J
| 1/1, 2/1
| 0
|-
| 2
| 92.31
| Kb
| 17/16, 18/17, 19/18, 20/19, 21/20, 22/21
| +1
|-
| 3
| 369.23
| L
| 5/4, 11/9, 16/13, 26/21
| +2
|-
| 4
| 461.54
| M
| 13/10, 17/13, 21/16, 22/17
| +3
|-
| 5
| 738.46
| Ob
| 17/11, 20/13, 26/17, 32/21
| +4
|-
| 6
| 830.77
| Pb
| 8/5, 13/8, 18/11, 21/13
| -2
|-
| 7
| 1107.69
| Q
| 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| -1
|}


== Other stuff ==
{{Navbox scale gallery}}


todo: try added fifths or tritones, describe chords with two additions or more
[[Category:13edo]]
[[Category:Lists of scales]]
[[Category:Guitar]]
Retrieved from "https://en.xen.wiki/w/13edo_scales"