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My ([[User:IlL|Inthar lus Lăneaf's]]) subjective perception of the relative consonance of different intervals from the most consonant to the most dissonant (octave equivalents are not taken into account):
My ([[User:IlL|Inthar's]]) subjective perception of the relative consonance of different intervals from the most consonant to the most dissonant (octave equivalents are not taken into account):
*''Basals'' (the most consonant): major second, major and minor thirds
*''Basals'' (the most consonant): major second, major and minor thirds
*''Glitterers'' (intermediate, buzzy consonance): major and minor fourths, major and minor sixths, major and minor sevenths, minor ninth
*''Glitterers'' (intermediate, buzzy consonance): major and minor fourths, major and minor sixths, major and minor sevenths, minor ninth
*''Flarers'' (the most dissonant): minor and major fifths, the most dissonant and categorically ambiguous intervals.
*''Flarers'' (the most dissonant): minor and major fifths, the most dissonant and categorically ambiguous intervals.


Cheat sheet of important [[MOS]] scale types with L = major second, s = minor second:
Cheat sheet of important [[MOS]] scale types with 9 notes or fewer:


{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2"
Line 82: Line 82:
! Most common consonant tetrad(s)
! Most common consonant tetrad(s)
|-
|-
| archeotonic (LLLLLLs)
| archeotonic (2222221)
| major second
| major second (2\13)
| 4:5:9
| 4:5:9
| 4:5:9:11, 4:5:9:13
| 4:5:9:11, 4:5:9:13
|-
|-
| oneirotonic (LLsLsLLs)
| Father pentatonic (32323)
| minor fourth
| minor fourth (5\13)
| 4:9:21. Also important: 4:5:9 and its minor counterpart 4:19:9.
|
|
|-
| oneirotonic (21221221)
| minor fourth (5\13)
| 4:9:21. Also important: 4:5:9 and its minor counterpart 0-3-15.
| Basic triads with added 6ths and 7ths
| Basic triads with added 6ths and 7ths
|-
|-
| nonatonic (LsLsLsLss)
| Lovecraft nonatonic (212121211)
| minor third
| minor third (3\13)
| 4:11:13
| 4:11:13
| 4:11:13:19, 4:9:11:13
| 4:9:11:13
|-
|-
| decatonic (LssLsssLss)
| Sephiroth decatonic (1313131)
| major third
| major third (4\13)
| 4:5:13
| 4:5:13
| 4:5:13:21
| 4:5:13:21
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 168: Line 173:
The root-major third-major ninth (approximating 4:5:9; J-L-K in Kentaku notation) and its minor equivalent root-minor third-major ninth (J-Lb-K in Kentaku notation) may be considered equivalents of root-third-fifth chords in diatonic music. Archeotonic scales have 6 such triads, 5 "major" and 1 "minor". The 11th and 13th harmonics are also plentiful, as already noted by Cryptic Ruse; 4 roots have the 11th harmonic over them and 5 roots have the 13th harmonic over them.  
The root-major third-major ninth (approximating 4:5:9; J-L-K in Kentaku notation) and its minor equivalent root-minor third-major ninth (J-Lb-K in Kentaku notation) may be considered equivalents of root-third-fifth chords in diatonic music. Archeotonic scales have 6 such triads, 5 "major" and 1 "minor". The 11th and 13th harmonics are also plentiful, as already noted by Cryptic Ruse; 4 roots have the 11th harmonic over them and 5 roots have the 13th harmonic over them.  


The chord spelled root-major third-major fourth-minor sixth in archeotonic nomenclature occurs twice in archeotonic and I call it The Beloved Extra Special Tetrad (BEST). The reason it's beloved and extra special is that it can be interpreted both as an 8:10:11:13 and as a 13:16:18:21 (which can be revoiced as 8:9:13:21), thanks to the way 13edo conflates higher-limit JI intervals together.
The chord spelled root-major third-major fourth-minor sixth in archeotonic nomenclature occurs twice in archeotonic. It can be interpreted both as an 8:10:11:13 and as a 13:16:18:21 (which can be revoiced as 8:9:13:21), thanks to the way 13edo conflates higher-limit JI intervals together.


Archeotonic offers fairly familiar-sounding chord progressions by major seconds, thirds, and (both major and minor) fourths. One example is root-major third-two major thirds-root (spelled J major - L major - N# major - J major in J Ryonian), where the (two major thirds) is a 21/16 minor fourth away from the root.
Archeotonic offers fairly familiar-sounding chord progressions by major seconds, thirds, and (both major and minor) fourths. One example is root-major third-two major thirds-root (spelled J major - L major - N# major - J major in J Ryonian), where the (two major thirds) is a 21/16 minor fourth away from the root.
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 (the oneirotonic Dorian)
#Mnarian: 2 1 2 1 2 2 1 2 (half-diminished + Dorian)
#Kadathian: 1 2 2 1 2 2 1 2 (Locrian + Dorian)
#Hlanithian: 1 2 2 1 2 1 2 2 (Locrian + natural minor)
#Sarnathian: 1 2 1 2 2 1 2 2 (diminished + natural minor)
 
==== 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...
 
=== Tetrachordal 8-note scales ===
 
You can also view oneirotonic as scales made of two 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 1 1] [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).
**Harmonically this will give you an 8:10:13 over the first degree, an 8:10:11 over the second degree, a "minor" key and an 8:9:10:11:13 over the fourth degree, an 8:9:10:11 over the fifth degree and an 8:9:10:13 over the seventh degree.
**Melodically you can play tricks by going up 5 scale steps which will be a fifth instead of a sixth, the same note as down 3 steps.
 
=== Samples ===
 
[[File:Oneirotonic 3 part sample.mp3]]


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


== Switching between archeo- and oneirotonic ==
== Switching between archeo- and oneirotonic ==
Line 296: 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 310: 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 329: Line 236:
| 0.00, 1200.00
| 0.00, 1200.00
| J
| J
| '''1/1''', '''2/1'''
| 1/1, 2/1
| 0
| 0
|-
|-
Line 335: 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 341: 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 347: 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 353: 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 365: 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 391: Line 298:
[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) ==
== Sephiroth heptatonic (3L 4s) ==
 
The symmetric 1313131 mode:
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 following is a sortable table of LsssLssLss. (Harmonics are in bold; this is useful for seeing a chord's complexity when you sort the intervals according to the generator chain.) 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.
 
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|-
|-
! style="text-align:right" | Degree
! Degree
! Cents
! Cents
! Note name on J
! Note name on J
! Approximate ratios
! Approximate ratios
! # generators up
! #Gens up
|-
|-
| 1, 11
| 1, 8
| 0.00, 1200.00
| 0.00, 1200.00
| J
| J
| '''1/1''', '''2/1'''
| 1/1, 2/1
| 0
| 0
|-
|-
| 2
| 2
| 184.62
| 92.31
| K
| Kb
| '''9/8''', 10/9, 11/10, 19/17, 21/19
| 17/16, 18/17, 19/18, 20/19, 21/20, 22/21  
| +7
| +1
|-
|-
| 3
| 3
| 276.92
| Lb
| 7/6, 13/11, 20/17, '''19/16''', 22/19
| +4
|-
| 4
| 369.23
| 369.23
| L
| L
| '''5/4''', 11/9, 16/13, 26/21  
| 5/4, 11/9, 16/13, 26/21
| +1
| +2
|-
|-
| 5
| 4
| 461.54
| 461.54
| M
| M
| 13/10, 17/13, '''21/16''', 22/17  
| 13/10, 17/13, 21/16, 22/17
| -2
| +3
|-
|-
| 6
| 5
| 646.15
| N
| 16/11, 13/9, 19/13
| +5
|-
| 7
| 738.46
| 738.46
| Ob
| Ob
| 17/11, 20/13, 26/17, 32/21  
| 17/11, 20/13, 26/17, 32/21  
| +2
| +4
|-
|-
| 8
| 6
| 830.77
| 830.77
| O
| Pb
| 8/5, '''13/8''', 18/11, 21/13  
| 8/5, 13/8, 18/11, 21/13
| -1
| -2
|-
|-
| 9
| 7
| 1015.38
| 1107.69  
| P#
| 9/5, 16/9, 20/11, 34/19, 38/21
| +6
|-
| 10
| 1107.69
| Q
| Q
| 17/9, 19/10, 21/11, 32/17, 36/19, 40/21  
| 17/9, 19/10, 21/11, 32/17, 36/19, 40/21
| +3
| -1
|}
|}


=== Melodic properties ===
{{Navbox scale gallery}}
 
The decatonic scale can be viewed as containing archeotonic or oneirotonic scales with possible chromatic alterations, containing different degrees of some intervals. For example, the LsssLssLss mode can be viewed as:
 
*From a 7-tone POV: P1 - M2 - m3~M3 - m4 - m5~M5 - m6 - m7~M7 - P8
*From an 8-tone POV: P1 - M2 - m3~M3 - m4 - Mᴛ - M5 - m6 - m7~M7 - P8 or P1 - M2 - m3~M3 - m4 - Mᴛ - M5~A5 - A6 - M7 - P8, thus containing chromatically altered versions of Dylathian, Ilarnekian, Ultharian, and Celephaïsian
 
The 3113131 subset (P1-A2-M3-m4-M5-m6-M7-P8), is more important from the regular temperament POV, in that you can modulate up by major thirds by using 4:5:13 triads on it. The subset is also melodically interesting and pleasing.
 
== Other stuff ==


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"