Xenharmonic Wiki

User:VectorGraphics/16edo theory: Difference between revisions

← Older edit
VisualWikitext
m deploying new cat
m added link to xenverse
 
(5 intermediate revisions by 2 users not shown)
Line 1: Line 1:
Vector's 16edo theory was created for Earth#16.
{{breadcrumb|16edo}}
 
Vector's 16edo theory was created for [[Beyond the Xenverse|Earth#16]]. 16edo is an extremely easy equal scale to tune, since it ultimately just requires taking lots of square roots, which can be done by geometric methods.


==Introduction: note names and intervals==
==Introduction: note names and intervals==
Line 5: Line 7:


To avoid ambiguity, the VQDMN interval names will be used throughout the entire page. For example, the interval closest to 5/4 will be a "minor fourth", and the interval closest to 3/2 will be a "perfect sixth". However, chords will be named after the mediant's functional name, for familiarity with major and minor chords.
To avoid ambiguity, the VQDMN interval names will be used throughout the entire page. For example, the interval closest to 5/4 will be a "minor fourth", and the interval closest to 3/2 will be a "perfect sixth". However, chords will be named after the mediant's functional name, for familiarity with major and minor chords.
Additionally, to avoid confusion regarding the term "semitone", the 75-cent interval will be called the "eka", as in Armodue theory.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 14: Line 18:
!Interval name (functional)
!Interval name (functional)
!Interval name (VQDMN)
!Interval name (VQDMN)
!Other interval names
!Type
!Type
|-
|-
Line 23: Line 28:
|Unison
|Unison
|Unison
|Unison
|
|Trivial
|Trivial
|-
|-
Line 32: Line 38:
|Minor second
|Minor second
|Minor second
|Minor second
|Eka, third-tone
|Dissonance
|Dissonance
|-
|-
Line 41: Line 48:
|Major second
|Major second
|Major second
|Major second
|2/3-tone
|Dissonance
|Dissonance
|-
|-
Line 50: Line 58:
|Arto third
|Arto third
|Minor third
|Minor third
|
|Secondary consonance
|Secondary consonance
|-
|-
Line 59: Line 68:
|Minor third
|Minor third
|Major third
|Major third
|
|Primary consonance
|Primary consonance
|-
|-
Line 68: Line 78:
|Major third
|Major third
|Minor fourth
|Minor fourth
|
|Primary consonance
|Primary consonance
|-
|-
Line 77: Line 88:
|Tendo third
|Tendo third
|Major fourth
|Major fourth
|
|Secondary consonance
|Secondary consonance
|-
|-
Line 86: Line 98:
|Perfect fourth
|Perfect fourth
|Perfect fifth
|Perfect fifth
|
|Primary consonance
|Primary consonance
|-
|-
Line 95: Line 108:
|Tritone
|Tritone
|Semidecim / augmented fifth / diminished sixth
|Semidecim / augmented fifth / diminished sixth
|Secondary consonance
|
|Dissonance
|-
|-
|675
|675
Line 104: Line 118:
|Perfect fifth
|Perfect fifth
|Perfect sixth
|Perfect sixth
|
|Primary consonance
|Primary consonance
|-
|-
Line 113: Line 128:
|Arto sixth
|Arto sixth
|Minor seventh
|Minor seventh
|
|Secondary consonance
|Secondary consonance
|-
|-
Line 122: Line 138:
|Minor sixth
|Minor sixth
|Major seventh
|Major seventh
|
|Primary consonance
|Primary consonance
|-
|-
Line 131: Line 148:
|Major sixth
|Major sixth
|Minor eighth
|Minor eighth
|
|Primary consonance
|Primary consonance
|-
|-
Line 140: Line 158:
|Tendo sixth
|Tendo sixth
|Major eighth
|Major eighth
|
|Secondary consonance
|Secondary consonance
|-
|-
Line 149: Line 168:
|Minor seventh
|Minor seventh
|Minor ninth
|Minor ninth
|
|Dissonance
|Dissonance
|-
|-
Line 158: Line 178:
|Major seventh
|Major seventh
|Major ninth
|Major ninth
|
|Dissonance
|Dissonance
|-
|-
Line 167: Line 188:
|Octave
|Octave
|Decim
|Decim
|
|Trivial
|Trivial
|}
|}
While not shown here, there are 2 accidentals used, as in VQDMN normally: # raises a note by a single edostep, and b lowers a note by a single edostep.
While not shown here, there are 2 accidentals used, as in VQDMN normally: # raises a note by a single eka, and b lowers a note by a single eka.


There are two scales in use in 16edo: antidiatonic ("heptatonic") and armotonic ("nonatonic").  
There are two scales in use in 16edo: antidiatonic ("heptatonic") and armotonic ("nonatonic").  
Line 184: Line 206:




This contains 7 white keys corresponding to a heptatonic scale starting on C, 7 black keys representing the same mode starting on G#, as well as two "blue notes" covered by neither, and which can be added to either to produce a nonatonic scale. Any nonatonic can be thought of as a parent heptatonic plus these two "blue notes" (on either side of the heptatonic on the chain of sixths), which are used significantly more in harmony than in melody, as they make available more possible chord types.
This contains 7 white keys corresponding to a heptatonic scale starting on C, 7 black keys representing the same mode starting on G#, as well as two "blue notes" covered by neither, which form a a semidecim with each other and which can be added to either to produce a nonatonic scale. Any nonatonic can be thought of as a parent heptatonic plus these two "blue notes" (on either side of the heptatonic on the chain of sixths), which are used significantly more in harmony than in melody, as they make available more possible chord types.




While the modes of a normal diatonic system can be categorized into major and minor based on the size of the third (since the third is the only mediant), in the case of armotonic, there are two mediants to keep track of (thirds and fourths), so the nonatonic modes are categorized based on what ''pair'' of third and fourth is available (of which there are 3 possibilities).
While the modes of a normal diatonic system can be categorized into major and minor based on the size of the third (since the third is the only mediant), in the case of armotonic, there are two mediants to keep track of (thirds and fourths), so the nonatonic modes are categorized based on what ''pair'' of third and fourth is available (of which there are 3 possibilities).
A heptatonic can be expanded by adding a "blue note" into each of the 3-eka gaps in the scale. There are 4 possible ways to do this for each heptatonic, 3 of which lead to nonatonic modes, and the fourth leads to a "modmos" similar to melodic minor. Scales that belong to the same parent heptatonic can be used in the same song, much like how you can switch between natural, harmonic, and melodic minor in a song in 12edo. Each nonatonic scale can be reached from three heptatonics.
Here are the scales arranged in a way that makes it clear what corresponds to what: by adding blue notes to a heptatonic, the nonatonic to its left as well as the nonatonics above and below that can be reached, and by removing notes from a nonatonic, the heptatonic to its right as well as the scales above and below that can be reached. A heptatonic can be transformed to its corresponding modmos mode, or the modmos mode to the heptatonic.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 197: Line 223:
!Third
!Third
!Fourth
!Fourth
!Heptatonic mode name
!Heptatonic pattern
!Heptatonic pattern
!Heptatonic pattern (16edo notes)
!Heptatonic pattern (16edo notes)
!Modmos scale
|-
|-
|Lydian
|Lydian
Line 208: Line 236:
|'''Major'''
|'''Major'''
| -
| -
| -
| -
|  -
|
|-
|-
|Ionian
|Ionian
Line 217: Line 247:
|'''Major'''
|'''Major'''
|'''Major'''
|'''Major'''
|Alsatian
|sssLssL
|sssLssL
|#_#_#_#__#_#_#__#
|#_#_#_#__#_#_#__#
|#_#_#_#_##_#_##_#
|-
|-
|Mixolydian
|Mixolydian
Line 227: Line 259:
|'''Major'''
|'''Major'''
|'''Major'''
|'''Major'''
|Provencal
|ssLsssL
|ssLsssL
|#_#_#__#_#_#_#__#
|#_#_#__#_#_#_#__#
|#_#_##_#_#_#_#_##
|-
|-
|Corinthian
|Corinthian
Line 237: Line 271:
|'''Major'''
|'''Major'''
|Minor
|Minor
|Norman
|ssLssLs
|ssLssLs
|#_#_#__#_#_#__#_#
|#_#_#__#_#_#__#_#
|#_#_#_##_#_##_#_#
|-
|-
|Olympian
|Olympian
Line 247: Line 283:
|'''Major'''
|'''Major'''
|Minor
|Minor
|Picardian
|sLsssLs
|sLsssLs
|#_#__#_#_#_#__#_#
|#_#__#_#_#_#__#_#
|#_##_#_#_#_#_##_#
|-
|-
|Dorian
|Dorian
Line 257: Line 295:
|Minor
|Minor
|Minor
|Minor
|Burgundian
|sLssLss
|sLssLss
|#_#__#_#_#__#_#_#
|#_#__#_#_#__#_#_#
|#_#_##_#_##_#_#_#
|-
|-
|Aeolian
|Aeolian
Line 267: Line 307:
|Minor
|Minor
|Minor
|Minor
|Breton
|LsssLss
|LsssLss
|#__#_#_#_#__#_#_#
|#__#_#_#_#__#_#_#
|##_#_#_#_#_##_#_#
|-
|-
|Phrygian
|Phrygian
Line 277: Line 319:
|Minor
|Minor
|Minor
|Minor
|Corsican
|LssLsss
|LssLsss
|#__#_#_#__#_#_#_#
|#__#_#_#__#_#_#_#
|#_##_#_##_#_#_#_#
|-
|-
|Locrian
|Locrian
Line 287: Line 331:
|Minor
|Minor
|Minor
|Minor
| -
|
| -
|  -
| -
|
|}
|}


Line 373: Line 419:


== Staff notation ==
== Staff notation ==
[[File:Staff 16.png|left|thumb|16edo staff]]This is a 16edo staff. There are 6 lines per staff, to accommodate the size of the decim in scale steps. The treble clef is the same G clef you're familiar with, but the bass clef is a new shape, because it marks the note H below middle C, as opposed to F (according to VQDMN convention for 9-note scales). The C clef is still the same as well
[[File:Staff 16.png|left|thumb|16edo staff]]This is a 16edo staff. There are 6 lines per staff, to accommodate the size of the decim in scale steps. The treble clef is the same G clef you're familiar with, but the bass clef is a new shape, because it marks the note H below middle C, as opposed to F (according to VQDMN convention for 9-note scales). The C clef is still the same as well.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
== Functional harmony ==
Functional harmony is a system for assigning functions to different triads in a scale. Chords that are separated by a third often have the same function in 12edo; this generalizes to 16edo, where the cycle of mediants is 18 entries long (due to the two options for mediants for each root).
 
The following table shows the functions for the Ionian nonatonic; more chords are available in Alsatian, however they are not covered here, as the difference between the Alsatian nonatonics is similar to the difference between minor scales in 12edo.
 
(xxxa = arto, xxx = minor, XXX=major, XXXT = tendo)
{| class="wikitable"
|+
!Function
!Chords with Alsatian root notes
!Other chords
!Resolves to
|-
|Function 1
|i, III, *VII
|
|Function 2, Function 5
|-
|Function 2
|iiia, VT, *II, v
|
|Function 3
|-
|Function 3
|*vii, VIT
|IXMdim, ixadim
|Function 4, Function 5
|-
|Function 4
|viii, IT
|iva
|Function 2, Function 5
|-
|Function 5
|*ii
|IV
|Function 6
|-
|Function 6
|vi, VIII
|
|Function 1
|}
<nowiki>*</nowiki>A resolution can also be done from a minor chord to its parallel major chord.
 
 
Functions 6 and 3 are similar to the dominant, functions 5 and 2 are similar to the subdominant, and functions 4 and 1 are similar to the tonic.
 
 
The functional names of notes are the same as in diatonic, except that "mediant" and "submediant" cover 2 pitch classes as opposed to 1.
[[Category:Method]]
[[Category:Method]]
[[Category:Approaches to tuning systems]]
[[Category:Approaches to tuning systems]]
Retrieved from "https://en.xen.wiki/w/User:VectorGraphics/16edo_theory"