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User:Overthink/22edo decatonic theory: Difference between revisions

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== Interval classification ==
== Interval classification ==
We will classify intervals based on the 2L 8s mos, with names "unison", "second", etc. However, since we have 10 note classes, the interval of equivalence is not an "octave" as in diatonic, but a "hendecave". In this system, 1sns, 2nds, 5ths, 6ths, 7ths, 10ths, and 11ves are perfect, while 3rds, 4ths, 8ths, and 9ths are imperfect. Pajara has a half-octave (600{{c}}) [[period]] representing both [[7/5]] and [[10/7]], and a [[~]][[3/2]] [[generator]] (~709{{c}} in [[22edo]]). The half-octave is a perfect 6th, and the [[3/2]] is a perfect 7th. (From now on, interval names are decatonic unless specified.) Here is a table of intervals that appear in the 2L 8s scale.
We will classify intervals based on the 2L 8s mos, with names "unison", "second", etc. However, since we have 10 note classes, the interval of equivalence is not an "octave" as in diatonic, but a "hendecave". In this system, 1sns, 2nds, 5ths, 6ths, 7ths, 10ths, and 11ves are perfect, while 3rds, 4ths, 8ths, and 9ths are imperfect. Pajara has a half-octave (600{{c}}) [[period]] representing both [[7/5]] and [[10/7]], and a [[~]][[3/2]] [[generator]] (~709{{c}} in [[22edo]]). The half-octave is a perfect 6th, and the [[3/2]] is a perfect 7th. (From now on, interval names are decatonic unless specified.) Here is a table of intervals classified by the 2L 8s scale.


{| class="wikitable"
{| class="wikitable right-all left-4"
|+ style="font-size: 105%" | Decatonic intervals
|+ style="font-size: 105%" | Decatonic intervals
|-
|-
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| Augmented 1sn || 54.5 || rowspan=2 | 1 || [[25/24]], [[28/27]], [[49/48]]
| Augmented 1sn || 54.5 || rowspan=2 | 1 || [[25/24]], [[28/27]], [[49/48]]
|-
|-
| Diminished 2nd || 54.5 || style="left" | [[36/35]], [[81/80]]
| Diminished 2nd || 54.5 || style="text-align: left;" | [[36/35]], [[81/80]]
|-
|-
| Perfect 2nd || 109.1 || 2 || [[16/15]], [[15/14]], [[21/20]]
| Perfect 2nd || 109.1 || 2 || [[16/15]], [[15/14]], [[21/20]]
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|-
|-
| Perfect 5th || 490.9 || 9 || [[4/3]], [[21/16]]
| Perfect 5th || 490.9 || 9 || [[4/3]], [[21/16]]
|-
| Augmented 5th || 545.5 || rowspan=2 | 10 || [[11/8]]
|-
| Diminished 6th || 545.5 || style="text-align: left" | [[27/20]], [[15/11]]
|-
|-
| Perfect 6th || 600.0 || 11 || [[7/5]], [[10/7]]
| Perfect 6th || 600.0 || 11 || [[7/5]], [[10/7]]
|-
| Augmented 6th || 654.5 || rowspan=2 | 12 || [[40/27]], [[22/15]]
|-
| Diminished 7th || 654.5 || style="text-align: left" | [[16/11]]
|-
|-
| Perfect 7th || 709.1 || 13 || [[3/2]], [[32/21]]
| Perfect 7th || 709.1 || 13 || [[3/2]], [[32/21]]
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| Augmented 10th || 1145.5 || rowspan=2 | 21 || [[27/14]], [[48/25]], [[96/49]]
| Augmented 10th || 1145.5 || rowspan=2 | 21 || [[27/14]], [[48/25]], [[96/49]]
|-
|-
| Diminished 11ve || 1145.5 || [[35/18]], [[160/81]]
| Diminished 11ve || 1145.5 || style="text-align: left;" | [[35/18]], [[160/81]]
|-
|-
| Perfect 11ve || 1200.0 || 22 || [[2/1]]
| Perfect 11ve || 1200.0 || 22 || [[2/1]]
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== The white keys ==
== The white keys ==
In diatonic, the white keys are in a mode such that, on C, there is a [[major triad]] on the tonic, dominant, and subdominant. Here, the main chords are ''tetrads'', with the '''major tetrad''' being P1−M4−P7−M9, approximating [[4:5:6:7]], and the '''minor tetrad''' being P1-m4-P7-m9, approximating [[70:84:105:120|1/(7:8:10:12)]]. Unfortunately, there is no mode of the pentachordal scale which places the major tetrad on the tonic, dominant, and subdominant of the same key, but the ssLsssLsss mode gets as close as possible. Hence, we will place the white keys on this mode above C.
In diatonic, the white keys are in a mode such that, on C, there is a [[major triad]] on the tonic, dominant, and subdominant. Here, the main chords are ''tetrads'', with the '''major tetrad''' being P1−M4−P7−M9, approximating [[4:5:6:7]], and the '''minor tetrad''' being P1−m4−P7−m9, approximating [[70:84:105:120|1/(7:8:10:12)]]. More about this will be discussed later. Unfortunately, there is no mode of the pentachordal scale which places the major tetrad on the tonic, dominant, and subdominant of the same key, but the ssLsssLsss mode gets as close as possible. Hence, we will place the white keys on this mode above C.


{| class="wikitable right-all"
{| class="wikitable right-3 right-4"
|+ style="font-size: 105%" | The white keys
|+ style="font-size: 105%" | The white keys
|-
|-
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|-
|-
| C || P11 || 1200.0 || 22
| C || P11 || 1200.0 || 22
|}
The notes are named so that C, D, E, F, G, A, and B represent the [[Zarlino]] diatonic scale, with R, S, and T filling the gaps.
== Modes ==
''Complete the table''
{| class="wikitable"
!Name
!Brightness
! Step pattern
!Intervals
!Note names on C
!Appears in white keys on
|-
| Scandium
| +5
| LsssLsssss
| P1, A2, M3, M4, P5, A6, A7, M8, M9, P10, P11
| C, R#, D#, E, F, T#, G#, A, S, B, C
| D
|-
| Titanium
| +3
| LsssssLsss
| P1, A2, M3, M4, P5, P6, P7, M8, M9, P10, P11
| C, R#, D#, E, F, T, G, A, S, B, C
| G
|-
| Vanadium
| +3
| sLsssLssss
| P1, P2, M3, M4, P5, P6, A7, M8, M9, P10, P11
| C, R, D#, E, F, T, G#, A, S, B, C
| R
|-
| Chromium
| +1
| sLsssssLss
| P1, P2, M3, M4, P5, P6, A7, M8, M9, P10, P11
| C, R, D#, E, F, T, G, Ab, S, B, C
| T
|-
| Manganese (Major)
| +1
| ssLsssLsss
| P1, P2, M3, M4, P5, P6, A7, M8, M9, P10, P11
| C, R, D, E, F, T, G, A, S, B, C
| C
|}
== Chords ==
''First explain what chords are there''
Using the Root−4th−7th−9th construction, we find the tetrads on each note.
{| class="wikitable"
|+ style="font-size: 105%" | Tetrads on white keys
|-
! Note !! Notes of chord !! Intervals of chord !! Name
|-
| C || C−E−G−S || P1−M4−P7−M9 || Major
|-
| R || R−F−A−B || P1−M4−A7−M9 || Aug.
|-
| D || D−T−S−C || P1−M4−A7−M9 || Aug.
|-
| E || E−G−B−R || P1−m4−P7−m9 || Minor
|-
| F || F−A−C−D || P1−M4−P7−m9 || Major−minor
|-
| T || T−S−R−E || P1−M4−P7−M9 || Major
|-
| G || G−B−D−F || P1−M4−P7−M9 || Major
|-
| A || A−C−E−T || P1−m4−P7−m9 || Minor
|-
| S || S−R−F−G || P1−m4−P7−m9 || Minor
|-
| B || B−D−T−A || P1−m4−P7−M9 || Minor−major
|}
|}
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