User:Overthink/22edo decatonic theory - Revision history

Overthink: /* Modes */ half-complete table

2026-01-28T03:02:41Z

Modes: half-complete table

← Older revision		Revision as of 03:02, 28 January 2026
Line 108:		Line 108:
	!Appears in white keys on		!Appears in white keys on
	\|-		\|-
	\|Scandium?		\| Scandium
	\| +5		\| +5
	\| LsssLsssss		\| LsssLsssss
	\|P1, A2, M3, M4, P5, A6, A7, M8, M9, P10, P11		\| P1, A2, M3, M4, P5, A6, A7, M8, M9, P10, P11
	\|C, R#, D#, E#, F, T#, G#, A, S, B, C		\| C, R#, D#, E, F, T#, G#, A, S, B, C
	\|D		\| 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
	\|}		\|}

Overthink: /* The white keys */ Lay out page, expand a bit

2026-01-19T06:27:46Z

The white keys: Lay out page, expand a bit

← Older revision		Revision as of 06:27, 19 January 2026
Line 32:		Line 32:
	\| 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 \|\| ~~placeholder~~		\| Augmented 5th \|\| 545.5 \|\| rowspan=2 \| 10 \|\| [[11/8]]
	\|-		\|-
	\| Diminished ~~5th~~ \|\| 545.5 \|\| style="text-align: left" \| ~~placeholder~~		\| 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 \|\| ~~placeholder~~		\| Augmented 6th \|\| 654.5 \|\| rowspan=2 \| 12 \|\| [[40/27]], [[22/15]]
	\|-		\|-
	\| Diminished 7th \|\| 654.5 \|\| style="text-align: left" \| ~~placeholder~~		\| 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]]
Line 66:		Line 66:

	== 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-3 right-4"		{\| class="wikitable right-3 right-4"
Line 96:		Line 96:
	\|}		\|}

	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. Using the Root−4th−7th−9th construction, we find the tetrads on each note.		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
			\|-
			\|
			\|
			\|
			\|
			\|
			\|
			\|}

			== Chords ==
			''First explain what chords are there''

			Using the Root−4th−7th−9th construction, we find the tetrads on each note.

	{\| class="wikitable"		{\| class="wikitable"

Overthink: /* Interval classification */ fix tables

2025-12-28T11:53:59Z

Interval classification: fix tables

← Older revision		Revision as of 11:53, 28 December 2025
Line 32:		Line 32:
	\| Perfect 5th \|\| 490.9 \|\| 9 \|\| [[4/3]], [[21/16]]		\| Perfect 5th \|\| 490.9 \|\| 9 \|\| [[4/3]], [[21/16]]
	\|-		\|-
	\| Augmented 5th \|\| 545.5 \|\| ~~colspan~~=2 \| 10 \|\| placeholder		\| Augmented 5th \|\| 545.5 \|\| rowspan=2 \| 10 \|\| placeholder
	\|-		\|-
	\| Diminished 5th \|\| 545.5 \|\| style="text-align: left" \| placeholder		\| Diminished 5th \|\| 545.5 \|\| style="text-align: left" \| placeholder
Line 38:		Line 38:
	\| Perfect 6th \|\| 600.0 \|\| 11 \|\| [[7/5]], [[10/7]]		\| Perfect 6th \|\| 600.0 \|\| 11 \|\| [[7/5]], [[10/7]]
	\|-		\|-
	\| Augmented 6th \|\| 654.5 \|\| ~~colspan~~=2 \| 12 \|\| placeholder		\| Augmented 6th \|\| 654.5 \|\| rowspan=2 \| 12 \|\| placeholder
	\|-		\|-
	\| Diminished 7th \|\| 654.5 \|\| style="text-align: left" \| placeholder		\| Diminished 7th \|\| 654.5 \|\| style="text-align: left" \| placeholder

Overthink: /* Interval classification */ + a few missing intervals

2025-12-28T11:52:56Z

Interval classification: + a few missing intervals

← Older revision		Revision as of 11:52, 28 December 2025
Line 31:		Line 31:
	\|-		\|-
	\| Perfect 5th \|\| 490.9 \|\| 9 \|\| [[4/3]], [[21/16]]		\| Perfect 5th \|\| 490.9 \|\| 9 \|\| [[4/3]], [[21/16]]
			\|-
			\| Augmented 5th \|\| 545.5 \|\| colspan=2 \| 10 \|\| placeholder
			\|-
			\| Diminished 5th \|\| 545.5 \|\| style="text-align: left" \| placeholder
	\|-		\|-
	\| Perfect 6th \|\| 600.0 \|\| 11 \|\| [[7/5]], [[10/7]]		\| Perfect 6th \|\| 600.0 \|\| 11 \|\| [[7/5]], [[10/7]]
			\|-
			\| Augmented 6th \|\| 654.5 \|\| colspan=2 \| 12 \|\| placeholder
			\|-
			\| Diminished 7th \|\| 654.5 \|\| style="text-align: left" \| placeholder
	\|-		\|-
	\| Perfect 7th \|\| 709.1 \|\| 13 \|\| [[3/2]], [[32/21]]		\| Perfect 7th \|\| 709.1 \|\| 13 \|\| [[3/2]], [[32/21]]

Overthink: /* Interval classification */ more correct (some aren't in 2L 8s)

2025-12-28T11:49:30Z

Interval classification: more correct (some aren't in 2L 8s)

← Older revision		Revision as of 11:49, 28 December 2025
Line 3:		Line 3:

	== 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 right-all left-4"		{\| class="wikitable right-all left-4"

Overthink: /* Interval classification */ alignment

2025-12-28T10:21:46Z

Interval classification: alignment

← Older revision		Revision as of 10:21, 28 December 2025
Line 5:		Line 5:
	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 that appear in the 2L 8s scale.

	{\| class="wikitable"		{\| class="wikitable right-all left-4"
	\|+ style="font-size: 105%" \| Decatonic intervals		\|+ style="font-size: 105%" \| Decatonic intervals
	\|-		\|-
Line 14:		Line 14:
	\| 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]]
Line 52:		Line 52:
	\| 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]]

Overthink: /* The white keys */ + Tetrads on each note, explanation of names

2025-12-28T10:19:15Z

The white keys: + Tetrads on each note, explanation of names

@@ Line 58: / Line 58: @@
 == 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)]]. 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
 |-
@@ Line 86: / Line 86: @@
 |-
 | C || P11 || 1200.0 || 22
 |}

Overthink: start page

2025-12-28T09:32:39Z

start page

New page

; Todo<nowiki>:</nowiki> Citation to Paul Erlich's paper, Add diagram
In classical music, we often use the [[5L 2s|diatonic]] scale, which is generated by [[meantone]] temperament. This gives us relatively simple access to [[5-limit]] harmony, while avoiding wolf intervals. There is a system many consider to be analogous to diatonic in the [[7-limit]]; the decatonic scale of [[pajara]] temperament. Strictly speaking, there are two decatonic scales: the symmetrical scale of the [[2L 8s]] [[mos]] pattern, with large and small steps in the pattern ssLssssLss; and the pentachordal near-mos scale, with steps in the pattern ssLsssLsss. Here we will mainly be focusing on the pentachordal scale.

== 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.

{| class="wikitable"
|+ style="font-size: 105%" | Decatonic intervals
|-
! Name !! Cents (22edo) !! Steps of 22edo !! JI ratios
|-
| Perfect 1sn || 0.0 || 0 || [[1/1]]
|-
| Augmented 1sn || 54.5 || rowspan=2 | 1 || [[25/24]], [[28/27]], [[49/48]]
|-
| Diminished 2nd || 54.5 || style="left" | [[36/35]], [[81/80]]
|-
| Perfect 2nd || 109.1 || 2 || [[16/15]], [[15/14]], [[21/20]]
|-
| Augmented 2nd || 163.6 || 3 || [[10/9]]
|-
| Minor 3rd || 218.2 || 4 || [[8/7]], [[9/8]]
|-
| Major 3rd || 272.7 || 5 || [[7/6]]
|-
| Minor 4th || 327.3 || 6 || [[6/5]]
|-
| Major 4th || 381.8 || 7 || [[5/4]]
|-
| Diminished 5th || 436.4 || 8 || [[9/7]], [[32/25]]
|-
| Perfect 5th || 490.9 || 9 || [[4/3]], [[21/16]]
|-
| Perfect 6th || 600.0 || 11 || [[7/5]], [[10/7]]
|-
| Perfect 7th || 709.1 || 13 || [[3/2]], [[32/21]]
|-
| Augmented 7th || 763.6 || 14 || [[14/9]], [[25/16]]
|-
| Minor 8th || 818.2 || 15 || [[8/5]]
|-
| Major 8th || 872.7 || 16 || [[5/3]]
|-
| Minor 9th || 927.3 || 17 || [[12/7]]
|-
| Major 9th || 981.8 || 18 || [[7/4]], [[16/9]]
|-
| Diminished 10th || 1036.4 || 19 || [[9/5]]
|-
| Perfect 10th || 1090.9 || 20 || [[15/8]], [[28/15]], [[40/21]]
|-
| Augmented 10th || 1145.5 || rowspan=2 | 21 || [[27/14]], [[48/25]], [[96/49]]
|-
| Diminished 11ve || 1145.5 || [[35/18]], [[160/81]]
|-
| Perfect 11ve || 1200.0 || 22 || [[2/1]]
|}

== 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.

{| class="wikitable right-all"
|+ style="font-size: 105%" | The white keys
|-
! Note !! Interval above C !! Cents !! Steps
|-
| C || P1 || 0.0 || 0
|-
| R || P2 || 109.1 || 2
|-
| D || m3 || 218.2 || 4
|-
| E || M4 || 381.8 || 7
|-
| F || P5 || 490.9 || 9
|-
| T || P6 || 600.0 || 11
|-
| G || P7 || 709.1 || 13
|-
| A || M8 || 872.7 || 16
|-
| S || M9 || 981.8 || 18
|-
| B || P10 || 1090.9 || 20
|-
| C || P11 || 1200.0 || 22
|}