Chthonic harmony - Revision history

Vector at 03:23, 8 April 2026

2026-04-08T03:23:18Z

← Older revision		Revision as of 03:23, 8 April 2026
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	Diatonic ordinals are typically notated by a symbol (P, M, m, etc.) to display its quality, plus a 1-indexed number (5, 3, 7, etc.) to display its place in the scale. The quality symbols can be maintained for describing interordinals. However, because these numbers are conventionally integers and notation has developed around that, it is difficult to slot interordinal intervals in between.		Diatonic ordinals are typically notated by a symbol (P, M, m, etc.) to display its quality, plus a 1-indexed number (5, 3, 7, etc.) to display its place in the scale. The quality symbols can be maintained for describing interordinals. However, because these numbers are conventionally integers and notation has developed around that, it is difficult to slot interordinal intervals in between.

	One solution uses decimal points (or equivalently, the symbol ½) and allows for intuitive placement of the intervals without much memorization. However, this comes with a number of issues when applied to staff notation, primarily with figured bass (where slashes already have a meaning and decimal points could be confused for two separate entries) and other aspects of notation where elements would need to be widened to fit a two- or three-character symbol, or symbols other than single-digit whole numbers could be ambiguous.		One solution uses decimal points (or equivalently, the symbol ½) and allows for intuitive placement of the intervals without much memorization. However, this comes with a number of issues when applied to staff notation, primarily with figured bass (where slashes already have a meaning and decimal points could be confused for two separate entries) and other aspects of notation where elements would need to be widened to fit a two- or three-character symbol, or symbols other than single-digit whole numbers could be ambiguous, such as reading and writing chords, or notating scale degrees.

	An alternate notation system used on this page, intended to be compatible with the constraints of figured bass, ~~uses~~ Greek ~~numerals~~ to represent ordinals and interordinals together, based on the degrees (1-indexed) of the 14-form; odd-numbered degrees will therefore be equivalent to ordinals of the 7-form, and as such can be notated with either Greek or Arabic numerals. ~~This has the advantage of being single characters;~~ however, the ordering of the intervals is not immediately obvious to someone who does not know the Greek alphabet and may require memorization.		An alternate notation system used on this page solves the issues with decimal notation, and is intended to be compatible with the constraints of figured bass and other conventional notations and avoid ambiguity by making all symbols single characters. It uses an alternate set of ordered symbols, specifically the Greek alphabet, to represent ordinals and interordinals together, based on the degrees (1-indexed) of the 14-form; odd-numbered degrees will therefore be equivalent to ordinals of the 7-form, and as such can be notated with either Greek or Arabic numerals. One potential problem with this notation, however, is that the ordering of the intervals is not immediately obvious to someone who does not know the Greek alphabet and may require memorization.

	Just as there is a clear circle created by a stack of fifths or thirds, so too is there one created by a stack of chthonics; two chthonics will create some type of fourth, three will create a [[cocytic]] (fifth-inter-sixth), four will create a seventh, etc.		Just as there is a clear circle created by a stack of fifths or thirds, so too is there one created by a stack of chthonics; two chthonics will create some type of fourth, three will create a [[cocytic]] (fifth-inter-sixth), four will create a seventh, etc.

Vector: make the notation section seem less like it's dismissing decimal notation

2026-04-08T03:16:26Z

make the notation section seem less like it's dismissing decimal notation

← Older revision		Revision as of 03:16, 8 April 2026
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	== On notation and stacking ==		== On notation and stacking ==
	Diatonic ordinals are typically notated by a symbol (P, M, m, etc.) to display its quality, plus a 1-indexed number (5, 3, 7, etc.) to display its place in the scale. The quality symbols can be maintained for describing interordinals. However, because these numbers are ~~necessarily~~ integers, it is difficult to slot interordinal intervals in between; while using decimal points has been suggested (and allows for intuitive placement of the intervals without much memorization), this comes with a number of issues, namely providing difficulty for reading and writing chords, figured bass, and scale degrees.		Diatonic ordinals are typically notated by a symbol (P, M, m, etc.) to display its quality, plus a 1-indexed number (5, 3, 7, etc.) to display its place in the scale. The quality symbols can be maintained for describing interordinals. However, because these numbers are conventionally integers and notation has developed around that, it is difficult to slot interordinal intervals in between.

	~~To fix~~ this ~~issue~~, ~~Greek numerals will~~ be used on this page to represent ordinals and interordinals together, based on the degrees (1-indexed) of the 14-form; odd-numbered degrees will therefore be equivalent to ordinals of the 7-form, and as such can be notated with either Greek or Arabic numerals. This has the advantage of being single characters; however, the ordering of the intervals is not immediately obvious to someone who does not know the Greek alphabet and may require memorization.		One solution uses decimal points (or equivalently, the symbol ½) and allows for intuitive placement of the intervals without much memorization. However, this comes with a number of issues when applied to staff notation, primarily with figured bass (where slashes already have a meaning and decimal points could be confused for two separate entries) and other aspects of notation where elements would need to be widened to fit a two- or three-character symbol, or symbols other than single-digit whole numbers could be ambiguous.

			An alternate notation system used on this page, intended to be compatible with the constraints of figured bass, uses Greek numerals to represent ordinals and interordinals together, based on the degrees (1-indexed) of the 14-form; odd-numbered degrees will therefore be equivalent to ordinals of the 7-form, and as such can be notated with either Greek or Arabic numerals. This has the advantage of being single characters; however, the ordering of the intervals is not immediately obvious to someone who does not know the Greek alphabet and may require memorization.

	Just as there is a clear circle created by a stack of fifths or thirds, so too is there one created by a stack of chthonics; two chthonics will create some type of fourth, three will create a [[cocytic]] (fifth-inter-sixth), four will create a seventh, etc.		Just as there is a clear circle created by a stack of fifths or thirds, so too is there one created by a stack of chthonics; two chthonics will create some type of fourth, three will create a [[cocytic]] (fifth-inter-sixth), four will create a seventh, etc.

Vector at 03:11, 8 April 2026

2026-04-08T03:11:20Z

← Older revision		Revision as of 03:11, 8 April 2026
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	\|+Cocytic (four-note) Chords		\|+Cocytic (four-note) Chords
	!Quality		!Quality
	!Intervals (~~Unque~~)		!Intervals (Greek)
	!Intervals (Decimal)		!Intervals (Decimal)
	!Formula (δ = 2½)		!Formula (δ = 2½)

Vector at 03:03, 8 April 2026

2026-04-08T03:03:17Z

← Older revision		Revision as of 03:03, 8 April 2026
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	\|+Cocytic (four-note) Chords		\|+Cocytic (four-note) Chords
	!Quality		!Quality
	!Intervals		!Intervals (Unque)
	!Formula		!Intervals (Decimal)
			!Formula (δ = 2½)
	\|-		\|-
	\|Major Cocytic		\|Major Cocytic
	\|Mδ, Pζ, Mι		\|Mδ, Pζ, Mι
			\|M2½, P4, M5½
	\|Mδ + mδ + Mδ		\|Mδ + mδ + Mδ
	\|-		\|-
	\|Minor Cocytic		\|Minor Cocytic
	\|mδ, Pζ, mι		\|mδ, Pζ, mι
			\|m2½, P4, m5½
	\|mδ + Mδ + mδ		\|mδ + Mδ + mδ
	\|-		\|-
	\|Major Minor Cocytic		\|Major Minor Cocytic
	\|Mδ, Pζ, mι		\|Mδ, Pζ, mι
			\|M2½, P4, m5½
	\|Mδ + mδ + mδ		\|Mδ + mδ + mδ
	\|-		\|-
	\|Minor Major Cocytic		\|Minor Major Cocytic
	\|mδ, Pζ, Mι		\|mδ, Pζ, Mι
			\|m2½, P4, M5½
	\|mδ + Mδ + Mδ		\|mδ + Mδ + Mδ
	\|-		\|-
	\|Half-Augmented Cocytic		\|Half-Augmented Cocytic
	\|Mδ, Aζ, Mι		\|Mδ, Aζ, Mι
			\|M2½, A4, M5½
	\|Mδ + Mδ + mδ		\|Mδ + Mδ + mδ
	\|-		\|-
	\|Augmented Cocytic		\|Augmented Cocytic
	\|Mδ, Aζ, Aι		\|Mδ, Aζ, Aι
			\|M2½, A4, A5½
	\|Mδ + Mδ + Mδ		\|Mδ + Mδ + Mδ
	\|-		\|-
	\|Half-Diminished Cocytic		\|Half-Diminished Cocytic
	\|mδ, dζ, mι		\|mδ, dζ, mι
			\|m2½, d4, m5½
	\|mδ + mδ + Mδ		\|mδ + mδ + Mδ
	\|-		\|-
	\|Diminished Cocytic		\|Diminished Cocytic
	\|mδ, dζ, dι		\|mδ, dζ, dι
			\|m2½, d4, d5½
	\|mδ + mδ + mδ		\|mδ + mδ + mδ
	\|}		\|}

Unque: The interval conversion between 7- and 14-form can be found two inches above the chord table if you just scroll up. Not to devalue the decimal notation scheme or anything, just that it takes up a lot of extra horizontal space here.

2026-04-07T22:02:05Z

The interval conversion between 7- and 14-form can be found two inches above the chord table if you just scroll up. Not to devalue the decimal notation scheme or anything, just that it takes up a lot of extra horizontal space here.

← Older revision		Revision as of 22:02, 7 April 2026
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	\|+Cocytic (four-note) Chords		\|+Cocytic (four-note) Chords
	!Quality		!Quality
	!Intervals ~~(δ = 2.5, ζ = 4, ι = 5.5)~~		!Intervals
	!Formula ~~(δ = 2.5)~~		!Formula
	\|-		\|-
	\|Major Cocytic		\|Major Cocytic

Unque: Chords need not be tuned to anything in particular; honestly I'm not entirely sure why I had the 36edo mention in there to begin with

2026-02-26T03:29:09Z

Chords need not be tuned to anything in particular; honestly I'm not entirely sure why I had the 36edo mention in there to begin with

← Older revision		Revision as of 03:29, 26 February 2026
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	Where tertian harmony is most practical with chords of three and four notes, chthonic harmony is most practical with chords of four or five notes. Because a chthonic added to a fourth will always yield a cocytic, chords can be named by the type of chthonic and the type of cocytic, just as tertian chords can be named for the quality of their third and seventh. Unless otherwise specified, the fourth is always assumed to be perfect, and the seventh minor.		Where tertian harmony is most practical with chords of three and four notes, chthonic harmony is most practical with chords of four or five notes. Because a chthonic added to a fourth will always yield a cocytic, chords can be named by the type of chthonic and the type of cocytic, just as tertian chords can be named for the quality of their third and seventh. Unless otherwise specified, the fourth is always assumed to be perfect, and the seventh minor.

	To display how these qualities propagate, this example will use only two types of chthonics~~, here tuned to 7\36 and 8\36 for simplicity~~.		To display how these qualities propagate, this example will use only two types of chthonics.
	{\| class="wikitable"		{\| class="wikitable"
	\|+Cocytic (four-note) Chords		\|+Cocytic (four-note) Chords
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	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness		! rowspan="2" \|Brightness
	~~Quotient~~
	! colspan="8" \|Degrees		! colspan="8" \|Degrees
	! rowspan="2" \|Cocytic Chord		! rowspan="2" \|Cocytic Chord
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	\|+Modes of Chthonic Chromatic		\|+Modes of Chthonic Chromatic
	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness ~~Quotient~~		! rowspan="2" \|Brightness
	! colspan="13" \|Degrees		! colspan="13" \|Degrees
	! rowspan="2" \|Diamode		! rowspan="2" \|Diamode

Unque: Undo revision 4287 by Vector (talk)

2026-02-25T17:49:07Z

Undo revision 4287 by Vector (talk)

← Older revision		Revision as of 17:49, 25 February 2026
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	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness		! rowspan="2" \|Brightness
			Quotient
	! colspan="8" \|Degrees		! colspan="8" \|Degrees
	! rowspan="2" \|Cocytic Chord		! rowspan="2" \|Cocytic Chord

Vector: give me a formula for the brightness that involves non-superfluous division and I'll retain it

2026-02-25T07:04:32Z

give me a formula for the brightness that involves non-superfluous division and I'll retain it

← Older revision		Revision as of 07:04, 25 February 2026
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	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness		! rowspan="2" \|Brightness
	~~Quotient~~
	! colspan="8" \|Degrees		! colspan="8" \|Degrees
	! rowspan="2" \|Cocytic Chord		! rowspan="2" \|Cocytic Chord

Unque: Quotient is established terminology in music theory for discussing brightness of modes. Don't forget we're talking about relative brightness, because there's no absolute brightness to be measured.

2026-02-25T04:31:20Z

Quotient is established terminology in music theory for discussing brightness of modes. Don't forget we're talking about *relative* brightness, because there's no absolute brightness to be measured.

← Older revision		Revision as of 04:31, 25 February 2026
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	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness		! rowspan="2" \|Brightness
			Quotient
	! colspan="8" \|Degrees		! colspan="8" \|Degrees
	! rowspan="2" \|Cocytic Chord		! rowspan="2" \|Cocytic Chord

Vector: stop calling things "quotient" to sound fancy

2026-02-25T03:24:29Z

stop calling things "quotient" to sound fancy

← Older revision		Revision as of 03:24, 25 February 2026
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	=== Superdiatonic ===		=== Superdiatonic ===
	One may note that the Diasem scale pattern produces an imperfect "wolf" interval on each step, which creates an awkward and discordant quality in some of the chords. One potential way to fix this issue is by equating the large wolf chthonic with the regular major chthonic; this may be seen as analogous to how Meantone temperament equates the wolf intervals of the Zarlino pattern with the more frequent forms of the ordinals.		One may note that the Diasem scale pattern produces an imperfect "wolf" interval on each step, which creates an awkward and discordant quality in some of the chords. One potential way to fix this issue is by equating the large wolf chthonic with the regular major chthonic; this may be seen as analogous to how Meantone temperament equates the wolf intervals of the Zarlino pattern with the more frequent forms of the ordinals. This is at the cost of severely sharpening both the major chthonic and the fourth.

	This scale is sometimes known as "superdiatonic," as it can be constructed by pentachords of LLLs in a similar manner to the diatonic scale's construction by tetrachords of LLs.		This scale is sometimes known as "superdiatonic," as it can be constructed by pentachords of LLLs in a similar manner to the diatonic scale's construction by tetrachords of LLs. It may also be called "armotonic", and is well-tuned in 16edo.
	{\| class="wikitable sortable"		{\| class="wikitable sortable"
	\|+Modes of Superdiatonic		\|+Modes of Superdiatonic
	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness		! rowspan="2" \|Brightness
	~~Quotient~~
	! colspan="8" \|Degrees		! colspan="8" \|Degrees
	! rowspan="2" \|Cocytic Chord		! rowspan="2" \|Cocytic Chord
Line 825:		Line 824:
	\|Dorian		\|Dorian
	\|}		\|}
			''todo: pajara''

← Older revision		Revision as of 03:29, 26 February 2026
Line 150:		Line 150:
	Where tertian harmony is most practical with chords of three and four notes, chthonic harmony is most practical with chords of four or five notes. Because a chthonic added to a fourth will always yield a cocytic, chords can be named by the type of chthonic and the type of cocytic, just as tertian chords can be named for the quality of their third and seventh. Unless otherwise specified, the fourth is always assumed to be perfect, and the seventh minor.		Where tertian harmony is most practical with chords of three and four notes, chthonic harmony is most practical with chords of four or five notes. Because a chthonic added to a fourth will always yield a cocytic, chords can be named by the type of chthonic and the type of cocytic, just as tertian chords can be named for the quality of their third and seventh. Unless otherwise specified, the fourth is always assumed to be perfect, and the seventh minor.

	To display how these qualities propagate, this example will use only two types of chthonics~~, here tuned to 7\36 and 8\36 for simplicity~~.		To display how these qualities propagate, this example will use only two types of chthonics.
	{\| class="wikitable"		{\| class="wikitable"
	\|+Cocytic (four-note) Chords		\|+Cocytic (four-note) Chords
Line 435:		Line 435:
	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness		! rowspan="2" \|Brightness
	~~Quotient~~
	! colspan="8" \|Degrees		! colspan="8" \|Degrees
	! rowspan="2" \|Cocytic Chord		! rowspan="2" \|Cocytic Chord
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	\|+Modes of Chthonic Chromatic		\|+Modes of Chthonic Chromatic
	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness ~~Quotient~~		! rowspan="2" \|Brightness
	! colspan="13" \|Degrees		! colspan="13" \|Degrees
	! rowspan="2" \|Diamode		! rowspan="2" \|Diamode

← Older revision		Revision as of 17:49, 25 February 2026
Line 435:		Line 435:
	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness		! rowspan="2" \|Brightness
			Quotient
	! colspan="8" \|Degrees		! colspan="8" \|Degrees
	! rowspan="2" \|Cocytic Chord		! rowspan="2" \|Cocytic Chord

← Older revision		Revision as of 07:04, 25 February 2026
Line 435:		Line 435:
	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness		! rowspan="2" \|Brightness
	~~Quotient~~
	! colspan="8" \|Degrees		! colspan="8" \|Degrees
	! rowspan="2" \|Cocytic Chord		! rowspan="2" \|Cocytic Chord

← Older revision		Revision as of 04:31, 25 February 2026
Line 435:		Line 435:
	! rowspan="2" \|Pattern		! rowspan="2" \|Pattern
	! rowspan="2" \|Brightness		! rowspan="2" \|Brightness
			Quotient
	! colspan="8" \|Degrees		! colspan="8" \|Degrees
	! rowspan="2" \|Cocytic Chord		! rowspan="2" \|Cocytic Chord