Chthonic harmony

Chthonic harmony, sometimes called semiquartal or latal harmony (latal/latus, pl. lati, means "1-step of the 5-form", and is borrowed from Latin lātus (1st declension) "carried"), is a type of chord structure useful in many styles of xenharmonic music. It serves as an analog for tertian harmony that splits the perfect fourth rather than the fifth.

Chthonic intervals

The term "chthonic" is derived from the name of an interordinal interval which splits the perfect fourth into two roughly-equal parts. When this division is not equal, one may call the larger chthonic "major" and the smaller one "minor," by analog to diatonic ordinals; when it is equal or nearly so, such as the case with 15/13, we may call it a "neutral" or "perfect" chthonic. Where further types of chthonics exist, one may use ups and downs alteration (in an EDO) or ADIN (in any tuning) to describe the qualities.

In just intonation

In JI structures, chthonic intervals will usually have either a numerator or denominator divisible by 3, which creates a relatively concordant triad bound by the perfect fourth.

Chthonic intervals in JI

Ratio	Cents	ADIN	Triad	Complement
15/13	247.7	Neutral/Perfect	39:45:52	52/45
7/6	266.9	Submajor	6:7:8	8/7
32/27	294.1	Nearmajor	27:32:36	9/8
6/5	315.6	Farmajor	15:18:20	10/9
11/9	347.4	Ultramajor	9:11:12	12/11

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.

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.

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.

Interordinal notation

δ up	Interval	Greek	Arabic	14-form
0	Unison	α	1	0\14
1	Chthonic	δ	2.5	3\14
2	Fourth	ζ	4	6\14
3	Cocytic	ι	5.5	9\14
4	Seventh	ιγ	7	12\14
5	Interizer†	β	1.5	1\14
6	Third	ε	3	4\14
7	Tritone	η	4.5	7\14
8	Sixth	ια	6	10\14
9	Antiïnterizer†	ιδ	7.5	13\14
10	Second	γ	2	2\14
11	Naiadic	ϛ	3.5	5\14
12	Fifth	θ	5	8\14
13	Ouranic	ιβ	6.5	11\14

^†The term "interizer" is coined by Inthar, as it is the interval which separates a diatonic ordinal from an interordinal; thus it can be used to "inter-ize" those ordinals. Because it is also half of a diatonic minor second, it may also be considered a quartertone, though this name may not be preferred, as four of them do not necessarily make a wholetone.

Chthonic chords

Chords can be constructed most intuitively by alternating a stack of a chthonic with its fourth complement, just as tertian chords tend to follow an alternating pattern of a third with its fifth complement.

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.

Cocytic (four-note) Chords

Quality	Intervals	Formula
Major Cocytic	Mδ, P4, Mι	Mδ + mδ + Mδ
Minor Cocytic	mδ, P4, mι	mδ + Mδ + mδ
Major Minor Cocytic	Mδ, P4, mι	Mδ + mδ + mδ
Minor Major Cocytic	mδ, P4, Mι	mδ + Mδ + Mδ
Half-Augmented Cocytic	Mδ, A4, Mι	Mδ + Mδ + mδ
Augmented Cocytic	Mδ, A4, Aι	Mδ + Mδ + Mδ
Half-Diminished Cocytic	mδ, d4, mι	mδ + mδ + Mδ
Diminished Cocytic	mδ, d4, dι	mδ + mδ + mδ

Note that the altered fourths of these chords are "diminished" and "augmented" by half of a diatonic chroma, not a full chroma.

Chthonic scales

The scales which most prominently feature chthonic harmony are, as one might suspect, those which are generated by chthonic intervals, just as the diatonic scale can be generated by an alternating stack of major and minor thirds. Also just like the diatonic scale, many chthonic scales have alternative constructions as well, analogous to the diatonic circle of fifths construction.

Diasem

See also: Quasi-diatonic aberrismic scales

The most fundamental of chthonic scales is Diasem, a 9-note scale created by alternating a stack of major and minor chthonic intervals, such as 7/6 with 8/7. This construction ensures a maximum number of major and minor cocytic chords.

Diasem has two possible chiralities, based on whether 8/7 or 7/6 was stacked first; each chirality has nine unique modes.

Left-Hand Diasem Modes

Pattern	Degrees
	1\9	2\9	3\9	4\9	5\9	6\9	7\9	8\9
LsLmLsLmL	9/8	8/7	9/7	4/3	3/2	32/21	12/7	16/9
sLmLsLmLL	64/63	8/7	32/27	4/3	256/189	32/21	128/81	16/9
LmLsLmLLs	9/8	7/6	21/16	4/3	3/2	14/9	7/4	63/32
mLsLmLLsL	28/27	7/6	32/27	4/3	112/81	14/9	7/4	16/9
LsLmLLsLm	9/8	8/7	9/7	4/3	3/2	27/16	12/7	27/14
sLmLLsLmL	64/63	8/7	32/27	4/3	3/2	32/21	12/7	16/9
LmLLsLmLs	9/8	7/6	21/16	189/128	3/2	27/16	7/4	63/32
mLLsLmLsL	28/27	7/6	21/16	4/3	3/2	14/9	7/4	16/9
LLsLmLsLm	9/8	81/64	9/7	81/56	3/2	27/16	12/7	27/14

Right-Hand Diasem Modes

Pattern	Degrees
	1\9	2\9	3\9	4\9	5\9	6\9	7\9	8\9
LmLsLmLsL	9/8	7/6	21/16	4/3	3/2	14/9	7/4	16/9
mLsLmLsLL	28/27	7/6	32/27	4/3	112/81	14/9	128/81	16/9
LsLmLsLLm	9/8	8/7	9/7	4/3	3/2	32/21	12/7	27/14
sLmLsLLmL	64/63	8/7	32/27	4/3	256/189	32/21	12/7	16/9
LmLsLLmLs	9/8	7/6	21/16	4/3	3/2	27/16	7/4	63/32
mLsLLmLsL	28/27	7/6	32/27	4/3	3/2	14/9	7/4	16/9
LsLLmLsLm	9/8	8/7	9/7	81/56	3/2	27/16	12/7	27/14
sLLmLsLmL	64/63	8/7	9/7	4/3	3/2	32/21	12/7	16/9
LLmLsLmLs	9/8	81/64	21/16	189/128	3/2	27/16	7/4	63/32

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.

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.

Modes of Superdiatonic

Pattern	Brightness Quotient	Degrees								Cocytic Chord
		1\9	2\9	3\9	4\9	5\9	6\9	7\9	8\9
LLLsLLLLs	+3	Maj.	Maj.	Maj.	Perf.	Perf.	Maj.	Maj.	Maj.	Major
LLsLLLLsL	+1	Maj.	Maj.	Min.	Perf.	Perf.	Maj.	Maj.	Min.	Major
LsLLLLsLL	-1	Maj.	Min.	Min.	Perf.	Perf.	Maj.	Min.	Min.	Minmaj
sLLLLsLLL	-3	Min.	Min.	Min.	Perf.	Perf.	Min.	Min.	Min.	Minor
LLLLsLLLs	+4	Maj.	Maj.	Maj.	Aug.	Perf.	Maj.	Maj.	Maj.	Half-Augmented
LLLsLLLsL	+2	Maj.	Maj.	Maj.	Perf.	Perf.	Maj.	Maj.	Min.	Major
LLsLLLsLL	0	Maj.	Maj.	Min.	Perf.	Perf.	Maj.	Min.	Min.	Major
LsLLLsLLL	-2	Maj.	Min.	Min.	Perf.	Perf.	Min.	Min.	Min.	Minor
sLLLsLLLL	-4	Min.	Min.	Min.	Perf.	Dim.	Min.	Min.	Min.	Minor

Semiquartal

Another way to simplify the structure of Diasem is to use a single type of neutral/perfect chthonic rather than the two alternating major and minor chthonics.

Todo: table

Chthonic chromatic

The Chthonic analog to the chromatic scale can be made by continuing the stack of neutral chthonics until you reach a scale of 14 notes. This scale has the pattern 5L 9s; because the step size is the interizer, the 14 notes of the scale can be considered to represent the seven diatonic ordinals plus the seven interordinal intervals.

Todo: table