User:Eufalesio/Fifth scale tree

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This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community. Terms: Many of the MOS pattern names are only found on this page.

This article is a mostly rewritten proposal for the Scale tree article, and more specifically, the scale tree pertaining to MOS scales with fifths as generators. Note that this article is full of idiosyncratic names, taken to be proposals to be considered. Acknowledgements to Kite Giedraitis for feedback.

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The scale tree is a Stern-Brocot tree that lists every possible interval in an equal tuning, assuming a given equave (on this page, the octave). It is commonly used in the context of MOS scales and regular temperament theory. The fifth scale tree is particularly important, both for historical and mathematical reasons, as the overwhelming amount of music theory today deals with fifth-based MOS scales and temperaments, and because temperaments built from octaves are tritaves are generally the easiest to conceptualize, as they involve two prime numbers instead of any rational or irrational number.

In the fifth scale tree, the octave is the equave and period, and the tritave or fifth is the generator. The octave is generally fixed to be pure, and so by changing the fifth, an infinitude of MOS scales and temperaments (including edos) can be described and classified.

Diagram

Here is a Desmos graph plotting edos, classified by sharpness (how many steps to reach one apotome), and patent val fifth size, which falls between the two reciprocal curves.

As seen on the diagram, fifths with sizes between 3\5 and 4\7 are diatonoid. Edos with fifths sharper than 5edo are oneirotonoid, and edos with fifths flatter than 7edo are antidiatonoid. Note the black line, which plots all convergents and semiconvergents of fifths, serving as boundaries for MOS scales.

MOS scales and fifth ranges.

A single MOS scale explicitly defines the ranges of a fifth, and describes a number of related temperaments, however, the fifth ranges can also be described with the EiE nomenclature. There are more descendants that are less notable. Also described are the MOSes generated by Pythagorean tuning in bold.

MOS-based adjectives

MOS-based names like diatonoid 3/2, sephirothish 5/4 or p-chro machinish 7/4 may be used, as they are also explicit in their ranges. If the MOS name ends in -ic, substitute by -oid (pentic -> pentoid). If the MOS name doesn't end in -ic or -oid, add -oid (lime -> limoid). If the MOS ends in -oid, recover original ending and add -ish, unless it ends in -us, in which case substitute. (sephiroid -> sephiroth -> sephirothish, dicoid -> dicot -> dicotish, helenoid -> helenus -> helenish).

Diatonic relationship	Scale Signature	TAMNAMS based name	L:s describes	Notes on mappings
self	5L 2s	diatonic	M2:m2	M2 and m2 are the major and minor seconds; A1 is the chroma, the apotome.
daughter	5L 7s	p-chromatic	A1:m2	d-2 is the chroma, the pythagorean comma. Inverted in m-chromatic (d2) where it is called meantone diesis.
	7L 5s	m-chromatic	m2:A1
granddaughter	5L 12s	s-enharmonic	d-2:m2	dd3 is the chroma, the gothic 17-comma. Inverted in s-enharmonic (dd-3).
	12L 5s	p-enharmonic	m2:d-2
	12L 7s	m-enharmonic	m2:d2	dd-2 is the chroma, the meantone kleisma. Inverted in f-enharmonic (dd2).
	7L 12s	f-enharmonic	d2:m2
3rd-descendant	12L 17s	pythagotonic	dd3:d-2	4d4 is the chroma, the mystery 29-comma. Inverted in gothitonic (4d-4).
	17L 12s	gothitonic	d-2:dd3
4th-descendant	12L 29s	pythamystonic	4d4:d-2	6d5 is the chroma, the countercomp 41-comma. Inverted in countermystonic (6d-5).
	29L 12s	countermystonic	d-2:4d4
5th-descendant	41L 12s	pythomerc	d-2:6d5	7d-6 is the chroma, the mercator 53-comma. Inverted in comptomerc (7d6).
	12L 41s	comptomerc	6d5:d-2
6th-descendant	41L 53s	garytonic	7d-6:6d5	13d10 is the chroma, the 94-comma. Inverted in garytonic (13d-10).
	53L 41s	acupyth	6d5:7d-6
	53L 12s	pontiacitonic	d-2:7d6	The chroma is 9d-7, the 65-comma. Inverted in comptograckle (9d7).
	12L 53s	comptograckle	7d6:d-2
. . .	53L 94s 53L 147s 53L 200s	p-chro acupyth s-enhar acupyth uha-acupyth	13d10:7d-6 21d15:7d-6 28d20:7d-6	Large steps are semiconvergent commas.
10th-descendant	53L 253s	qiantonic	36d25:7d-6
11th-descendant	306L 53s	m-chro qiantonic	7d-6:43d30	51d-35 and 43d30 are the large and small Qian commas respectively. The chroma is the satanic comma.
12th-descendant	306L 359s	picopyth	51d-35:43d30

Some notable MOS scales that diverge from the pythagorean line are:

Diatonic relationship	Scale Signature	TAMNAMS based name	EiE (3/2)	L:s describes	Notes on mappings
3rd-descendant (m-enharmonic)	19L 12s	aurotonic	31i19	d2:dd-2	4d3 is the chroma, the 31-comma. Reversed in comptomean (4d-3).
	12L 19s	meancomptonic	12i31	dd-2:d2
3rd-descendant (s-enharmonic)	5L 17s	reinhardic	5i22	dd-3:d-2	3d-4 is the chroma, the 22-comma. Reversed in protofractalic (3d4).
	17L 5s	protofractalic	22i17	d-2:dd-3
4th-descendant (aurotonic)	31L 19s	ultimeantonic	50i31	dd-2:4d3	7d-4 is the comma, the 50-comma.

Bolded MOS support a pythagorean generator. Bolded and underlined names are also of a record lowest hardness when that generator is used. Italic names only appear in this article. See User:Eufalesio/TAMNAMS Extensions for more info.