User:Eufalesio/Fifth scale tree

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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 and for designing the blueprint for the EiC (Edo-inter-Edo) nomenclature.

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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.

EiE nomenclature

Blueprinted by Kite, it is written as ~R AiB; where R is any interval, A and B are patent val approximations of edos, ~R of A > ~R of B. It describes a range of approximations of R, including A and B. Here, it is used to describe ranges of ~3/2, but without implicit knowledge, R has to be declared, Such as in ~5/4 28i41 or ~7/4 26i31. This would be read as "five over four twenty eight inter forty one" and "seven over four twenty six inter thirty one".

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	EiE (3/2)	L:s describes	Notes on mappings
self	5L 2s	diatonic	5i7	M2:m2	M2 and m2 are the major and minor seconds; A1 is the chroma, the apotome.
daughter	5L 7s	p-chromatic	5i12	A1:m2	d-2 is the chroma, the pythagorean comma.
	7L 5s	m-chromatic	12i7	m2:A1
granddaughter	5L 12s	s-enharmonic	5i17	d-2:m2	dd3 is the chroma, the gothic 17-comma.
	12L 5s	p-enharmonic	17i12	m2:d-2
	12L 7s	m-enharmonic	12i19	m2:d2	d2 is the meantone diesis; dd-2 is the chroma, the meantone kleisma.
	7L 12s	f-enharmonic	19i7	d2:m2
3rd-descendant	12L 17s	pythagotonic	29i12	dd3:d-2	4d4 is the chroma, the mystery 29-comma.
	17L 12s	gothitonic	29i17	d-2:dd3
4th-descendant	12L 29s	pythamystonic	41i12	4d4:d-2	6d5 is the chroma, the countercomp 41-comma.
	29L 12s	countermystonic	41i29	d-2:4d4
5th-descendant	41L 12s	pythomerc	41i53	d-2:6d5	7d-6 is the chroma, the mercator 53-comma.
	12L 41s	comptomerc	53i12	6d5:d-2
6th-descendant	41L 53s	garytonic	41i94	7d-6:6d5	13d10 is the chroma, the 94-comma.
	53L 41s	acupyth	94i53	6d5:7d-6
	53L 12s	pontiacitonic	53i65	d-2:7d6	7d6 is the inverse mercator comma. The chroma is 9d-7, the 65-comma.
	12L 53s	comptograc	65i12	7d6:d-2
. . .	53L 94s 53L 147s 53L 200s	p-chro acupyth s-enhar acupyth uha acupyth	147i53 200i53 253i53	13d10:7d-6 21d15:7d-6 28d20:7d-6	Large steps are semiconvergent commas.
10th-descendant	53L 253s	qiantonic	306i53	36d25:7d-6
11th-descendant	306L 53s	m-chro qiantonic	359i306	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	306i665	51d-35:43d30

The names of the MOSes are coined as follows:

• countermystonic comes from countercomp and mystery, the two temperaments that converge in this MOS.
• comptomerc comes from compton, as 12edo can also generate this MOS, albeit trivially.
• garytonic comes from gary, the temperament that tempers the garischisma.
• acupyth comes from acus (needle), judging by the accuracy of the fifths in this range.
• pontiacitonic comes from pontiac, which generates this MOS.
• comptograckle comes from compton and grackle, which can both generate this MOS.
• qiantonic comes from Qian, the man who discovered the 306- and 359- comma.
• picopyth comes from pico-, a SI prefix denoting 10^-12, a minute order of magnitude.