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 |
Fifth ranges
in edos |
Fifth ranges
in cents |
TAMNAMS based name |
L:s describes |
|---|---|---|---|---|---|
| self | 5L 2s | 5-7 | 34.285 | diatonic | M2:m2 |
| daughter | 5L 7s | 5-12 | 20.000 | p-chromatic | A1:m2 |
| 7L 5s | 7-12 | 14.285 | m-chromatic | m2:A1 | |
| granddaughter | 5L 12s | 5-17 | 14.117 | s-enharmonic | d-2:m2 |
| 12L 5s | 12-17 | 5.882 | p-enharmonic | m2:d-2 | |
| 12L 7s | 12-19 | 5.263 | m-enharmonic | m2:d2 | |
| 7L 12s | 7-19 | 9.022 | f-enharmonic | d2:m2 | |
| 3rd-descendant | 12L 17s | 12-29 | 3.448 | pythagotonic | dd3:d-2 |
| 17L 12s | 17-29 | 2.434 | gothitonic | d-2:dd3 | |
| 4th-descendant | 12L 29s | 12-41 | 2.439 | pythamystonic | 4d4:d-2 |
| 29L 12s | 29-41 | 1.009 | countermystonic | d-2:4d4 | |
| 5th-descendant | 41L 12s | 41-53 | 0.552 | pythomerc | d-2:6d5 |
| 12L 41s | 12-53 | 1.886 | comptomerc | 6d5:d-2 | |
| 6th-descendant | 41L 53s | 41-94 | 0.311 | garytonic | 7d-6:6d5 |
| 53L 41s | 53-94 | 0.240 | acupyth | 6d5:7d-6 | |
| 53L 12s | 53-65 | 0.348 | pontiacitonic | d-2:7d6 | |
| 12L 53s | 12-65 | 1.538 | comptograckle | 7d6:d-2 | |
| . . . |
53L 94s 53L 147s 53L 200s |
53-147
53-200 53-253 |
p-chro acupyth s-enhar acupyth uha-acupyth |
13d10:7d-6 21d15:7d-6 28d20:7d-6 | |
| 10th-descendant | 53L 253s | 53-306 | qiantonic | 36d25:7d-6 | |
| 11th-descendant | 306L 53s | 306-359 | m-chro qiantonic | 7d-6:43d30 | |
| 12th-descendant | 306L 359s | 306-665 | picopyth | 51d-35:43d30 |
Some notable MOS scales that diverge from the pythagorean line are:
| Diatonic relationship |
Scale Signature |
Fifth ranges
in edos |
Fifth ranges
in cents |
TAMNAMS based name |
L:s describes |
|---|---|---|---|---|---|
| 3rd-descendant
(m-enharmonic) |
19L 12s | 19-31 | aurotonic | d2:dd-2 | |
| 12L 19s | 12-31 | meancomptonic | dd-2:d2 | ||
| 3rd-descendant
(s-enharmonic) |
5L 17s | 5-22 | reinhardic | dd-3:d-2 | |
| 17L 5s | 17-22 | protofractalic | d-2:dd-3 | ||
| 4th-descendant
(aurotonic) |
31L 19s | 31-50 | ultimeantonic | dd-2:4d3 | |
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.