User:Eufalesio/Fifth scale tree
|
This user page is editable by any wiki editor.
As a general rule, most users expect their user space to be edited only by themselves, except for minor edits (e.g. maintenance), undoing obviously harmful edits such as vandalism or disruptive editing, and user talk pages. However, by including this message box, the author of this user page has indicated that this page is open to contributions from other users (e.g. content-related edits). |
|
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 and for designing the blueprint for the EiE (Edo-inter-Edo) nomenclature.
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
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. Inverted in m-chromatic (d2) where it is called meantone diesis. |
| 7L 5s | m-chromatic | 12i7 | m2:A1 | ||
| granddaughter | 5L 12s | s-enharmonic | 5i17 | d-2:m2 | dd3 is the chroma, the gothic 17-comma.
Inverted in s-enharmonic (dd-3). |
| 12L 5s | p-enharmonic | 17i12 | m2:d-2 | ||
| 12L 7s | m-enharmonic | 12i19 | m2:d2 | dd-2 is the chroma, the meantone kleisma.
Inverted in f-enharmonic (dd2). | |
| 7L 12s | f-enharmonic | 19i7 | d2:m2 | ||
| 3rd-descendant | 12L 17s | pythagotonic | 29i12 | dd3:d-2 | 4d4 is the chroma, the mystery 29-comma.
Inverted in gothitonic (4d-4). |
| 17L 12s | gothitonic | 29i17 | d-2:dd3 | ||
| 4th-descendant | 12L 29s | pythamystonic | 41i12 | 4d4:d-2 | 6d5 is the chroma, the countercomp 41-comma.
Inverted in countermystonic (6d-5). |
| 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.
Inverted in comptomerc (7d6). |
| 12L 41s | comptomerc | 53i12 | 6d5:d-2 | ||
| 6th-descendant | 41L 53s | garytonic | 41i94 | 7d-6:6d5 | 13d10 is the chroma, the 94-comma.
Inverted in garytonic (13d-10). |
| 53L 41s | acupyth | 94i53 | 6d5:7d-6 | ||
| 53L 12s | pontiacitonic | 53i65 | d-2:7d6 | The chroma is 9d-7, the 65-comma.
Inverted in comptograckle (9d7). | |
| 12L 53s | comptograckle | 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 |
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.
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 Qián Lèzhī, the ancient Chinese astronomer and instrument maker who discovered the 306- and 359- comma.
- picopyth comes from pico-, a SI prefix denoting 10^-12, a minute order of magnitude.
- aurotonic comes from the golden meantone, as it supports a generator of 696.214 cents.
- meancomptonic comes from compton and meantone, as the ranges are between sharper meantone and compton.
- ultimeantonic comes from the fact that the basic tuning 47\81 is the last patent val fifth that supports golden meantone.