User:Eufalesio/Fifth scale tree: Difference between revisions

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{{Idiosyncratic terms|Many of the MOS pattern names are only found on this page.}}
{{Idiosyncratic 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 [[3/2|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 ('''E'''do-'''i'''nter-'''E'''do) nomenclature.
This article is a mostly rewritten proposal for the [[Scale tree]] article, and more specifically, the scale tree pertaining to MOS scales with [[3/2|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.


–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Line 17: Line 17:
== MOS scales and fifth ranges. ==
== 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.
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 adjectives ===
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! Scale<br>Signature
! Scale<br>Signature
! TAMNAMS<br>based name
! TAMNAMS<br>based name
! EiE (3/2)
! L:s describes
! L:s describes
! Notes on mappings
! Notes on mappings
Line 36: Line 32:
| '''[[5L 2s]]'''
| '''[[5L 2s]]'''
| '''diatonic'''
| '''diatonic'''
| 5i7
| M2:m2
| M2:m2
| M2 and m2 are the major and minor seconds;<br>A1 is the chroma, the [[apotome]].
| M2 and m2 are the major and minor seconds;<br>A1 is the chroma, the [[apotome]].
Line 43: Line 38:
| '''[[5L 7s]]'''
| '''[[5L 7s]]'''
| '''<u>p-chromatic</u>'''
| '''<u>p-chromatic</u>'''
| 5i12
| A1:m2
| A1:m2
| rowspan="2" | d-2 is the chroma, the [[pythagorean comma]]. Inverted in m-chromatic (d2) where it is called ''meantone diesis''.
| rowspan="2" | d-2 is the chroma, the [[pythagorean comma]]. Inverted in m-chromatic (d2) where it is called ''meantone diesis''.
Line 49: Line 43:
| [[7L 5s]]
| [[7L 5s]]
| m-chromatic
| m-chromatic
| 12i7
| m2:A1
| m2:A1
|-
|-
Line 55: Line 48:
| [[5L 12s]]
| [[5L 12s]]
| s-enharmonic
| s-enharmonic
| 5i17
| d-2:m2
| d-2:m2
| rowspan="2" | dd3 is the chroma, the ''gothic'' [[17-comma]].  
| rowspan="2" | dd3 is the chroma, the ''gothic'' [[17-comma]].  
Line 62: Line 54:
| '''[[12L 5s]]'''
| '''[[12L 5s]]'''
| '''p-enharmonic'''
| '''p-enharmonic'''
| 17i12
| m2:d-2
| m2:d-2
|-
|-
| 12L 7s
| 12L 7s
| m-enharmonic
| m-enharmonic
| 12i19
| m2:d2
| m2:d2
| rowspan="2" | dd-2 is the chroma, the meantone kleisma.  
| rowspan="2" | dd-2 is the chroma, the meantone kleisma.  
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| 7L 12s
| 7L 12s
| f-enharmonic
| f-enharmonic
| 19i7
| d2:m2
| d2:m2
|-
|-
Line 80: Line 69:
| '''12L 17s'''
| '''12L 17s'''
| '''pythagotonic'''
| '''pythagotonic'''
| 29i12
| dd3:d-2
| dd3:d-2
| rowspan="2" | 4d4 is the chroma, the ''mystery'' [[29-comma]].  
| rowspan="2" | 4d4 is the chroma, the ''mystery'' [[29-comma]].  
Line 87: Line 75:
| 17L 12s
| 17L 12s
| gothitonic
| gothitonic
| 29i17
| d-2:dd3
| d-2:dd3
|-
|-
Line 93: Line 80:
| 12L 29s
| 12L 29s
| '''pythamystonic'''
| '''pythamystonic'''
| 41i12
| 4d4:d-2
| 4d4:d-2
| rowspan="2" | 6d5 is the chroma, the ''countercomp'' [[41-comma]].
| rowspan="2" | 6d5 is the chroma, the ''countercomp'' [[41-comma]].
Line 100: Line 86:
| 29L 12s
| 29L 12s
| ''countermystonic''
| ''countermystonic''
| 41i29
| d-2:4d4
| d-2:4d4
|-
|-
Line 106: Line 91:
| '''41L 12s'''
| '''41L 12s'''
| '''<u>pythomerc</u>'''
| '''<u>pythomerc</u>'''
| 41i53
| d-2:6d5
| d-2:6d5
| rowspan="2" | 7d-6 is the chroma, the ''mercator'' [[53-comma]].
| rowspan="2" | 7d-6 is the chroma, the ''mercator'' [[53-comma]].
Line 113: Line 97:
| 12L 41s
| 12L 41s
| ''comptomerc''
| ''comptomerc''
| 53i12
| 6d5:d-2
| 6d5:d-2
|-
|-
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| 41L 53s
| 41L 53s
| ''garytonic''
| ''garytonic''
| 41i94
| 7d-6:6d5
| 7d-6:6d5
| rowspan="2" | 13d10 is the chroma, the [[94-comma]].
| rowspan="2" | 13d10 is the chroma, the [[94-comma]].
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| '''53L 41s'''
| '''53L 41s'''
| '''''acupyth'''''
| '''''acupyth'''''
| 94i53
| 6d5:7d-6
| 6d5:7d-6
|-
|-
| 53L 12s
| 53L 12s
| ''pontiacitonic''
| ''pontiacitonic''
| 53i65
| d-2:7d6
| d-2:7d6
| rowspan="2" | The chroma is 9d-7, the [[65-comma|65-comma.]]
| rowspan="2" | The chroma is 9d-7, the [[65-comma|65-comma.]]
Line 138: Line 118:
| 12L 53s
| 12L 53s
| ''comptograckle''
| ''comptograckle''
| 65i12
| 7d6:d-2
| 7d6:d-2
|-
|-
Line 144: Line 123:
| <small>'''53L 94s'''<br>'''53L 147s'''<br>'''53L 200s'''</small>
| <small>'''53L 94s'''<br>'''53L 147s'''<br>'''53L 200s'''</small>
| <small>'''''p-chro acupyth'''''<br>'''''s-enhar acupyth'''''<br>'''''uha-acupyth'''''</small>
| <small>'''''p-chro acupyth'''''<br>'''''s-enhar acupyth'''''<br>'''''uha-acupyth'''''</small>
| 147i53<br>200i53<br>253i53
| 13d10:7d-6<br>21d15:7d-6<br>28d20:7d-6
| 13d10:7d-6<br>21d15:7d-6<br>28d20:7d-6
| rowspan="2" | Large steps are semiconvergent commas.
| rowspan="2" | Large steps are semiconvergent commas.
Line 151: Line 129:
| '''53L 253s'''
| '''53L 253s'''
| '''''qiantonic'''''
| '''''qiantonic'''''
| 306i53
| 36d25:7d-6
| 36d25:7d-6
|-
|-
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| '''306L 53s'''
| '''306L 53s'''
| '''''<small>m-chro qiantonic</small>'''''
| '''''<small>m-chro qiantonic</small>'''''
| 359i306
| 7d-6:43d30
| 7d-6:43d30
| rowspan="2" | 51d-35 and 43d30 are the [[359-comma|large]]<br>and [[306-comma|small]] Qian commas respectively.<br>The chroma is the [[satanic comma]].
| rowspan="2" | 51d-35 and 43d30 are the [[359-comma|large]]<br>and [[306-comma|small]] Qian commas respectively.<br>The chroma is the [[satanic comma]].
Line 164: Line 140:
| '''306L 359s'''
| '''306L 359s'''
| '''''<u>picopyth</u>'''''
| '''''<u>picopyth</u>'''''
| 306i665
| 51d-35:43d30
| 51d-35:43d30
|}
|}
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|}
|}


 
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.
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.

Revision as of 18:04, 28 February 2026

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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.

–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

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.

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