Fractional-octave temperaments

Fractional-octave temperaments, when viewed from a regular temperament theory perspective, are temperaments which have a period which corresponds to a just interval mapped to a fraction of the octave, that is one step of an EDO.

Theory

Fractional-octave temperaments are valuable with regards to polysystemicism and polychromatics. They are acoustically significant with regards to containing modes of limited transposition, as well as their ability to expand on the harmony of the equal division they are a superset of. Such temperaments are also a way of introducing less common and harmonically less performing equal divisions into music that prefers consonance and is based on regular temperament theory.

Terminology

The terminology was developed by Eliora. The equal division containing the mos scale of such a temperament, starting from the tonic, is referred to as a wireframe, and individual notes of that equal division are called hinges. Thus in this context, the wireframe is the tuning consisting of only stacks of the period and no stacks of the generator. Temperament-agnostically, this can be used to refer to any structure embedded in an (x,y)-ET which repeats y times within that period, its "wireframe" is y-ET.

The most common way to produce a fractional-octave temperament is through an excellent approximation of an interval relative to the size of the wireframe edo. For example, compton family tempers out the Pythagorean comma and maps 7 steps of 12edo to 3/2. Likewise, a lot of 10th-octave temperaments have a 13/8 as 7\10, and 26th-octave temperaments often have a 7/4 for 21\26.

Disagreement between regular temperament theory and fractional-octave practice

Traditional regular temperament perspective on periods and generators has a shortcoming when it comes to handling fractional-octave temperaments, as it treats divisions of periods (for example, what hemiennealimmal is to ennealimmal) as extensions of a temperament with a subset period. However fractional-octave temperaments and scales are sought for being able to treat an each equal division as an entity in its own right, so a composer might find hemiennealimmal to be a drastically different system to ennealimmal in line with 18edo being very different from 9edo.

A particularly strong offender of this is the landscape microtemperaments list, which features temperaments which are all supersets of 3edo, but from a composer's perspective it contains wildly different temperaments due to the fact that edo multiples of 3 themselves are different. For example, magnesium (12), and zinc (30), are both landscape systems due to being multiples of 3, but 30edo is drastically different from 12edo in terms of composition, and therefore such temperaments are not alike at all.

Individual pages of temperaments by equal division

2 to 40

Many pages are yet to be created.

	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40

40 and up

41, 44, 53*, 56, 60, 61, 65, 80, 91, 111, 118

<nowiki>*</nowiki> Mercator family equated with 53rd-octave temperaments until otherwise discovered

Temperaments discussed elsewhere

Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include:

• 1\2 period temperaments
  • Diaschismic temperaments
  • Vishnuzmic temperaments
  • Jubilismic temperaments
  • Varunismic temperaments
  • Lokismic temperaments
• 1\3 period temperaments
  • Augmented temperaments
  • Misty temperaments
  • Landscape temperaments
• 1\4 period temperaments
  • Diminished temperaments
  • Undim temperaments
• 1\5 period temperaments
  • Pental temperaments
  • Qintosec temperaments
  • Trisedodge temperaments
  • Cloudy temperaments
  • Limmic temperaments
• 1\6 period temperaments
  • Hexe
  • Sextile
  • Stearnscape
• Akjaysmic temperaments (1\7 period)
  • Brahmagupta
  • Septant
  • Whitewood
  • Sevond
  • Neutron
• Octoid, octant (1\8 period)
• Tritrizo temperaments (1\9 period)
  • Ennealimmal
  • Niner
  • Enneaportent
  • Novemkleismic
• Linus temperaments (1\10 period)
  • Decoid
  • Deca
  • Decic
  • Decistearn
  • Decal
  • Decavish
  • Decimetra
• Hendecatonic, undeka (1\11 period)
• Compton, atomic (1\12 period)
• Triskaidekic, tridecatonic, trideci, aluminium (1\13 period)
• Silicon (1\14 period)
• Pentadecal, quindecic (1\15 period)
• Hexadecoid, sedecic (1\16 period)
• Chlorine (1\17 period)
• Hemiennealimmal (1\18 period)
• Enneadecal, meanmag (1\19 period)
• Degrees (1\20 period)
• Akjayland (1\21 period)
• Icosidillic (1\22 period)
• Icositritonic (1\23 period)
• Hours, chromium (1\24 period)
• Trinealimmal, cobalt (1\27 period)
• Oquatonic (1\28 period)
• Mystery, copper (1\29 period)
• Birds (1\31 period)
• Decades (1\36 period)
• Rubidium, dzelic (1\37 period)
• Hemienneadecal, semihemienneadecal (1\38 period)
• Counterpyth temperaments, niobium (1\41 period)
• Meridic (1\43 period)
• Palladium (1\46 period)
• Mercator temperaments (1\53 period)
• Omicronbeta, hafnium (1\72 period)
• Iridium (1\77 period)
• Octogintic, (1\80 period)
• Garistearn (1\94 period)
• Undecentic (1\99 period)
• Schisennealimmal (1\171 period)
• Lunennealimmal (1\441 period)