Fractional-octave temperaments

Fractional-octave temperaments 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) equal temperament which repeats y times within that period, its "wireframe" is y equal temperament. If an equal division is a subset of a temperament, it is said to subtend the temperament, just how hinges on a ferris wheel subtend the structure to make it rotate and function.

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.

However, an equal division does not have to be harmonically decent to be a wireframe for a fractional-octave temperament. If an equal division has multiples which are high in consistency or are zeta equal divisions or otherwise harmonically strong, it can produce a lot of such temperaments—notable examples being 20edo or 32edo. Likewise, proximity of a step of equal division to a comma is often a source of these temperaments—for example 56edo's step being directly close to 81/80, and 44edo's step being extremely close to 64/63.

Disagreement between temperament catalog policy and fractional-octave practice

Vanilla regular temperament theory does not distinguish periods and generators, so it treats divisions of periods (for example, what hemiennealimmal is to ennealimmal) as extensions of a temperament with a subset period, just like how it treats divisions of generators. However, fractional-octave temperaments and scales are sometimes sought for being able to treat 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.

Besides, on this wiki, temperament collection pages are used to collect temperaments that temper out a common comma. For example, the landscape microtemperaments list features temperaments which all temper out the landscape comma, as they are all related by that. A fractional-octave temperament user might consider that 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.

Octave-splitting comma

An octave-splitting comma is a rational interval which induces a fractional-octave temperament. If tempering out the comma leads to splitting the octave into N equal parts (N > 1), the comma is an octave-splitting comma which induces an Nth-octave temperament. This terminology was developed by Xenllium.

Property

While a comma is given in the p-limit monzo form [a₂ a₃ a₅ … a_p⟩, as a rational interval, it is an octave-splitting comma if and only if GCD(a₂, a₃, a₅, …, a_p) = 1 and N = GCD(a₃, a₅, …, a_p) > 1, and leads to splitting the octave into N equal parts.

Examples

Below is a list of octave-splitting commas for common temperaments:

Comma	Associated temperament	Harmonic limit	Splitting order
256/243	Blackwood	3	5
2187/2048	Whitewood	3	7
531441/524288	Compton	3	12
2048/2025	Diaschismic	5	2
128/125	Augmented	5	3
648/625	Diminished	5	4
50/49	Jubilismic	7	2
250047/250000	Landscape	7	3
9801/9800	Kalismic	11	2
289/288	Semitonismic	17	2

Individual pages of temperaments by subtending equal division

2 to 100

	2–4			5	6	7	8	9	10
11	12/C	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
41/CC	42	43	44	45	46	47	48	49	50
51	52	53/M	54	55	56	57	58	59	60
61	62	63	64	65	66	67	68	69	70
71	72	73	74	75	76	77	78	79	80
81	82	83	84	85	86	87	88	89	90
91	92	93	94	95	96	97	98	99	100

• C = compton family
• CC = countercomp family
• M = mercator family equated with 53rd-octave temperaments until otherwise documented, also contains 106th-octave temperaments

101 and up

111, 118, 159, 400, 665

Temperaments discussed elsewhere

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

• 1\5 period temperaments
  • Quintile temperaments
  • Quintosec temperaments
  • Trisedodge temperaments
  • Cloudy temperaments
  • Blackwood family
• Octoid, octant (1\8 period)
• Septiennealimmal temperaments (1\9 period)
  • Ennealimmal
  • Niner
  • Enneaportent
  • Novemkleismic
• Linus temperaments (1\10 period)
  • Decoid
  • Deca
  • Decic
  • Decistearn
  • Decile
  • Decavish
  • Decimetra
• Hendecatonic (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)
• Gamelstearn (1\36 period)
• Hemienneadecal, semihemienneadecal (1\38 period)
• Countercomp temperaments, niobium (1\41 period)
• Meridic (1\43 period)
• Palladium (1\46 period)
• Omicronbeta (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)

See also

• Map of rank-2 temperaments: Visual map of many of the temperaments listed here.