Interval size measure

Interval size measure or interval size unit means the distance between pitches. Intervals can be measured logarithmic or by frequency ratios.

Logarithmic

All logarithmic measures can be combined by adding and subtracting them.

Backslash notation

A common shorthand in use in the microtonal community is k\N, written with a backslash (\) instead of a forwardslash (/), to refer to an interval with a frequency ratio of 2^k/N. k\N is pronounced "k steps of N edo", and can be derived from the meaning of "steps" in the context of edos (unless talking about steps of specific subsets/scales of some edo).

Steps are linear in the log-frequency domain, so expressions like 11\19 − 6\19 = 5\19 hold. In general, we have

a\N + b\N = (a + b)\N

which expresses the same thing as 2^a/N × 2^b/N = 2^{(a + b)/N}.

Or equivalently, for subtraction/division:

a\N − b\N = (a − b)\N

which expresses the same thing as 2^a/N / 2^b/N = 2^{(a - b)/N}.

Backslash notation can be extended to support nonoctave equal tunings by writing the tuning in full after the backslash. For example, 11\13edt means 11 steps of 13edt, 14\9edf means 14 steps of 9edf, and 7\12ed12/5 means 7 steps of 12ed12/5.

Gross

The octave and the decade are common coarse units for interval sizes. The decibel, being a relative logarithmic-scale unit for power or root-power quantities, is inappropriate for measuring intervals; the decade is used instead. Similarly, the neper (Np) and the dineper (dNp), like the decibel, should not be used. However, in the absence of a substitute, dinepers have an application in logarithmic approximants.

Intervals are sometimes expressed in the number of scale steps between them. These steps can be of different size, compare for example the names of the major scale in the classic music. An early unit for measuring intervals is the "tone" which dates back to classic Greece.

In serial music, all intervals were measured by the number of 12edo semitones. In analogy, the relative interval measure is the number of steps between two pitches of an equal tuning, sometimes called "degrees". These measures can be written using backslash notation if the degree itself isn't sufficiently clear in context.

Fine

The cent (¢), 1\1200 octave, is the classic measure for intervals when more precision than 12edo is required. Some people object to it on the grounds that it is too (obviously) closely related to 12 equal.

Octave-based fine measures

The following table demonstrates a list of measures derived from the logarithmic division of the octave:

List of Octave-Based Fine Measures (Logarithmic)

Unit Name (Symbol):	Divisions of Octave	Prime Factors	Origin / Significance
Eka	16	24	From Sanskrit eka: one, unit; chromatic unit of Armodue 16edo Theory.
Normal diesis	31	31 (prime)	See the dedicated page.
Méride	43	43 (prime)	Proposed by Joseph Sauveur, as 7 heptaméride units.
Holdrian comma	53	53 (prime)	See the dedicated page.
Mercator’s old comma	55	5 x 11	Not to be confused with Mercator's comma.
Decitone	60	22 × 3 × 5
Morion	72	23 × 32	See the dedicated page.
Farab	144	24 × 32	1/12 of 12edo semitone; Proposed by al-Farabi in 10th century.
Mem	205	5 × 41	Unit used by H-Pi Instruments.
Tredek	270	2 × 33 × 5	Proposed by Joseph Monzo (2013).
Savart*	300	22 × 3 × 52	Alexander Wood's definition of the Savart, containing 12edo.
Heptaméride / eptaméride / savart*	301	7 × 43	301 ≃ 1,000 × log102; 1/7 of Méride unit; proposed by Joseph Sauveur (1701), advocated by Félix Savart.
Gene	311	311 (prime)	Proposed by Joseph Monzo (2007).
Dröbisch Angle	360	23 × 32 × 5	Proposed as angle by Moritz Dröbisch in the 19th century, later by Andrew Pikler as the current name in Logarithmic Frequency Systems (1966).
Squb	494	2 × 13 × 19	⁠ ⁠[citation needed ]
Great iring / centitone	500	22 × 53	⁠ ⁠[citation needed ]
Dexl	540	22 × 33 × 5	Proposed by Joseph Monzo (2023).
Iring / centitone	600	23 × 3 × 52	Relative cent of 6edo (12edo tone); Proposed by Widogast Iring (1898), later by Joseph Yasser as a "centitone" (1932).
Skisma (Sk)	612	22 × 32 × 17	Edo representation of Sagittal's Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina".
Delfi	665	5 × 7 × 19
Small iring / centitone	700	22 × 52 x 7	⁠ ⁠[citation needed ]
Woolhouse	730	2 × 5 × 73	Proposed by Wesley S.B. Woolhouse (1835).
Millioctave (moct)	1000	23 × 53	See the dedicated page.
Cent (¢)	1200	24 × 3 × 52	See the dedicated page.
Greater muon	1224	23 × 32 × 17	⁠ ⁠[citation needed ]
Triangular cent	1260	22 × 32 × 5 × 7	⁠ ⁠[citation needed ]
Pion	1272	23 × 3 × 53	⁠ ⁠[citation needed ]
Pound	1344	26 × 3 × 7	⁠ ⁠[citation needed ]
Neutron	1392	24 × 3 × 29	⁠ ⁠[citation needed ]
Lesser muon	1428	22 × 3 × 7 × 17	⁠ ⁠[citation needed ]
Decifarab	1440	25 × 32 × 5	1/10 of Farab unit.
Quadratic cent	1452	22 × 3 × 112	⁠ ⁠[citation needed ]
Ksion	1476	22 × 32 × 41	⁠ ⁠[citation needed ]
Cubic cent	1500	22 × 3 × 53	⁠ ⁠[citation needed ]
Heptamu (7mu)	1536	29 × 3	Seventh MIDI-resolution unit, 1/128 (1/(27)) of 12edo semitone
Rhoon	1560	23 × 3 × 5 × 13	⁠ ⁠[citation needed ]
śata	1600	26 × 52	From Sanskrit śatam: hundred; Relative cent of Armodue 16edo Theory⁠ ⁠[citation needed ]
Tile	1632	25 × 3 × 17	⁠ ⁠[citation needed ]
Iota	1700	22 × 52 × 17	Relative cent of 17edo; proposed by Margo Schulter (2002) and George Secor.
Harmos	1728	26 × 33	1728 = 123; 1/144 of 12edo semitone; Proposed by Paul Beaver.
Hind śat / Indian cent	2200	23 × 11 × 52	⁠ ⁠[citation needed ]
Mina	2460	22 × 3 × 5 × 41	Abbreviation of "schismina", edo representation of Sagittal's Extreme (Olympian) precision level JI notation (233eda).
Centidiesis	3100	22 × 52 x 31	⁠ ⁠[citation needed ]
Centiméride	4300	22 × 52 x 43	⁠ ⁠[citation needed ]
Major tina	8269	8269 (prime)	Proposed by Flora Canou (2021).
Tina	8539	8539 (prime)	Provides good approximations for 41-limit primes except 37; named by Dave Keenan and George Secor; edo representation of Sagittal's Insane (Magrathean) precision level JI notation (809eda).
Purdal	9900	22 × 32 × 52 × 11	Relative cent of 99edo; Suggested by Osmiorisbendi, advocated by Tútim Dennsuul Wafiil. See the dedicated page.
Türk sent / Turkish cent	10600	23 × 52 × 53	Relative cent of 106edo, 1/200 of 53edo; invented by M. Ekrem Karadeniz (1965), influenced by Abdülkadir Töre.
Prima	12276	22 × 32 × 11 × 31	Proposed by Erv Wilson, Gene Ward Smith and Gavin Putland.
Jinn	16808	23 × 11 × 191	See the dedicated page.
Jot	30103	30103 (prime)	30103 ≃ 100,000 × log102; Proposed by Augustus de Morgan (1864).
Imp	31920	24 × 3 × 5 × 7 × 19
Flu	46032	24 × 3 × 7 × 137	Proposed by Gene Ward Smith (2005).
Normal atom	78005	5 × 15601	Name proposed by Tristan Bay in 2023; 78005edo consistently maps Kirnberger's atom to 1 edostep and is a very strong 5-limit system. ⁠ ⁠[citation needed ]
MIDI Tuning Standard unit (14mu)	196608	216 × 3	Fourteenth MIDI-resolution unit, 1/16384 (1/(214)) of 12edo semitone.

* More to be added regarding the Heptaméride/Savart units

Non-octave fine measures

There are other fine measurements based upon the logarithmic division of other intervals (e.g. 3/1 (twelfth)), a few of which are listed below:

List of Non-Octave Fine Measures (Logarithmic)

Unit Name (Symbol):	Base Interval:	Parts of Base Interval:	Origin/Significance
Hekt	3/1 (twelfth)	1300	1/100 of 13edt (Bohlen-Pierce) scale step
Euhekt	3/1 (twelfth)	1900	1/100 of 19edt (OnlyPure) scale step
Grad	531441/524288 (Pythagorean comma)	12	12edo flattens 3/2 by this amount
Tuning unit	531441/524288 (Pythagorean comma)	720

To convert hekts, which is quite common in EDT systems, into cents, use following formula: c = h*12/13*math.log(3)/math.log(2)

Relative measures

Within a given equal-stepped tonal system, the relative cent (rct, r¢) can be used to describe properties of pitches (for instance the approximation of JI intervals). It is defined as on 100th (or 1 percent) of the interval between two neighbouring pitches in the used equal tuning.

Ratio

Intervals can be measured also giving their ratio. For instance the major third as 5/4 or the pure fifth 3/2. When combining sizes given in ratios, you have to multiply or divide:

a pure fifth increased by a major third gives the major seventh 3/2 × 5/4 = 15/8,

which is a diatonic semitone below an octave (2/1) / (15/8) = 2/1 × 8/15 = 16/15.

Another notation for ratios is a vector of prime factor exponents, often called a monzo, such as [-4 4 -1⟩ (for the syntonic comma, 2⁻⁴ × 3⁴ × 5⁻¹), which builds a bridge back to the logarithmic measure: intervals can be combined by component-wise addition or subtraction of their vectors.

See also

• Interval span

Articles

• Why the Kirnberger Kernel Is So Small by Don N. Page