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:
Unit Name (Symbol): | Divisions of Octave | Prime Factors | Origin / Significance |
---|---|---|---|
Eka | 16 | 2^{4} | From Sanskrit eka: one, unit; chromatic unit of Armodue 16edo Theory ^{[citation needed ]}. |
Normal diesis | 31 | 31 (prime) | See the dedicated page. |
Méride | 43 | 43 (prime) | Proposed by Joseph Sauveur, as 7 heptaméride units^{[1]}^{[2]}. |
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 | 2^{2} × 3 × 5 | |
Morion | 72 | 2^{3} × 3^{2} | See the dedicated page. |
Farab | 144 | 2^{4} × 3^{2} | 1/12 of 12edo semitone; Proposed by al-Farabi in 10th century^{[1]}^{[3]}. |
Mem | 205 | 5 × 41 | Unit used by H-Pi Instruments^{[1]}^{[4]}^{[5]}. |
Tredek | 270 | 2 × 3^{3} × 5 | Proposed by Joseph Monzo (2013)^{[6]}. |
Savart* | 300 | 2^{2} × 3 × 5^{2} | Alexander Wood's definition of the Savart^{[7]}, containing 12edo. |
Heptaméride / eptaméride / savart* | 301 | 7 × 43 | 301 ≃ 1,000 × log_{10}2; 1/7 of Méride unit; proposed by Joseph Sauveur (1701), advocated by Félix Savart^{[1]}^{[8]}. |
Gene | 311 | 311 (prime) | Proposed by Joseph Monzo (2007)^{[9]}. |
Dröbisch Angle | 360 | 2^{3} × 3^{2} × 5 | Proposed as angle by Moritz Dröbisch in the 19th century, later by Andrew Pikler as the current name in Logarithmic Frequency Systems (1966)^{[1]}. |
Squb | 494 | 2 × 13 × 19 | ^{[citation needed ]} |
Great iring / centitone | 500 | 2^{2} × 5^{3} | ^{[citation needed ]} |
Dexl | 540 | 2^{2} × 3^{3} × 5 | Proposed by Joseph Monzo (2023)^{[10]}. |
Iring / centitone | 600 | 2^{3} × 3 × 5^{2} | Relative cent of 6edo (12edo tone); Proposed by Widogast Iring (1898), later by Joseph Yasser as a "centitone" (1932)^{[1]}^{[11]}. |
Skisma (Sk) | 612 | 2^{2} × 3^{2} × 17 | Edo representation of Sagittal's Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina"^{[1]}^{[12]}. |
Delfi | 665 | 5 × 7 × 19 | ^{[1]} |
Small iring / centitone | 700 | 2^{2} × 5^{2} x 7 | ^{[citation needed ]} |
Woolhouse | 730 | 2 × 5 × 73 | Proposed by Wesley S.B. Woolhouse (1835)^{[13]}. |
Millioctave (moct) | 1000 | 2^{3} × 5^{3} | See the dedicated page. |
Cent (¢) | 1200 | 2^{4} × 3 × 5^{2} | See the dedicated page. |
Greater muon | 1224 | 2^{3} × 3^{2} × 17 | ^{[citation needed ]} |
Triangular cent | 1260 | 2^{2} × 3^{2} × 5 × 7 | ^{[citation needed ]} |
Pion | 1272 | 2^{3} × 3 × 53 | ^{[citation needed ]} |
Pound | 1344 | 2^{6} × 3 × 7 | ^{[citation needed ]} |
Neutron | 1392 | 2^{4} × 3 × 29 | ^{[citation needed ]} |
Lesser muon | 1428 | 2^{2} × 3 × 7 × 17 | ^{[citation needed ]} |
Decifarab | 1440 | 2^{5} × 3^{2} × 5 | 1/10 of Farab unit^{[1]}. |
Quadratic cent | 1452 | 2^{2} × 3 × 11^{2} | ^{[citation needed ]} |
Ksion | 1476 | 2^{2} × 3^{2} × 41 | ^{[citation needed ]} |
Cubic cent | 1500 | 2^{2} × 3 × 5^{3} | ^{[citation needed ]} |
Heptamu (7mu) | 1536 | 2^{9} × 3 | Seventh MIDI-resolution unit, 1/128 (1/(2^{7})) of 12edo semitone^{[14]} |
Rhoon | 1560 | 2^{3} × 3 × 5 × 13 | ^{[citation needed ]} |
śata | 1600 | 2^{6} × 5^{2} | From Sanskrit śatam: hundred; Relative cent of Armodue 16edo Theory ^{[citation needed ]} |
Tile | 1632 | 2^{5} × 3 × 17 | ^{[citation needed ]} |
Iota | 1700 | 2^{2} × 5^{2} × 17 | Relative cent of 17edo; proposed by Margo Schulter (2002) and George Secor^{[1]}. |
Harmos | 1728 | 2^{6} × 3^{3} | 1728 = 12^{3}; 1/144 of 12edo semitone; Proposed by Paul Beaver^{[1]}^{[15]}. |
Hind śat / Indian cent | 2200 | 2^{3} × 11 × 5^{2} | ^{[citation needed ]} |
Mina | 2460 | 2^{2} × 3 × 5 × 41 | Abbreviation of "schismina", edo representation of Sagittal's Extreme (Olympian) precision level JI notation (233eda)^{[1]}^{[16]}. |
Centidiesis | 3100 | 2^{2} × 5^{2} x 31 | ^{[citation needed ]} |
Centiméride | 4300 | 2^{2} × 5^{2} x 43 | ^{[citation needed ]} |
Major tina | 8269 | 8269 (prime) | Proposed by Flora Canou (2021)^{[17]}. |
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)^{[1]}^{[18]}. |
Purdal | 9900 | 2^{2} × 3^{2} × 5^{2} × 11 | Relative cent of 99edo; Suggested by Osmiorisbendi, advocated by Tútim Dennsuul Wafiil. See the dedicated page. |
Türk sent / Turkish cent | 10600 | 2^{3} × 5^{2} × 53 | Relative cent of 106edo, 1/200 of 53edo; invented by M. Ekrem Karadeniz (1965), influenced by Abdülkadir Töre^{[1]}^{[19]}^{[20]}. |
Prima | 12276 | 2^{2} × 3^{2} × 11 × 31 | Proposed by Erv Wilson, Gene Ward Smith and Gavin Putland^{[1]}. |
Jinn | 16808 | 2^{3} × 11 × 191 | See the dedicated page. |
Jot | 30103 | 30103 (prime) | 30103 ≃ 100,000 × log_{10}2; Proposed by Augustus de Morgan (1864)^{[1]}^{[21]}^{[15]}. |
Imp | 31920 | 2^{4} × 3 × 5 × 7 × 19 | ^{[1]} |
Flu | 46032 | 2^{4} × 3 × 7 × 137 | Proposed by Gene Ward Smith (2005)^{[1]}^{[22]}. |
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 | 2^{16} × 3 | Fourteenth MIDI-resolution unit, 1/16384 (1/(2^{14})) of 12edo semitone^{[1]}. |
* 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:
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} × 3^{4} × 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
Articles
Notes
- ↑ ^{1.00} ^{1.01} ^{1.02} ^{1.03} ^{1.04} ^{1.05} ^{1.06} ^{1.07} ^{1.08} ^{1.09} ^{1.10} ^{1.11} ^{1.12} ^{1.13} ^{1.14} ^{1.15} ^{1.16} ^{1.17} ^{1.18} Stichting Huygens-Fokker: Logarithmic Interval Measures
- ↑ Tonalsoft | Méride / 43-ed2 / 43-edo / 43-ET / 43-tone equal-temperament
- ↑ Tonalsoft | Farab.
- ↑ H-Pi Instruments | Hunt Theoretical System
- ↑ Tonalsoft | Mem, 205-edo
- ↑ Tonalsoft | Tredek, 270-edo
- ↑ The Physics of Music, Alexander Wood, 1944.
- ↑ Tonalsoft | Heptaméride
- ↑ Tonalsoft | Gene, 311-edo
- ↑ Tonalsoft | Dexl, 540-edo
- ↑ Tonalsoft | Centitone, iring
- ↑ Tonalsoft | Sk, 612-edo
- ↑ Essay on musical intervals, harmonics, and the temperament of the musical scale, &c, Wesley S.B. Woolhouse.
- ↑ Tonalsoft | 7mu / heptamu
- ↑ ^{15.0} ^{15.1} Tonalsoft | Equal temperaments
- ↑ Tonalsoft | Mina
- ↑ The Sagittal Forum | Definition of the tina reviewed
- ↑ Tonalsoft | Tina
- ↑ Tonalsoft | Türk-sent
- ↑ 79-Tone Tuning & Theory for Turkish Maqam Music, Ozan Yarman.
- ↑ Tonalsoft | Jot
- ↑ Tonalsoft | Flu