Comparison between tunings: Pythagorean, equal-tempered, quarter-comma meantone, and others. For each, the common origin is arbitrarily chosen as C. The degrees are arranged in the order or the cycle of fifths; as in each of these tunings except just intonation all fifths are of the same size, the tunings appear as straight lines, the slope indicating the relative tempering with respect to Pythagorean, which has pure fifths (3:2, 702 cents). The Pythagorean A♭ (at the left) is at 792 cents, G♯ (at the right) at 816 cents; the difference is the Pythagorean comma. Equal temperament by definition is such that A♭ and G♯ are at the same level. 1⁄4-comma meantone produces the "just" major third (5:4, 386 cents, a syntonic comma lower than the Pythagorean one of 408 cents). 1⁄3-comma meantone produces the "just" minor third (6:5, 316 cents, a syntonic comma higher than the Pythagorean one of 294 cents). In both these meantone temperaments, the enharmony, here the difference between A♭ and G♯, is much larger than in Pythagorean, and with the flat degree higher than the sharp one.Comparison of two sets of musical intervals. The equal-tempered intervals are black; the Pythagorean intervals are green.
Below is a list of intervals expressible in terms of a prime limit (see Terminology), completed by a choice of intervals in various equal subdivisions of the octave or of other intervals.
For commonly encountered harmonic or melodic intervals between pairs of notes in contemporary Western music theory, without consideration of the way in which they are tuned, see Interval (music) § Main intervals.
Terminology
The prime limit[1] henceforth referred to simply as the limit, is the largest prime number occurring in the factorizations of the numerator and denominator of the frequency ratio describing a rational interval. For instance, the limit of the just perfect fourth (4:3) is 3, but the just minor tone (10:9) has a limit of 5, because 10 can be factored into 2 × 5 (and 9 into 3 × 3). There exists another type of limit, the odd limit, a concept used by Harry Partch (bigger of odd numbers obtained after dividing numerator and denominator by highest possible powers of 2), but it is not used here. The term "limit" was devised by Partch.[1]
By definition, every interval in a given limit can also be part of a limit of higher order. For instance, a 3-limit unit can also be part of a 5-limit tuning and so on. By sorting the limit columns in the table below, all intervals of a given limit can be brought together (sort backwards by clicking the button twice).
Just intonation means 5-limit intonation—a ratio of numbers with prime factors no higher than five.
Septimal, undecimal, tridecimal, and septendecimal mean, respectively, 7, 11, 13, and 17-limit intonation.
Meantone refers to meantone temperament, where the whole tone is the mean of the major third. In general, a meantone is constructed in the same way as Pythagorean tuning, as a stack of fifths: the tone is reached after two fifths, the major third after four, so that as all fifths are the same, the tone is the mean of the third. In a meantone temperament, each fifth is narrowed ("tempered") by the same small amount. The most common of meantone temperaments is the quarter-comma meantone, in which each fifth is tempered by 1⁄4 of the syntonic comma, so that after four steps the major third (as C-G-D-A-E) is a full syntonic comma lower than the Pythagorean one. The extremes of the meantone systems encountered in historical practice are the Pythagorean tuning, where the whole tone corresponds to 9:8, i.e. (3:2)2/2, the mean of the major third (3:2)4/4, and the fifth (3:2) is not tempered; and the 1⁄3-comma meantone, where the fifth is tempered to the extent that three ascending fifths produce a pure minor third.(See meantone temperaments). The music program Logic Pro uses also 1⁄2-comma meantone temperament.
Equal-tempered refers to X-tone equal temperament with intervals corresponding to X divisions per octave.
Tempered intervals however cannot be expressed in terms of prime limits and, unless exceptions, are not found in the table below.
The table can also be sorted by frequency ratio, by cents, or alphabetically.
Superparticular ratios are intervals that can be expressed as the ratio of two consecutive integers.
^ a bFox, Christopher (2003). "Microtones and Microtonalities", Contemporary Music Review, v. 22, pt. 1–2. (Abingdon, Oxfordshire, UK: Routledge): p. 13.
^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay az ba bb bc bd be bf bg bh biFonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music 29, no. 2 (Summer): 106–137.
^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay az ba bb bc bd be bf bg bh bi bj bk bl bm bn bo bp bq br bs bt bu bv bw bx by bz ca cb cc cd ce cf cg ch ci cj ck cl cm cn co cp cq cr cs ct cu cv cw cx cy cz da db dc dd de df dg dh di"List of intervals", Huygens-Fokker Foundation. The Foundation uses "classic" to indicate "just" or leaves off any adjective, as in "major sixth".
^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay az ba bb bc bd be bf bg bh bi bj bk bl bm bn bo bp bq br bs bt bu bv bw bx by bz ca cb cc cd ce cf cg ch ci cj ck cl cm cn co cp cq cr cs ct cu cv cw cx cy cz da db dc dd de df dg dh di dj dk dl dm dn do dp dq dr ds dt du dv dw dx dy dz"Anatomy of an Octave", Kyle Gann (1998). Gann leaves off "just" but includes "5-limit". He uses "median" for "neutral".
^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ayHaluška, Ján (2003). The Mathematical Theory of Tone Systems, pp. xxv–xxix. ISBN 978-0-8247-4714-5.
^William Smythe Babcock Mathews (1895). Pronouncing Dictionary and Condensed Encyclopedia of Musical Terms, p. 13. ISBN 1-112-44188-3.
^ a b c d e fAnger, Joseph Humfrey (1912). A Treatise on Harmony, with Exercises, Volume 3, pp. xiv–xv. W. Tyrrell.
^ a b c d e f g h i j k l m n oHermann Ludwig F. von Helmholtz (Alexander John Ellis, trans.) (1875). "Additions by the translator", On the sensations of tone as a physiological basis for the theory of music, p. 644. [ISBN unspecified]
^A. R. Meuss (2004). Intervals, Scales, Tones and the Concert Pitch C. Temple Lodge Publishing. p. 15. ISBN 1902636465.
^ a b c d e f g h i j k l m n o p q r s t u v w x yPaul, Oscar (1885). A Manual of Harmony for Use in Music-schools and Seminaries and for Self-instruction, p. 165. Theodore Baker, trans. G. Schirmer. Paul uses "natural" for "just".
^ a b"13th-harmonic", 31et.com.
^Brabner, John H. F. (1884). The National Encyclopaedia, vol. 13, p. 182. London. [ISBN unspecified]
