Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited


[Greek: "thru tones"]

  • 1. diatonic scale: An adjective referring to a scale composed of five tones and two semitones, such as the Pythagorean diatonic or the familiar 12-tone version.

    See Diatonon

    [from John Chalmers, Divisions of the Tetrachord]

    . . . . . . . . . . . . .

    (The latter is also referred to as a '7oo12' or '7-out-of-12' scale.)

  • 2. diatonic genus: In ancient Greek theory, one of the three basic types of genus. It had a characteristic interval of approximately a "tone" at the top of the tetrachord, then two successive intervals of approximately a "tone" and then a semitone at the bottom, making up a 4/3 "perfect 4th". [see my Tutorial on ancient Greek tetrachord-theory]

  • 3. diatonic semitone: In several different theories of the medieval, Renaissance, Baroque, and Classical periods, a distinction was made between chromatic and diatonic semitones, as there were two different sizes of semitone in many tuning systems. (Two good examples are Pythagorean and meantone.)

    The diatonic semitone is the interval between two notes of a diatonic scale which have different letter-names, whether or not one of them is qualified by an accidental, as in B:C or A:Bb. The implication is that both tones are bonafide diatonic scale-degrees. This interval is called the "minor 2nd" and may be found abbreviated as min2, m2, b2, or -2 . See also chromatic.

    In the Pythagorean [3-Limit] system, the diatonic semitone is also called the leimma or limma.

    Note that in positive systems (typical example: Pythagorean tuning), the chromatic semitone is larger than the diatonic, while in negative systems (typical example: any meantone tuning) it is exactly the opposite and the diatonic is the larger.

    Marchetto specified three types of semitone and called the third, naturally enough, the enharmonic; however, his names were matched with the intervals differently than in traditional theory.

    General comments about diatonic scales / semitones

    The 5-limit JI diatonic scale is a periodicity-block whose unison-vectors are the syntonic comma and the chromatic semitone 25:24. Here is a Monzo lattice of the most typical version of the scale in the major mode:

         81:80   =  [ 4 -1]  syntonic comma
         25:24   =  [-1  2]  Ellis's "small semitone", Rameau's "minor semitone"

    The diatonic semitone in this scale is the ratio 16:15 = 243-15-1 = ~111.7312853 cents. It occurs in this scale between the pairs of ratios ( 5/4 : 4/3 ) and ( 15/8 : 1/1 ) ; if we call the 1/1 "C", this is between E:F and B:C.

    By moving the boundaries of the periodicity-block as a group, it may be transposed to other keys or also to the minor mode in various keys, and retain all of its essential properties.

    Below is an interval-matrix of the just-intonation diatonic scale given in the above lattice, in cents:

    If the lattice shown above were to be cut along the top and bottom (diesis unison-vector) boundaries of the periodicity-block, extended infinitely, and then wrapped into a cylinder so that those two edges meet (i.e, so that 9:8 and 10:9 would occupy the same point, etc.), and then snipped along the left and right edges (which are theoretically infinitely distant) according to how many pitches are contained in the given system, this would describe perfectly the harmonic tonal relationships avialable in the various meantone tunings, whose diatonic scales are the basis of the standard repertoire of "common-practice" (c. 1600-1900) Eurocentric music and music-theory.

    In a meantone tuning, the diatonic semitone is technically -5 generators. Here are the sizes of the diatonic semitone in various historical meantone tunings:

                meantone                 equivalent EDO
                        ~cents                 ~cents
         1/3-comma    126.0688117     2(2/19)  126.3157895
         2/7-comma    120.9482665     2(5/50)  120.0
         7/26-comma   119.1757701        "      "
         1/4-comma    117.1078577     2(3/31)  116.1290323
         2/9-comma    114.120873      2(7/74)  113.5135135
         1/5-comma    111.7312853     2(4/43)  111.627907
         1/6-comma    108.1469037     2(5/55)  109.0909091
         1/11-comma   100.0005819  ~= 2(7/12)  100.0

    Note also that the MIRACLE generator, the secor (~116.7 cents), is also in the typical size range of a diatonic semitone, altho it may not function as one.

    See also:

    [From Joe Monzo, JustMusic: A New Harmony]

    . . . . . . . . . . . . .

    An adjective referring to octave scales consisting of 7 tones and composed at most of three semitones.


    Classical examples with three different semitones:

  • JI major and minor scales
  • Helmholtz (= gypsy) major and minor scales
  • Redfield major and minor scales
  • Classical examples with two different semitones:

  • Greek modes (in the frame of Pythagorean Tuning)
  • scales of the fifth circle in the frame of the 1/4-comma meantone tuning
  • Classical example with only one semitone:

  • heptatonic (e.g. Messien's) modes in the frame of 12-TET
  • [From Jan Haluska, personal communication]


2003.02.09 -- added interval matrix of JI diatonic scale

(to download a zip file of the entire Dictionary, click here)

  • For many more diagrams and explanations of historical tunings, see my book.
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