Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited


schisma


    A term coined by Alexander Ellis in his translation of Helmhotz's On the Sensations of Tone, and originally spelled skhisma. It designates an extremely small interval, just barely discernible to human pitch-detection.

    Unqualified, it is the difference between
    51[= 5/4 = 3.86 Semitone = ~ 386.313 cents] and
    3-8 [= 8192/6561 = 3.84 Semitone = ~ 384.359 cents]
    and has an interval size of approximately
    0.02 Semitone [= 3851 = 32805/32768 = ~ 1.953 cents].

    As no combination of different prime numbers will ever produce ratios which have exactly the same interval size, if cycles of a particular ratio are calculated far enough, very small intervals like this eventually appear between ratios having different sets of prime factors. When these are under consideration, the term schisma is qualified with a latin word designating the higher prime, with the assumption that the other prime being compared is a more familiar one, almost always 3.

    Thus, we get the septimal schisma, which is the difference between
    71[= 7/4 = 9.69 Semitone = ~ 968.825 cents] and
    3-14 [= 8388608/4782969 = 9.73 Semitone = ~ 972.629 cents]
    and has an interval size of approximately
    0.04 Semitones [= 3-147-1 = 33554432/33480783 = ~ 3.804 cents].

    Likewise, there is the nondecimal schisma, which is the difference between
    191 [= 19/16 = 2.98 Semitone = ~ 297.513 cents]
    and the 'standard' Pythagorean "minor 3rd"
    3-3 [= 32/27 = 2.94 Semitone = ~ 294.134 cents],
    and which also has an interval size of approximately
    0.04 Semitones [33191 = 513/512 = ~ 3.378 cents].

    Schismas are a key element in my concept of bridging. I refer to the three schismas explained here as the 3==5, 3==7, and 3==19 bridges, respectively.

    see also anomaly

    [from Joe Monzo, JustMusic: A New Harmony]


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