GOLDEN MEANTONE TUNING

Developed by Thorvald Kornerup and advocated by Jacques Dudon, this tuning is based on Phi, the golden mean, equal to (51/2+1) / 2 or 1.61803398875. (Phi is a unique constant. It is the ratio of two quantities, so that the smaller is to the greater as the greater is to the sum of the greater and the smaller. It is one larger than its own reciprocal. It is the mean of quotients of adjacent elements in a Fibonacci series. Phi is the ratio of width to height of many Greek temples, because ancient architects considered it pleasant to the eye. Whether it is as pleasing to the ear is left to listeners' taste.)

The whole tone is 20.160357456, or 1.117564002.

  • Closure to equal temperament is roughly approached along a Fibonacci series of {5, 7, 12, 19, 31, 50, 81, . . .}. (See definition and comparative data.)

    Mathematical Specifications

    Name Ratio
    Fifth 1.49503
    Fourth 1.33776
    Whole Tone 1.11756
    Diatonic Semitone 1.07111
    Chroma (chromatic semitone) 1.04337
    R = (ratio of whole tone to diatonic semitone) 1.618033989
    1 / R = (ratio of diatonic semitone to whole tone) 0.618033989
    1 / (R-1) = (ratio of diatonic semitone to chroma) 1.618033989
    R - 1 = (ratio of chroma to diatonic semitone) 0.618033989

    Golden Meantone lends itself to scales of various sizes. After the first few tones, each additional increment divides the existing intervals in a ratio close to 3:2 or 5:3, efficiently filling gaps. (Most regular tunings add new tones close to old ones, leaving uneven gaps.) Golden Meantone may provide advantages in combining different tuning systems into one ensemble.

    Intervals

    Interval (above tonic)
    Ratio
    Value in diapasons
    Value in cents
    Corresponding just interval
    Difference in diapasons from just interval
    Difference in cents from just interval
    5th 1.49503 0.58018 696.214 3 / 2 -0.00478 -5.741
    4th 1.33776 0.41982 503.786 4 / 3 +0.00478 +5.741
    Major 2nd 1.11756 0.16036 192.429 9 / 8 -0.00957 -11.481
    Minor 7th 1.78961 0.83964 1007.571 16 / 9 +0.00957 +11.481
    (End of 5-tone scale on D)
    Major 6th 1.67080 0.74054 888.643 5 / 3 +0.00357 +4.285
    Minor 3rd 1.19703 0.25946 311.357 6 / 5 -0.00357 -4.285
    (End of 7-tone scale on D)
    Major 3rd 1.24895 0.32071 384.858 5 / 4 -0.00121 -1.456
    Minor 6th 1.60135 0.67929 815.142 8 / 5 +0.00121 +1.456
    Major 7th 1.86722 0.90089 1081.072 15 / 8 -0.00600 -7.196
    Minor 2nd 1.07111 0.09911 118.928 16 / 15 +0.00600 +7.196
    Augmented 4th 1.39578 0.48107 577.287 7 / 5 -0.00435 -5.225
    (End of 12-tone scale)
    Diminished 5th 1.43289 0.51893 622.713 10 / 7 +0.00435 +5.225
    Augmented 1st 1.04337 0.06125 73.501 25 / 24 +0.00236 +2.829
    Diminished 8th 1.91687 0.93875 1126.499 48 / 25 -0.00236 -2.829
    Augmented 5th 1.55987 0.64143 769.716 14 / 9 +0.00400 +4.800
    Diminished 4th 1.28215 0.35857 430.284 9 / 7 -0.00400 -4.800
    Augmented 2nd 1.16603 0.22161 265.930 7 / 6 -0.00078 -0.941
    Diminished 7th 1.71522 0.77839 934.070 12 / 7 +0.00078 +0.941
    (End of 19-tone scale)
    Augmented 6th 1.74326 0.80179 962.145 7 / 4 -0.00557 -6.681
    Diminished 3rd 1.14728 0.19821 237.855 8 / 7 +0.00557 +6.681
    Augmented 3rd 1.30312 0.38197 458.359 64 / 49 -0.00332 -3.989
    Diminished 6th 1.53478 0.61803 741.641 49 / 32 +0.00332 +3.989
    Augmented 7th 1.94820 0.96214 1154.574 35 / 18 +0.00279 +3.344
    Diminished 2nd 1.02659 0.03786 45.426 36 / 35 -0.00279 -3.344
    Double-augmented 4th 1.45632 0.54232 650.788 16 / 11 +0.00176 +2.106
    Double-diminished 5th 1.37333 0.45768 549.212 11 / 8 -0.00176 -2.106
    Neutral 2nd 1.08862 0.12250 147.003 12 / 11 -0.00303 -3.634
    Neutral 7th 1.83719 0.87750 1052.997 11 / 6 +0.00303 +3.634
    Neutral 6th 1.62753 0.70268 843.217 18 / 11 -0.00781 -9.375
    Neutral 3rd 1.22886 0.29732 356.783 11 / 9 +0.00781 +9.375
    (End of 31-tone scale)
    Interval (above tonic)
    Ratio
    Value in diapasons
    Value in cents
    Corresponding just interval
    Difference in diapasons from just interval
    Difference in cents from just interval


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    Last revised: 17 March 2020

    Last edited: 17 March 2020

    visitors since 23 February 1999