Pianotech

  • 1.  Ranking Intervals by LCM of Ratio's Numbers, Harmonic Entropy and Consonance

    Posted 05-04-2019 15:56
      |   view attached

    Ranking Intervals by LCM of Ratio's Numbers + Harmonic Entropy + Consonance


    Compare the rankings below with the following chart on Harmonic Entropy



    Source: http://tonalsoft.com/enc/h/harmonic-entropy.aspx



    Rank Ratio Number 1 Number 2 LCM Number 1 + Number 2 + LCM
    1 1/1 1 1 1 3
    2 2/1 2 1 2 5
    3 3/1 3 1 3 7
    4 4/1 4 1 4 9
    5 3/2 3 2 6 11
    6 5/2 5 2 10 17
    7 4/3 4 3 12 19
    8 7/2 7 2 14 23
    9 5/3 5 3 15 23
    10 5/4 5 4 20 29
    11 7/3 7 3 21 31
    12 8/3 8 3 24 35
    13 7/4 7 4 28 39
    14 6/5 6 5 30 41
    15 10/3 10 3 30 43
    16 11/3 11 3 33 47
    17 7/5 7 5 35 47
    18 9/4 9 4 36 49
    19 8/5 8 5 40 53
    20 7/6 7 6 42 55
    21 11/4 11 4 44 59
    22 9/5 9 5 45 59
    23 13/4 13 4 52 69
    24 11/5 11 5 55 71
    25 8/7 8 7 56 71
    26 12/5 12 5 60 77
    27 15/4 15 4 60 79
    28 9/7 9 7 63 79
    29 13/5 13 5 65 83
    30 11/6 11 6 66 83
    31 10/7 10 7 70 87
    32 14/5 14 5 70 89



    Compare the rankings below with the following chart on Consonance



    Source: http://www.huygens-fokker.org/bpsite/consonance.html



    Rank Ratio Number 1 Number 2 LCM Number 1 + Number 2 + LCM
    1 1/1 1 1 1 3
    2 2/1 2 1 2 5
    3 3/1 3 1 3 7
    4 3/2 3 2 6 11
    5 5/2 5 2 10 17
    6 4/3 4 3 12 19
    7 5/3 5 3 15 23
    8 5/4 5 4 20 29
    9 7/3 7 3 21 31
    10 8/3 8 3 24 35
    11 7/4 7 4 28 39
    12 6/5 6 5 30 41
    13 7/5 7 5 35 47
    14 9/4 9 4 36 49
    15 8/5 8 5 40 53
    16 7/6 7 6 42 55
    17 11/4 11 4 44 59
    18 9/5 9 5 45 59
    19 11/5 11 5 55 71
    20 9/7 9 7 63 79


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    Roshan Kakiya
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  • 2.  RE: Ranking Intervals by LCM of Ratio's Numbers, Harmonic Entropy and Consonance

    Posted 05-06-2019 12:59
      |   view attached
    I have attached an updated Excel file to this post. I have also updated both of the tables in my original post in accordance with the details mentioned below.

    The following column has been added to both of the tables:

    • A column for Number 1 + Number 2 + LCM. This column can be used to rank the ratios that have the same LCM. For example, LCM = 30 for 6/5 and 10/3, LCM = 60 for 12/5 and 15/4 and LCM = 70 for 10/7 and 14/5.

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    Roshan Kakiya
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    Original Message


  • 3.  RE: Ranking Intervals by LCM of Ratio's Numbers, Harmonic Entropy and Consonance

    Posted 05-07-2019 08:02

    Equations for LCM of Ratio's Numbers Ranking System


    Ratio = Number 1 / Number 2.

    Each Ratio must be reduced to its simplest form. For example, 4/2 must be reduced to 2/1.


    LCM = Number 1 × Number 2.

    LCM stands for Lowest Common Multiple. For example, the LCM of 3 and 2 is 6. The lower the LCM, the higher the rank of the Ratio.


    If LCM = LCM, use Number 1 + Number 2 + LCM.

    Some Ratios have the same LCM. For example, 6/5 and 10/3 have the same LCM because the LCM of 6 and 5 is 30 and the LCM of 10 and 3 is 30. Therefore, 6/5 and 10/3 cannot be ranked by LCM so they must be ranked by Number 1 + Number 2 + LCM instead. The lower the result of Number 1 + Number 2 + LCM, the higher the rank of the Ratio.


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    Roshan Kakiya
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    Original Message