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Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

  • 1.  Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma

    Posted 12-06-2018 11:29
    Edited by Roshan Kakiya 12-07-2018 12:30
      |   view attached

    Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma
    Roshan Kakiya's Well Temperament (created by correcting the Pythagorean comma):

    http://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=db956071-2c21-439d-8ae5-be761bfbf01e&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf&tab=digestviewer&ReturnUrl=%2fcommunities%2fcommunity-home%2fdigestviewer%3fcommunitykey%3d6265a40b-9fd2-4152-a628-bd7c7d770cbf%26tab%3ddigestviewer


    Just Intonation Dilemma (partially solved by correcting the Syntonic comma):

    http://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=3ce711a8-1939-4ab8-9f88-c85435a71885&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf&tab=digestviewer&ReturnUrl=%2fcommunities%2fcommunity-home%2fdigestviewer%3fCommunityKey%3d6265a40b-9fd2-4152-a628-bd7c7d770cbf


    Steps Required to Completely Solve the Just Intonation Dilemma

    1. Correct the Syntonic comma:

    The Syntonic comma can be corrected by keeping C-G (Fifth) pure, narrowing C-D (Major Second) by 1/3 Syntonic comma, narrowing F-G (Major Second) by 1/3 Syntonic comma and widening D-F (Minor Third) by 2/3 Syntonic comma.


    2. Establish the Fifth-Fourth relationship and the Fourth-Fifth relationship:

    Fifth-Fourth relationship = Pure Fifth-Tempered Fourth = C-G-D-A-E-B-F#

    Pure Fifth = 701.96 cents
    Tempered Fourth = 505.21 cents

    Fourth-Fifth relationship = Pure Fourth-Tempered Fifth = F#-C#-G#-D#-A#-F-C

    Pure Fourth = 498.04 cents
    Tempered Fifth = 694.79 cents


    3. Use the relationships established in Step 2 to create a temperament:

    C-G (Pure Fifth)
    G-D (Tempered Fourth)
    D-A (Pure Fifth)
    A-E (Tempered Fourth)
    E-B (Pure Fifth)
    B-F# (Tempered Fourth)
    F#-C# (Pure Fourth)
    C#-G# (Tempered Fifth)
    G#-D# (Pure Fourth)
    D#-A# (Tempered Fifth)
    A#-F (Pure Fourth)
    F-C (Tempered Fifth)

    C 0.00
    C# 92.18
    D 196.74
    D# 288.92
    E 393.48
    F 485.66
    F# 590.22
    G 701.96
    G# 786.96
    A 898.70
    A# 983.71
    B 1095.44
    C 1180.45

    The Octave should have a value of 1200 cents. However, the correction of the Syntonic comma has led to the creation of another comma called the Diaschisma which has a value of 19.55 cents (1200 cents - 1180.45 cents).


    4. Correcting the Diaschisma:

    Pure Fifth = 701.96 cents
    Tempered Fifth = 694.79 cents

    There are 6 Pure Fifths and 6 Tempered Fifths. The Circle of Fifths is not complete because of the Diaschisma. The 6 Pure Fifths can remain intact. Each of the 6 Tempered Fifths must be widened in order to correct the Diaschisma. The Diaschisma can be corrected by widening each Tempered Fifth by 1/6 Diaschisma.

    Revised Tempered Fifth = 698.04 cents


    5. Producing the Final Temperament:

    Here is a list of all the intervals that are needed to produce the Final Temperament:

    Pure Fifth = 701.96 cents
    Revised Tempered Fifth = 698.04 cents
    Pure Fourth = 498.04 cents
    Revised Tempered Fourth = 501.96 cents

    C-G (Pure Fifth)
    G-D (Revised Tempered Fourth)
    D-A (Pure Fifth)
    A-E (Revised Tempered Fourth)
    E-B (Pure Fifth)
    B-F# (Revised Tempered Fourth)
    F#-C# (Pure Fourth)
    C#-G# (Revised Tempered Fifth)
    G#-D# (Pure Fourth)
    D#-A# (Revised Tempered Fifth)
    A#-F (Pure Fourth)
    F-C (Revised Tempered Fifth)

    C 0.00
    C# 101.96
    D 200.00
    D# 301.96
    E 400.00
    F 501.96
    F# 600.00
    G 701.96
    G# 800.00
    A 901.96
    A# 1000.00
    B 1101.96
    C 1200.00


    The Final Temperament is Roshan Kakiya's Well Temperament!

    Conclusion: Roshan Kakiya's Well Temperament Completely Solves the Just Intonation Dilemma.

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    Roshan Kakiya
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