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12-Tone Equal Temperament: An Alternative Approach

  • 1.  12-Tone Equal Temperament: An Alternative Approach

    Posted 01-19-2019 16:16
    Edited by Roshan Kakiya 01-19-2019 19:41
      |   view attached

    Standard 12-Tone Equal Temperament

    Semitone = 100 cents.
    12 Narrowed Fifths = 8400 cents.
    7 Pure Octaves = 8400 cents.

    Note Cents
    C 0.00
    C# 100.00
    D 200.00
    D# 300.00
    E 400.00
    F 500.00
    F# 600.00
    G 700.00
    G# 800.00
    A 900.00
    A# 1000.00
    B 1100.00
    C 1200.00


    Alternative 12-Tone Equal Temperament

    Revised Semitone = 100.28 cents.
    12 Pure Fifths = 8423.46 cents.
    7 Widened Octaves = 8423.46 cents.

    Note Cents
    C 0.00
    C# 100.28
    D 200.56
    D# 300.84
    E 401.12
    F 501.40
    F# 601.68
    G 701.96
    G# 802.23
    A 902.51
    A# 1002.79
    B 1103.07
    C 1203.35


    The inharmonicity of a piano's strings needs to be dealt with in order to make a piano sound in tune. Piano tuners deal with the inharmonicity of a piano's strings by stretching Octaves.

    In theory, the standard version of 12-Tone Equal Temperament corrects the Pythagorean Comma by narrowing all 12 Pure Fifths whilst preserving the purity of all 7 Octaves. In practice, the Octaves are stretched which means that the Octaves will not be pure in reality. This means that the theory of the standard version of 12-Tone Equal Temperament does not match the practice of stretching Octaves.

    In theory, the alternative version of 12-Tone Equal Temperament corrects the Pythagorean Comma by widening all 7 Octaves whilst preserving the purity of all 12 Pure Fifths. This means that the theory of the alternative version of 12-Tone Equal Temperament does match the practice of stretching Octaves.

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    Roshan Kakiya
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  • 2.  RE: 12-Tone Equal Temperament: An Alternative Approach

    Posted 01-20-2019 12:57
    Roshan: Do you imagine all piano octaves stretch the same (3.35 cents) or only the temperament?


  • 3.  RE: 12-Tone Equal Temperament: An Alternative Approach

    Posted 01-20-2019 13:24
    Edited by Roshan Kakiya 01-20-2019 14:18
    The inharmonicity of a piano's strings increases as we move up the keyboard:

    http://daffy.uah.edu/piano/page4/page8/index.html

    Therefore, all octaves should not be stretched by the same amount.

    The alternative version of 12-TET only contains octaves of equal size in order to theoretically fulfil the definition of equal temperament (every semitone must be equal in terms of size).

    I have designed the alternative version of 12-TET to account for the practice of stretching octaves which means it can be used as a framework for controlling inharmonicity. The size of each octave can be adjusted in accordance with the specific inharmonicity of a piano.
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    Roshan Kakiya
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  • 4.  RE: 12-Tone Equal Temperament: An Alternative Approach

    Posted 01-20-2019 16:32
    A framework, yes. I understand you depict one average generic piano octave not all, and maybe not even any in particular. Thanks for your interesting post and explanation.


  • 5.  RE: 12-Tone Equal Temperament: An Alternative Approach

    Posted 01-20-2019 15:26
    Edited by Jon Page 01-20-2019 16:27
    This might be good in the digital realm but a piano is tuned with the specific partial makeup of each note, pianos are not tuned the same because each has differing partial strengths (inharmonicity). I can't see forcing this on a piano would have an acceptable outcome.

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    Regards,

    Jon Page
    mailto:jonpage@pianocapecod.com
    http://www.pianocapecod.com
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  • 6.  RE: 12-Tone Equal Temperament: An Alternative Approach

    Posted 01-20-2019 18:13
    The Circle of Fifths of Standard 12-Tone Equal Temperament and Alternative 12-Tone Equal Temperament.

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    Roshan Kakiya
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  • 7.  RE: 12-Tone Equal Temperament: An Alternative Approach

    Posted 01-27-2019 07:26
    Alternatives to ET are available, effective, musical and provide coherence to piano sound.

    Best wishes

    David P

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    David Pinnegar BSc ARCS
    Curator and House Tuner - Hammerwood Park, East Grinstead, Sussex UK
    antespam@gmail.com

    Call for papers - Seminar 6th May 2019 - "Restoring emotion to classical music through tuning."
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