Pianotech

How to calculate the value of the semitones of any well temperament

  • 1.  How to calculate the value of the semitones of any well temperament

    Posted 4 days ago
    Edited by Roshan Kakiya 4 days ago

    Aims

    To show how the value of the semitones, in cents, of any well temperament can be calculated by using the Circle of Fifths and Fourths.

    This post is completely about the theoretical/mathematical aspects of tuning and their practical application with respect to the creation of scales, with the use of a software called Scala, for the purpose of retuning MIDI files.

     

    Essential Knowledge

    The section entitled “Well Temperament” of the following post:

    http://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=a46d662f-d447-4feb-9d22-f97ee1e7342b&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf&tab=digestviewer&ReturnUrl=%2fcommunities%2fcommunity-home%2fdigestviewer%3fcommunitykey%3d6265a40b-9fd2-4152-a628-bd7c7d770cbf%26tab%3ddigestviewer



    Thomas Young’s Second Temperament

    This tuning will be used as an example for the rest of this post. Here is its Circle of Fifths and Fourths:


    A Scala file of this tuning can probably be found online and used. However, the purpose of this post is to show how these Scala files can be created so that the skills needed to create a Scala file of any well temperament can be acquired.


    Why Have I Chosen Well Temperament?

    Pure intonation, which is based on fractions, is supposedly the most harmonious form of tuning. However, it is very difficult to create pure tunings that are practical and do not contain wolf fifths that deviate massively from pure fifths. Usable pure tunings do exist though:

    http://forum.pianoworld.com/ubbthreads.php/topics/2722008/3.html

    On the other hand, 12-tone equal temperament (12-TET) is not considered to be as harmonious as pure intonation, although it corrects the Pythagorean comma. This is because every 12-TET interval except for the octave has been tempered.

    Well temperament is the best compromise since it contains pure and tempered intervals and it also corrects the Pythagorean comma which ensures its practicality. Additionally, the unequal semitones of well temperament seemingly provide key colour which seems to be a feature that is missing from 12-TET due to its equal semitones. Fortunately, Scala files can contain the value of each interval within, and including, the octave to a high degree of mathematical precision. Therefore, MIDI files can be retuned precisely. Multiple well temperaments have been created. Thomas Young’s Second Temperament is one of them.


    Fifths and Fourths

    First, the values of the fifths and the fourths must be calculated. My whole method is based on the Circle of Fifths and Fourths.

     

    Tempered fifths and fourths:

    C-G-D-A-E-B-Gb.

     

    Tempered fifth = 698.05 cents.

    Tempered fourth = 501.95 cents.

     

    Pure fifths and fourths:

    Gb-Db-Ab-Eb-Bb-F-C.

     

    Pure fifth = 701.96 cents.

    Pure fourth = 498.04 cents.

     

    Whole Tones

    Whole Tone = Fifth – Fourth.

     

    C-D = C-G – D-G = 698.05 cents – 501.95 cents = 196.10 cents.

    D-E = D-A – E-A = 698.05 cents – 501.95 cents = 196.10 cents.

    F-G = F-C – G-C = 701.96 cents – 501.95 cents = 200.01 cents.

    G-A = D-A – D-G = 698.05 cents – 501.95 cents = 196.10 cents.

    A-B = E-B – E-A = 698.05 cents – 501.95 cents = 196.10 cents.

     

    Db-Eb = Db-Ab – Eb-Ab = 701.96 cents – 498.04 cents = 203.92 cents.

    Gb-Ab = Db-Ab – Db-Gb = 701.96 cents – 498.04 cents = 203.92 cents.

    Ab-Bb = Eb-Bb – Eb-Ab = 701.96 cents – 498.04 cents = 203.92 cents.

    Bb-C = F-C – Bb-F = 701.96 cents – 498.04 cents = 203.92 cents.

     


    Semitones (Stage 1)

    Semitone = Fourth – (Whole Tone + Whole Tone).

     

    E-F = C-F – (C-D + D-E) = 498.04 cents – (196.10 cents + 196.10 cents) = 105.84 cents.

    B-C = G-C – (G-A + A-B) = 501.95 cents – (196.10 cents + 196.10 cents) = 109.75 cents.

     

    Semitone = Octave – (Semitone + Fourth + Fourth).

     

    C-Db = C-C – (B-C + Gb-B + Db-Gb) = 1200 cents – (109.75 cents + 501.95 cents + 498.04 cents) = 90.26 cents.

     


    Semitones (Stage 2)

    Semitone = Whole Tone – Semitone.

     

    Db-D = C-D – C-Db = 196.10 cents – 90.26 cents = 105.84 cents.

    D-Eb = Db-Eb – Db-D = 203.92 cents – 105.84 cents = 98.08 cents.

    Eb-E = D-E – D-Eb = 196.10 cents – 98.08 cents = 98.02 cents.

     

    Bb-B = Bb-C – B-C = 203.92 cents – 109.75 cents = 94.17 cents.

    A-Bb = A-B – Bb-B = 196.10 cents – 94.17 cents = 101.93 cents.

    Ab-A = Ab-Bb – A-Bb = 203.92 cents – 101.93 cents = 101.99 cents.

    G-Ab = G-A – Ab-A = 196.10 cents – 101.99 cents = 94.11 cents.

    Gb-G = Gb-Ab – G-Ab = 203.92 cents – 94.11 cents = 109.81 cents.

    F-Gb = F-G – Gb-G = 200.01 cents – 109.81 cents = 90.20 cents.

     


    Semitones (Summary)

    C-Db = 90.26 cents.

    Db-D = 105.84 cents.

    D-Eb = 98.08 cents.

    Eb-E = 98.02 cents.

    E-F = 105.84 cents.

    F-Gb = 90.20 cents.

    Gb-G = 109.81 cents.

    G-Ab = 94.11 cents.

    Ab-A = 101.99 cents.

    A-Bb = 101.93 cents.

    Bb-B = 94.17 cents.

    B-C = 109.75 cents.

     


    Scale (Thomas Young’s Second Temperament)

    C = 0.00 cents.

    Db = 90.26 cents.

    D = 196.10 cents.

    Eb = 294.18 cents.

    E = 392.20 cents.

    F = 498.04 cents.

    Gb = 588.24 cents.

    G = 698.05 cents.

    Ab = 792.16 cents.

    A = 894.15 cents.

    Bb = 996.08 cents.

    B = 1090.25 cents.

    C = 1200.00 cents.

     


    Verification

    I have tested this data with Scala and the results are convincing.

    The results:

    Pure fifths reduced by 1/6 of the Pythagorean comma = 698 cents. C-G-D-A-E-B-Gb.

    Pure fifths = 702 cents. Gb-Db-Ab-Eb-Bb-F-C.

     

    1. Press “Edit” and then enter the value of each interval. C = 0.00 cents is already there by default. Start with Db = 90.26 cents.
    2. Press “Analyse” -> “Tone circle” -> “Temperament Radar” to view the Circle of Fifths and Fourths. Round the values to the nearest whole number. The results should match the results above.

     


    Retuning MIDI Files (Final Stage)

    Use MIDI files that only contain one instrument such as the piano. MIDI files that contain orchestras cannot usually be successfully retuned because all the instruments eventually become out of sync somehow.

     

    1. Press “Tools” -> “Retune MIDI file” and then place the original MIDI file in the “Input file” section.
    2. Choose a destination for the retuned MIDI file in the “Output file” section, for example, Desktop. Give the “Output file” a short name without any spaces. I have encountered problems when I have used a long name with spaces. Alternatively, use the underscore sign to indicate spaces, for example, Young_2.

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    Roshan Kakiya
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