Pianotech

  • 1.  Ranking the Intervals of Just Intonation by Lowest First Number then Lowest Second Number of Ratio

    Posted 05-02-2019 10:14
      |   view attached

    UPDATE


    https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=67b6d4f7-737f-451d-965e-5d2797234896&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf



    Previous Ranking System: Lowest Common Multiple of Ratio's Numbers

    https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=82c96d81-b20a-41fb-93c8-12d8e0f06c60&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

    This ranking system breaks down for intervals beyond the Octave such as the Major Seventeenth. Therefore, I have devised a new ranking system that overcomes this problem.


    Frequency Ratios 

    Semitones
    		 Note Interval Ratio
    0 C Unison 1:1
    1 C# Minor Second 16:15
    2 D Major Second 9:8
    3 D# Minor Third 6:5
    4 E Major Third 5:4
    5 F Perfect Fourth 4:3
    6 F# Tritone 25:18
    7 G Perfect Fifth 3:2
    8 G# Minor Sixth 8:5
    9 A Major Sixth 5:3
    10 A# Minor Seventh 9:5
    11 B Major Seventh 15:8
    12 C Octave 2:1

    Source: https://fundamentals-of-piano-practice.readthedocs.io/en/latest/chapter2/CH2.2.html


    New Ranking System: Lowest 1st Number then Lowest 2nd Number of Ratio


    Rank Interval Ratio Number 1 Number 2
    1 Octave 2/1 2 1
    2 Perfect Twelfth 3/1 3 1
    3 Perfect Fifth 3/2 3 2
    4 Perfect Fourth 4/3 4 3
    5 Major Seventeenth 5/1 5 1
    6 Major Sixth 5/3 5 3
    7 Major Third 5/4 5 4
    8 Minor Third 6/5 6 5
    9 Minor Sixth 8/5 8 5
    10 Minor Seventh 9/5 9 5
    11 Major Second 9/8 9 8
    12 Major Seventh 15/8 15 8
    13 Minor Second 16/15 16 15
    14 Tritone 25/18 25 18


    This new system is compatible with the order in which harmonics coincide.


    Examples


    1. 2nd harmonic of Unison coincides with 1st harmonic of Octave.

    2. 3rd harmonic of Unison coincides with 1st harmonic of Perfect Twelfth.

    3. 3rd harmonic of Unison coincides with 2nd harmonic of Perfect Fifth.

    4. 4th harmonic of Unison coincides with 3rd harmonic of Perfect Fourth.

    5. 5th harmonic of Unison coincides with 1st harmonic of Major Seventeenth.

    6. 5th harmonic of Unison coincides with 3rd harmonic of Major Sixth.

    7. 5th harmonic of Unison coincides with 4th harmonic of Major Third.

    8. 6th harmonic of Unison coincides with 5th harmonic of Minor Third.

    9. 8th harmonic of Unison coincides with 5th harmonic of Minor Sixth.

    10. 9th harmonic of Unison coincides with 5th harmonic of Minor Seventh.

    11. 9th harmonic of Unison coincides with 8th harmonic of Major Second.

    12. 15th harmonic of Unison coincides with 8th harmonic of Major Seventh.

    13. 16th harmonic of Unison coincides with 15th harmonic of Minor Second.

    14. 25th harmonic of Unison coincides with 18th harmonic of Tritone.


    ------------------------------
    Roshan Kakiya
    ------------------------------


  • 2.  RE: Ranking the Intervals of Just Intonation by Lowest First Number then Lowest Second Number of Ratio

    Posted 05-03-2019 15:35
      |   view attached
    I have added more information to the table in my original post:

    1. A column for Frequency 1 (Hz). Frequency 1 is the same for all the intervals. Frequency 1 = 300 Hz.
    2. A column for Frequency 2 (Hz). Frequency 2 has been calculated for each interval by multiplying Frequency 1 by the frequency ratio. For example, the frequency ratio of the Octave is 2/1 so Frequency 2 = 300 Hz × 2/1 = 600 Hz.
    3. A column for LCM (Hz). The lowest common multiple of Frequency 1 and Frequency 2 has been calculated. LCM (Hz) is the frequency at which Frequency 1 and Frequency 2 coincide. For example, Frequency 1 (300 Hz) and Frequency 2 (600 Hz) coincide at LCM (600 Hz) for the Octave.

    14 Intervals Ranked by Lowest 1st Number then Lowest 2nd Number of Ratio + Analysis of LCM (Hz)


    Rank Interval Ratio Number 1 Number 2 F1 (Hz) F2 (Hz) LCM (Hz)
    1 Octave 2/1 2 1 300 600 600
    2 Perfect Twelfth 3/1 3 1 300 900 900
    3 Perfect Fifth 3/2 3 2 300 450 900
    4 Perfect Fourth 4/3 4 3 300 400 1200
    5 Major Seventeenth 5/1 5 1 300 1500 1500
    6 Major Sixth 5/3 5 3 300 500 1500
    7 Major Third 5/4 5 4 300 375 1500
    8 Minor Third 6/5 6 5 300 360 1800
    9 Minor Sixth 8/5 8 5 300 480 2400
    10 Minor Seventh 9/5 9 5 300 540 2700
    11 Major Second 9/8 9 8 300 338 2700
    12 Major Seventh 15/8 15 8 300 563 4500
    13 Minor Second 16/15 16 15 300 320 4800
    14 Tritone 25/18 25 18 300 417 7500


    ------------------------------
    Roshan Kakiya
    ------------------------------

    Original Message