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The Relationship Between the Lowest Common Multiple and Harmonics

  • 1.  The Relationship Between the Lowest Common Multiple and Harmonics

    Posted 04-30-2019 16:16
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    UPDATE


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



    I have been interested in ranking the intervals of just intonation in terms of harmony ever since I began experimenting with different tuning systems and producing temperaments.

    I have noticed a pattern that has finally enabled me to do this with conviction.


    The following three intervals are usually ranked in the following order from most harmonious to least harmonious:

    1. Octave (Ratio: 2/1).
    2. Perfect Fifth (Ratio: 3/2).
    3. Perfect Fourth (Ratio: 4/3).


    These intervals can be ranked in the same order in terms of the lowest common multiple (LCM) of the numbers that form their ratio from lowest to highest:

    1. Octave (LCM of 2 and 1 is 2).
    2. Perfect Fifth (LCM of 3 and 2 is 6).
    3. Perfect Fourth (LCM of 4 and 3 is 12).

    Therefore, it seems as if the lower the LCM, the more harmonious the interval.

    I suppose this is because the harmonics of the two frequencies that form an interval coincide more frequently as the LCM decreases. This relationship only applies to intervals within the Octave.


    For example, if we call 300 Hz the first frequency, the second frequency would be:

    • 600 Hz for the Octave.
    • 450 Hz for the Perfect Fifth.
    • 400 Hz for the Perfect Fourth.

    This means that the harmonics of the first frequency and the second frequency would coincide every:

    • 600 Hz for the Octave (LCM of 300 Hz and 600 Hz is 600 Hz).
    • 900 Hz for the Perfect Fifth (LCM of 300 Hz and 450 Hz is 900 Hz).
    • 1200 Hz for the Perfect Fourth (LCM of 300 Hz and 400 Hz is 1200 Hz).

    This information indicates that the Octave's harmonics coincide more frequently than those of the Perfect Fifth and the Perfect Fifth's harmonics coincide more frequently than those of the Perfect Fourth.

    I will use this pattern to rank all 12 intervals from most harmonious to least harmonious.

    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

    12 Intervals Ranked by Lowest Common Multiple from Most Harmonious to Least Harmonious

    Rank Interval Ratio Number 1 Number 2 LCM
    1 Octave 2/1 2 1 2
    2 Perfect Fifth 3/2 3 2 6
    3 Perfect Fourth 4/3 4 3 12
    4 Major Sixth 5/3 5 3 15
    5 Major Third 5/4 5 4 20
    6 Minor Third 6/5 6 5 30
    7 Minor Sixth 8/5 8 5 40
    8 Minor Seventh 9/5 9 5 45
    9 Major Second 9/8 9 8 72
    10 Major Seventh 15/8 15 8 120
    11 Minor Second 16/15 16 15 240
    12 Tritone 25/18 25 18 450


    ------------------------------
    Roshan Kakiya
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  • 2.  RE: The Relationship Between the Lowest Common Multiple and Harmonics

    Posted 04-30-2019 17:32
    Umm . . . I think you're confusing time domain and frequency domain in places 

    "This means that the harmonics of the first frequency and the second frequency would coincide every:
    • 600 Hz for the Octave (LCM of 300 Hz and 600 Hz is 600 Hz).
    • 900 Hz for the Perfect Fifth (LCM of 300 Hz and 450 Hz is 900 Hz).
    • 1200 Hz for the Perfect Fourth (LCM of 300 Hz and 400 Hz is 1200 Hz)."
    No. The octave will coincide every 300Hz. The Fourth 200Hz and the Fifth 150Hz

    When mixing two frequencies it's the difference between them which is the frequency at which the vibrations coincide. Effectively the two frequencies become harmonics of the difference frequency.

    Best wishes

    David P

     

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    David Pinnegar, B.Sc., A.R.C.S.
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    +44 1342 850594





  • 3.  RE: The Relationship Between the Lowest Common Multiple and Harmonics

    Posted 04-30-2019 19:29
    David,

    300 Hz is the frequency of the Unison. Therefore, 300 Hz is the frequency of the 1st harmonic.


    I have multiplied 300 Hz to calculate the frequency of the Octave, Perfect Fifth and Perfect Fourth:

    • Octave = 300 Hz × 2/1 = 600 Hz.
    • Perfect Fifth = 300 Hz × 3/2 = 450 Hz.
    • Perfect Fourth = 300 Hz × 4/3 = 400 Hz.


    Therefore, the following harmonics of the Unison coincide with the following harmonics of these intervals:

    • 2nd harmonic of Unison (600 Hz) coincides with 1st harmonic of Octave (600 Hz).
    • 3rd harmonic of Unison (900 Hz) coincides with 2nd harmonic of Perfect Fifth (900 Hz).
    • 4th harmonic of Unison (1200 Hz) coincides with 3rd harmonic of Perfect Fourth (1200 Hz).

    These relationships will repeat every 600 Hz for Octaves, 900 Hz for Perfect Fifths and 1200 Hz for Perfect Fourths at higher harmonics.


    Examples:

    • 4th harmonic of Unison (1200 Hz) coincides with 2nd harmonic of Octave (1200 Hz).
    • 6th harmonic of Unison (1800 Hz) coincides with 4th harmonic of Perfect Fifth (1800 Hz).
    • 8th harmonic of Unison (2400 Hz) coincides with 6th harmonic of Perfect Fourth (2400 Hz).


    ------------------------------
    Roshan Kakiya
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  • 4.  RE: The Relationship Between the Lowest Common Multiple and Harmonics

    Posted 04-30-2019 19:24
    And what of the perfect twelfth, with its “LCM” of, uh, 3?






  • 5.  RE: The Relationship Between the Lowest Common Multiple and Harmonics

    Posted 04-30-2019 19:44
    Kent,

    The ratio of the Perfect Twelfth is 3/1 and the LCM of 3 and 1 is 3.

    Based on the ranking system that I am using (lowest common multiple), the Perfect Twelfth is less harmonious than the Octave and more harmonious than the Perfect Fifth.

    Unison = 300 Hz.
    Perfect Twelfth = 300 Hz × 3/1 = 900 Hz.

    3rd harmonic of Unison (900 Hz) coincides with 1st harmonic of Perfect Twelfth (900 Hz).

    This relationship will repeat every 900 Hz for Perfect Twelfths at higher harmonics.

    For example, 6th harmonic of Unison (1800 Hz) coincides with 2nd harmonic of Perfect Twelfth (1800 Hz).

    The Perfect Twelfth seems to be more harmonious than the Perfect Fifth because the Perfect Twelfth directly coincides with the 3rd harmonic of the Unison, whereas the 2nd harmonic of the Perfect Fifth coincides with the 3rd harmonic of the Unison.


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



  • 6.  RE: The Relationship Between the Lowest Common Multiple and Harmonics

    Posted 04-30-2019 21:11
    The ranking of the intervals can be modified as follows:

    1. Octave (1200 cents).
    2. Perfect Twelfth (1901.96 cents).
    3. Perfect Fifth (701.96 cents).
    4. Perfect Fourth (498.04 cents).


    I think it would be interesting to compare Pure Octave ET, Pure 12th ET and Pure 5th ET.


    Pure Octave Equal Temperament:

    • Pure Octaves (1200 cents).
    • Tempered 12ths (1900 cents).
    • Tempered 5ths (700 cents).
    • Tempered 4ths (500 cents).

    Overall deviation from Just Intonation (absolute value) = 0.00 cents + 1.96 cents + 1.96 cents + 1.96 cents = 5.88 cents.



    Pure 12th Equal Temperament:

    • Tempered Octaves (1201.23 cents).
    • Pure 12ths (1901.96 cents).
    • Tempered 5ths (700.72 cents).
    • Tempered 4ths (500.51 cents).

    Overall deviation from Just Intonation (absolute value) = 1.23 cents + 0.00 cents + 1.24 cents + 2.47 cents = 4.94 cents.



    Pure 5th Equal Temperament:

    • Tempered Octaves (1203.35 cents).
    • Tempered 12ths (1905.31 cents).
    • Pure 5ths (701.96 cents).
    • Tempered 4ths (501.40 cents).

    Overall deviation from Just Intonation (absolute value) = 3.35 cents + 3.35 cents + 0.00 cents + 3.36 cents = 10.06 cents.



    Therefore, based on the information above, Pure Octave ET, Pure 12th ET and Pure 5th ET can be ranked as follows from most harmonious to least harmonious:

    1. Pure 12th ET (Overall deviation from JI = 4.94 cents).
    2. Pure Octave ET (Overall deviation from JI = 5.88 cents).
    3. Pure 5th ET (Overall deviation from JI = 10.06 cents).


    ------------------------------
    Roshan Kakiya
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  • 7.  RE: The Relationship Between the Lowest Common Multiple and Harmonics

    Posted 05-01-2019 10:46
    What about the M17 at 5:1?

    ------------------------------
    Jason Leininger
    Pittsburgh PA
    412-874-6992
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  • 8.  RE: The Relationship Between the Lowest Common Multiple and Harmonics

    Posted 05-01-2019 12:45
    Jason,

    The harmonics of the two frequencies that form an interval coincide more frequently as the LCM decreases. This pattern/relationship applies to intervals within the Octave.

    However, this pattern/relationship seems to break down for intervals beyond the Octave such as the Major Seventeenth.


    The ratio of the Major Seventeenth is 5/1 and the LCM of 5 and 1 is 5.

    Based on the ranking system that I am using (lowest common multiple), the Major Seventeenth (LCM = 5) is less harmonious than the Perfect Twelfth (LCM = 3) and more harmonious than the Perfect Fifth (LCM = 6).

    Unison = 300 Hz.
    Major Seventeenth = 300 Hz × 5/1 = 1500 Hz.

    5th harmonic of Unison (1500 Hz) coincides with 1st harmonic of Major Seventeenth (1500 Hz).

    This relationship will repeat every 1500 Hz for Major Seventeenths at higher harmonics.

    For example, 10th harmonic of Unison (3000 Hz) coincides with 2nd harmonic of Major Seventeenth (3000 Hz).


    The pattern/relationship mentioned above has broken down because the Major Seventeenth's LCM (5) is lower than the Perfect Fifth's LCM (6) but the Major Seventeenth's harmonics coincide less frequently (every 1500 Hz) with those of the Unison in comparison with the Perfect Fifth (every 900 Hz).

    Therefore, in the case of the Major Seventeenth, the LCM has decreased but the harmonics of its two frequencies (300 Hz and 1500 Hz) coincide less frequently rather than more frequently.


    ------------------------------
    Roshan Kakiya
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  • 9.  RE: The Relationship Between the Lowest Common Multiple and Harmonics

    Posted 05-01-2019 11:16
    I have compared the overall deviation of Pure Octave ET, Pure 12th ET and Pure 5th ET from Just Intonation.

    Summary of the results:

    3 Equal Temperaments Ranked by Overall Deviation from Lowest (Most Harmonious) to Highest (Least Harmonious)

    Rank

    Equal Temperament

    Overall Deviation

    1

    Pure 12th

    140.46 Cents

    2

    Pure Octave

    140.77 Cents

    3

    Pure 5th

    146.64 Cents



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