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Thread: An application of abstract algebra to music theory

  1. #1 An application of abstract algebra to music theory 
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    In music theory, if you start with any note on an equal-tempered scale and go upwards by intervals of a perfect fifth, you will eventually pass through every other note in the chromatic scale before ending back on the same note. For example, if you start with the note C and move upwards in perfect fifths, you will generate the following sequence of notes:

    C G D A E B F♯/G♭ C♯/D♭ G♯/A♭ D♯/E♭ A♯/B♭ F C

    This is called the circle of fifths.

    There are three other ways achieve a similar full circle of notes: by moving upwards in perfect-fourth, major-seventh, and minor-second intervals. No other intervals will achieve a full circle of notes. For example, moving upwards by intervals of a minor third will only produce the following notes starting with C:

    C E♭ G♭ A C

    The reason has to do with the group-theoretic structure of set of all chromatic intervals. Each interval can be defined by the number of semitone intervals it contains: we denote by x the interval that contains precisely x semitones. For example, the semitone (minor-second) interval is 1, the whole-tone (major-second) interval is 2, the minor-third interval is 3, the major-third interval is 4, and so on, up to the major-seventh interval 11. The unison interval 0 is also included. Intervals can be added by simply adding the individual semitones modulo 12 – e.g. 3 + 4 = 7, 8 + 6 = 2, etc. Then:

    The set Χ = {0, 1, …, 11} is a cyclic group of order 12 with respect to addition of intervals.

    (I use the symbol Χ (chi) for “chromatic”. ) Now we can see why the circles of fifths, fourths, major sevenths, and minor seconds work. These are the intervals 7, 5, 11, and 1. And the numbers 1, 5, 7, 11 are precisely the positive integers that are coprime with 12 (i.e. {n∊ℤ<sup>+</sup>: gcd(n,12) = 1} = {1, 5, 7, 11}) – hence these are precisely the generators of the order-12 cyclic group ℤ/12ℤ (to which Χ is isomorphic).

    Finally, note that the incomplete circle of major seconds corresponds to the whole-tone scale much favoured by the composer Claude Debussy. There are just two cycles of whole-tone notes in the scale:

    M2<sub>1</sub> = {C, D, E, F♯, G♯, A♯}, M2<sub>2</sub> = {C♯, D♯, F, G, A, B}

    If you start with any note in either of these cycles and move upwards by whole-tone intervals, you will generate all the notes in the same cycle but none of the notes in the other cycle. Reason: The subset of Χ generated by 2 is a subgroup of index 2 in Χ and M2<sub>1</sub> and M2<sub>2</sub> correspond to the two cosets of the subgroup <2> in Χ.

    In the same way, since <3> is a subgroup of index 3 in Χ, there are three mutually exclusive cycles of minor-third notes:

    m3<sub>1</sub> = {C, E♭, G♭, A}, m3<sub>2</sub> = {D♭, E, G, B♭}, m3<sub>3</sub> = {D, F, A♭, B}


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    Forum Professor sunshinewarrior's Avatar
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    Thanks for that, but I've never quite understood, in music, what a 'fifth' or a 'seventh' or a 'third' might mean. I'm still of the suspicion that it might be jargon and could be explained/labelled more explicitly. Could you explain? Please?


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    Quote Originally Posted by sunshinewarrior
    Thanks for that, but I've never quite understood, in music, what a 'fifth' or a 'seventh' or a 'third' might mean. I'm still of the suspicion that it might be jargon and could be explained/labelled more explicitly. Could you explain? Please?
    The notes within a scale. C D E F G A H C. By jumping 5 notes from C, then 5 notes from G, 5 notes from D etc, you get the circle of fifths. In norway (I don't now if it's the same elsewhere) we count intervals (distance from one note to another) like this:

    1 Prim: C C
    2 Sekund: C D
    3 Ters: C E
    4 Kvart: C F
    5 Kvint: C G
    6 Sekst: C A
    7 Septim: C H
    8 Oktav: C C2

    The "kvint" is the same as fifth. This is kinda simplified version, but I hope it helps.

    And also thanks to JaneBennet for putting this up. I wanted to respond earlier, but I'm still mildly confused by what you've done :? ^^
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    An interval is a “gap” between two notes. For example, if you play the note middle C on a piano and then the note G to the right of it, you produce an interval. There are seven semitones in this interval: C – C sharp or D flat – D – D sharp or E flat – E – F – F sharp or G flat – G. (The dashes represent the semitones.)

    The interval is called a (perfect) fifth because the note G is the fifth note in the scale of C major (C–D–E–F–G–A–B–C).

    Similarly, from C to E flat is another interval, one made up of three semitones. This is called the interval of a minor third (because E flat is the third note in any scale of C minor – e.g. C minor harmonic: C–D–E♭–F–G–A♭–B♮–C).

    Note that I define an interval as always starting from the lower-pitched note and counting up – thus, if you play the note middle C on a piano and then the note G to the left of it, this counts as an interval from G to C. It’s a five-semitone interval – what’s called the interval of a (perfect) fourth.

    By the way, a semitone is the interval between two adjacent keys on a keyboard – i.e. a white key and a black key next to it, or two white keys in the case of E–F and B–C. The semitone is the smallest interval in traditional Western music and in the chromatic music of composers like Arnold Schoenberg.
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    Quote Originally Posted by JaneBennet
    An interval is a “gap” between two notes. For example, if you play the note middle C on a piano and then the note G to the right of it, you produce an interval. There are seven semitones in this interval: C – C sharp or D flat – D – D sharp or E flat – E – F – F sharp or G flat – G. (The dashes represent the semitones.)

    The interval is called a (perfect) fifth because the note G is the fifth note in the scale of C major (C–D–E–F–G–A–B–C).

    Similarly, from C to E flat is another interval, one made up of three semitones. This is called the interval of a minor third (because E flat is the third note in any scale of C minor – e.g. C minor harmonic: C–D–E♭–F–G–A♭–B♮–C).

    Note that I define an interval as always starting from the lower-pitched note and counting up – thus, if you play the note middle C on a piano and then the note G to the left of it, this counts as an interval from G to C. It’s a five-semitone interval – what’s called the interval of a (perfect) fourth.

    By the way, a semitone is the interval between two adjacent keys on a keyboard – i.e. a white key and a black key next to it, or two white keys in the case of E–F and B–C. The semitone is the smallest interval in traditional Western music and in the chromatic music of composers like Arnold Schoenberg.
    See! That's why I call it jargon.

    A fourth has five semitones, a fifth seven. Why can't they call it by the actual number of semitones? Well... because the scales were devised without reference to them or Bach's well-tempered clavier, or whatever.

    Now, if we take the notion of the well-tempered scale (12th root of 2 ad all that), then of course your premise is sound - there are twelve intervals, so numbers that are mutually prime with 12 (including 1) will do the round of all 12 notes in time. Others, with shared HCDs, will not.

    But each time you speak of these you have to convert, don't you, from the mathematical interval you speak of to the conventional name for it in music? Or have I misinterpreted what you said?
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    Quote Originally Posted by sunshinewarrior
    But each time you speak of these you have to convert, don't you, from the mathematical interval you speak of to the conventional name for it in music? Or have I misinterpreted what you said?
    The conventional names are always used when talking about intervals. The mathematical notation is just my own invention.
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    I feel pretty useless here

    But that might have something to do with me not understanding much :P
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    Maths in music ruins it. Ask divide by zero that.
    "If you wish to make an apple pie from scratch, you must first invent the universe". - Carl Sagan
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  10. #9  
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    Quote Originally Posted by 425 Chaotic Requisition
    Maths in music ruins it. Ask divide by zero that.
    Ruins what?
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  11. #10  
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    i'm with Shanks on this. Could you explain it in terms of physics? What is an interval, a difference in frequency?
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  12. #11  
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    I’ve already explained interval above. If you’re still in the dark, this article contains much more info on the subject: http://en.wikipedia.org/wiki/Interval_(music).

    Note that my theory is based on an equal-tempered scale of notes – so some of what the article says may have to be interpreted according to this. For example, one part of the article says, “The name of an interval cannot be determined by counting semitones alone.” This would be true if you’re using a non-equal temperament of notes – but I’ve stated that I’m am only considering equal termperament, in which case intervals can be classified by the number of semitones they contain.
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    I'm also interested but confused. Regarding the physics I found this:

    For the equal temperament scale, the frequency of each note in the chromatic scale is related to the frequency of the notes next to it by a factor of the twelfth root of 2 (1.0594630944....). For the Just scale, the notes are related to the fundamental by rational numbers and the semitones are not equally spaced.
    Why the 12th. root of 2? This is quite strange.

    http://www.phy.mtu.edu/~suits/scales.html
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  14. #13  
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    The 12th root of 2 is the ratio of the absolute pitch (frequency) of one note to that of the note one semitone below in an equal-tempered scale, where each note is taken to have a frequency twice that of the note an octave below.

    Proof: Let r be the ratio of the frequency of any note to that of the note one semitone below. Then, given a note with frequency f, the frequencies of that note and the twelve notes above it (in semitone increments) are

    f, fr, fr<sup>2</sup>, … fr<sup>11</sup>, fr<sup>12</sup>

    fr<sup>12</sup> is the frequency of the note an octave above the starting note. Setting fr<sup>12</sup> = 2f gives r = <sup>12</sup>√2.

    This irrational number only crops up when you’re dealing with absolute pitches (e.g. when tuning an instrument) in an equal-tempered scale. It does not feature in the mathematical theory of the circle of fifths because my theory only deals with relative pitches, not absolute pitches. 8)
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    I'm glad to see I'm not the only one here in musical kindergarten, Misery loves company. :-D
    Would it be fair to say that in the C major scale whre there are 7 semitones in an interval that each note is separated by a factor of the seventh root of 2?
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    The C major scale does not increase by one semitone at each step. In fact, it goes up two semitones between pair of notes each except between E and F and between B and C.
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  17. #16  
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    Quote Originally Posted by Harold14370
    I'm glad to see I'm not the only one here in musical kindergarten, Misery loves company. :-D
    Would it be fair to say that in the C major scale whre there are 7 semitones in an interval that each note is separated by a factor of the seventh root of 2?
    The major scale I’m referring to is what’s more technically called the diatonic major scale. (Sorry, I didn’t want to introduce more technical terms than I had to. :P) The intervals in a diatonic major scale are:

    whole tone – whole tone – semitone – whole tone – whole tone – semitone

    Thus, starting with the note D, you get this sequence of notes:

    D – E – F sharp – G – A – B – C sharp – D



    This is the diatonic major scale of D – usually just called D major for short.
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