Circle of fifths and number theory

Why does music have a circle of fifths but no circle of thirds or circle of sixths?

If you start at on any note and go up by fifths, you’ll cycle through the entire chromatic scale. For example: C, G, D, A, E, B, F#, C#, G#, D#, A#, F, C. If you go up by fourths, you’ll get the same sequences of notes but in the reverse order. So there’s a cycle of fifths and a cycle of fourths, but there are no other ways to cycle through the chromatic scale other than the chromatic scale itself.

If you start at C and go up by minor thirds, for example, you’ll only hit four distinct notes before returning to where you started: C, D#, F#, A, C. You don’t cycle through all 12 notes, only four of them. Instead of filling out a chromatic scale, you fill out a diminished chord. You could fill out two other diminished chords by starting on C# or on D. Going up by major sixths produces the same sequence of notes as going down by minor thirds.

What’s special about fourths and fifths that their cycles cover the chromatic scale while cycles of other intervals partition the chromatic scale into smaller groups of notes? A fourth is 5 chromatic steps and a fifth is 7 chromatic steps. The numbers 5 and 7 are relatively prime to 12, that is, they share no factors with 12 (other than 1, which doesn’t count).

The numbers less than 12 and relatively prime to 12 are 1, 5, 7, and 11. These intervals correspond to the ascending chromatic scale, the circle of fourths, the circle of fifths, and the descending chromatic scale.

The numbers less than 12 and not relatively prime to 12 are 2, 3, 4, 6, 8, 9, and 10. Going up by 2 chromatic steps produces a whole-tone scale. Going up by 10 steps produces the same sequence of notes but in the opposite order. Going up by 3  or 9 steps produces a diminished chord. Going up by 4 or 8 steps produces an augmented chord. Going up by 6 steps produces a tritone pair. (I’m used to jazz terminology which uses the term “tritone.” Classical musicians would more likely say “augmented fourth” or “diminished fifth.”)

Now imagine a non-traditional scale that divided the octave into some number of parts other than 12. Suppose this new scale has n notes. Cycling in steps of size m will cover all n notes if and only if m and n are relatively prime. For example, if we divide the scale into 15 parts, we could cover all 15 pitches if we went up 4 steps at a time. We could play notes 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15.

If m and n are not relatively prime, let d be their greatest common divisor, the largest number that divides both m and n. Then going up d parts at a time will cycle through m/d notes and there will be d distinct cycles. For example, if there were 15 notes in our scale and we went up in intervals of 10 notes, we would cover 3 distinct notes, and we could make 5 different such three-note chords. For example, one such chord would be notes 1, 11, and 6, and another would be notes 2, 12, and 7.

If a scale had a prime number of notes, then every interval (other than an octave) would cycle through all notes.

Why is the 12-note scale so common? There have been other systems, but these are mostly subsets (at least approximately) of the 12-note scale. The answer seems to have something to do with the fact that intervals in the 12-tone scale have simple frequency ratios. For example, a fifth is a ratio 3:2 and a forth is a ratio 3:4. (More on that here.) These intervals are pleasant to our ears. There was a prehistoric flute in the news a few weeks ago and it appears to have been based on the same musical intervals common in modern music.

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11 thoughts on “Circle of fifths and number theory

  1. Very cool.

    I’m trying to visualize the cycles in terms of powers, like in modular arithmetic. For example, for the circle of 5ths, I compute 2^(7/12) mod 2 (loosely speaking) :

    1. 1 * 2^(7/12) = 2^(7/12)
    2. 2^(7/12) * 2^(7/12) = 2^(14/12)
    3. 2^(2/12) * 2^(7/12) = 2^(9/12)
    4. 2^(9/12) * 2^(7/12) = 2^(16/12)
    5. 2^(4/12) * 2^(7/12) = 2^(11/12)
    6. 2^(11/12) * 2^(7/12) = 2^(18/12)
    7. 2^(6/12) * 2^(7/12) = 2^(13/12)
    8. 2^(1/12) * 2^(7/12) = 2^(8/12)
    9. 2^(8/12) * 2^(7/12) = 2^(15/12)
    10. 2^(3/12) * 2^(7/12) = 2^(10/12)
    11. 2^(10/12) * 2^(7/12) = 2^(17/12)
    12. 2^(5/12) * 2^(7/12) = 2^(12/12)
    13. 1 * 2^(7/12) = 2^(7/12)

    Is that how you think of it?

  2. Yes, this is very much like arithmetic mod 12, just like adding times on a clock.

    You could think of an infinite range of notes of the form k rn where r is the 12th root of 2 and k is your reference pitch, say 440 for A 440. Then map each note k rn to the remainder when n is divided by 12.

    In group theory terminology, 7 is a generator of the group Z12 and so you can solve 7x = b (mod 12) for any b. The generators are 1, 5, 7, and 11 and so these correspond to intervals that will generate all the notes in the chromatic scale. There’s no circle of major thirds, for example, because 4 is not a generator of Z12.

  3. Great overview of the twelve-tone scale. I wanted to add that I think the scale is common not just because the fourth and fifth are pleasant to the ear, but also because the physics behind those intervals is so easy to stumble upon and re-discover: harmonics on a plucked string (simple guitar), blowing on a bottle (simple flute), buzzing one’s lips on a (literal animal) horn or seashell (simple bugle or trumpet with no valves), etc. No wonder so many cultures discovered the basics of (some subset of) twelve-tone harmony.

    Are you familiar with the composer Arnold Schoenberg? He decided to create scales (“tone rows”) by more irregular permutations of all twelve tones and compose music from these new musical beasts. The math behind it may be of interest:

    Last, I’ll leave everyone with a deceptively simple puzzle to test their knowledge of basic harmony and the circle of fifths. What are the proper note names (not just the enharmonics–too easy!) for the chord D# major and in what key will you find it? What is its diatonic functionality in this key?

    It turns out that learning how to answer that question properly (for me) resulted in a month of study where I learned basic music theory (basic harmony). Of course to test how much I really understood I taught my knowledge to a computer:

    This guitar/piano chord finder turns this simple list of 75 guitar chord forms:

    into 928 guitar chords. (It also has the answer to the puzzle if you poke around enough.)

  4. Jared, yes I’m familiar with Schoenberg. I have a funny story about him.

    I went to an organ concert while I was in college. I forget who was performing, but he was amazing. As part of the concert, he improvised fugues on themes handed to him on stage that had been prepared earlier. As I was leaving the concert, I told a friend who had not been there about the improvisations. I mentioned that one of the themes sounded like a Schoenberg tone row. (I didn’t know for certain that it was a tone row, only that it sounded weird.) I thought that was a dirty trick, but the organist took it in stride and improvised a fugue to the theme. A music professor overheard our conversation and approached me. He was impressed that I recognized the tone row but disappointed when I told him I was not a music major.

  5. @jared: D# F## G# (V in the key of G#). Useful form for blues with Cm7 as root I. Slightly impractical for most common notation. Ab is more practical.

    Very nice web chord app, BTW. I use GuitarePro which also has excellent facilities for chord exploration particularly when paired to scale alternatives. A pet wish of mine is something similar for the raga systems.

  6. I think you mean A#? You need a perfect fifth between the root and, well, fifth. It’s still the V in G# though.

  7. @Ien, @AntiSlice: That’s right. It’s D# F## A#, it’s the V chord in the key of g# minor (i.e. harmonic minor).

  8. Hi folks,

    The intervals of twelve-tone equal temperament (12-TET) do not have simple frequency ratios! The best you can say about them is that they “approximate” (try to get close to) simple frequency ratios.

    This whole idea of a stack of intervals being able to wrap around to the starting pitch class (pitch class meaning any and all octaves of the same pitch) is purely an artifact of enforcing an artificial equal temperament. It’s a fun trick, but don’t get all woo-woo over it. It’s not some kind of mystical musical truth. Any equal temperament that divides the octave by an even number of divisions (on a log scale of frequency) causes its intervals to wrap. 12-TET is a practical compromise for the benefit of instrument manufacture and compositional simplicity at the expense of everything being slightly out of tune and keeping our thinking trapped in comfy pajamas (to borrow from Zappa).

    Natural intervals, the ones that truly do have simple frequency ratios, which are the intervals our ears seems to gravitate toward and recognize most easily, never wrap back to the same pitch class when stacked. That bears repeating:

    Natural intervals do not wrap.

    That’s a big insight that most music-theory students will never have the opportunity to grasp because our schools have unquestioningly accepted and reiterated 12-TET thinking for the past few hundred years. Even most non-music students have probably heard about the “circle of fifths” somewhere, without being warned that it’s a contrivance not found in nature.

    Natural intervals continue to generate new pitch classes indefinitely when stacked. That goes for all natural intervals, not only the fifths. The reason for that is related to the fact that no non-zero power of any prime number equals any non-zero power of any other prime number, and natural intervals are based on products of powers of primes. In the simple case of fifths versus octaves, no power of 3 (which is what fifths are based on) equals any power of 2 (which is what octaves are based on).

    Such is the nature of pitch as naturally perceived by humans. It’s a lot more complex and beautiful than the over-simplified, artificial system of 12-TET that is purposely bent in on itself in such a way as to limit the pitch classes to only 12 and make intervals loop when they shouldn’t.

    Here’s a comparison of only the fifths; people can explore the other intervals themselves:

    The 12-TET fifth = 2^(7/12))/1, which is an irrational number approximately equal to 1.498307076876681498799280732029… (it goes on forever, since it can’t be notated in a rational numerical system, such as base-10 or “decimal” notation, being irrational). You stack that interval 12 times, and you get the same pitch class you started on. That fifth is not an easy interval to reproduce by ear. It’s likely that most people asked to reproduce it would produce a natural fifth instead. You really have to admire the skill of singers who have learned to sing in 12-TET by overcoming and overriding their natural tendency to sign in tune.

    A natural or “just” fifth = 3/2, which is a nice simple rational number equal to exactly 1.5 that the ear seems to like and which any amateur after brief instruction can easily tune by eliminating beats. Stack that interval as many times as you like, and you’ll never get back to the starting pitch class. That’s not a bad thing. It gives a real sense of distance to harmonic progressions that roam far from the tonic. Using natural fifths as a basis of harmony instead of the 12-TET fifths creates a unique identity for every possible chord and avoids the confounding of (loss of distinction between) harmonic senses that 12-TET causes. For example, the D# F## A# triad mentioned before is confounded in 12-TET with the Eb G Bb triad. Natural or just harmony retains their distinction, and gives each one a slightly different pull or sense of direction as to the path they would most naturally take to get back to a given distant tonic, such as C Major.

  9. Oliver Clissold

    Love the maths explanation, but I cannot think why the fifth sequence also increased the black note count consistently, something to do with there being 5 black keys per octave? Was this deliberate piano layout?

  10. @OliverClissold I found this thread by googling the exact question you asked on 27 April 2020. Would love to read if there will be a response!

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