To give a microtonal answer again - it is based on a very simple idea.
Take any musical interval - doesn't matter if it is a fifth, or a third, or a fourth or whatever - and now repeat the same interval over and over.
So if you do that with major thirds you get
C E Ab C'
which is a kind of "circle" of major thirds.
Now that's with equal temperament. But if you do the same with pure harmonic series major thirds - the most harmonious ones you can get - the interval you will naturally sing if you sing over a long drone for instance - then each of the major thirds is a smidgen flat compared with twelve equal.
So, then, E is a bit flat, Ab is a bit flatter, and when you get to the final C', you end up with a note that is a long way flat of the octave above your original C.
Mathematically that's because the pure major third is the ratio of the fifth to the fourth harmonic (is the interval between them) so a frequency ratio of 5/4 - and our ears hear identical frequency ratios, rather than identical frequency differences, as the "same interval".
So when you add musical intervals, you multiply ratios.
So if you try your circle of major thirds it's (5/4) * (5/4) * (5/4) or 125/64.
That's way flat of the octave which would be 128/64. To get the major thirds of twelve equal, you need to tune all your thirds quite a bit sharp and then they can hit the octave.
The same thing also happens with fifths 3/2, but - after going up two fifths you are already well over the octave so you aren't going to hit the octave, but instead some multiple of an octave. As it turns out, you hit the nineteenth octave, after stacking twelve fifths.
And, just as we found before, for the 5/4s, you can't possibly hit the octave exactly. No matter how many times you multiply 3/2s together, the result will never be a power of 2 (when you go up an octave you double the frequency, so to hit the octave you have to hit a power of 2).
Turns out it ends up a bit sharp.
So - to get it to work, you have to make all the fifths of twelve equal a smidgen flat.
That's the circle of fifths as usually understood.
But you can keep going, because it didn't hit the octave exactly after all. Keep going and you find the next close miss is after 17 notes - but that's only a little bit closer.
The next really close near miss is at 53 notes, which gives you the cycle of 53 fifths used in a lot of Turkish and Arabic music theory.
And you could go even further.
Now there are intermediate moments as well when you get interesting tunings. These are known as "moments of symmetry".
So if we look at our circle of tempered twelve equal fifths - and start at C and keep stacking fifths, you get
C G
C G D
C G D A
C G D A E
Now some of these are of special interest. Look at the intervals when arranged in ascending order
C G C' (fifth, fourth)
C D G C' (whole tone, fourth, fourth)
C D G A C' (whole tone, fourth, whole tone, minor third)
C D E G A C' (whole tone, whole tone, minor third, whole tone, minor third)
C D E G A B C' (whole tone, whole tone, minor third, whole tone, whole tone, whole tone, semitone)
Then - most so far have only two interval sized, but
C D G A C' has three interval sizes for the successive steps, as does C G D E G A B C'.
Turns out - this gets harder to achieve from now on in. Most of these fragments have three interval sizes.
The next one to have only two interval sizes is
C D E F G A B C
our diatonic system
Then finally you get to twelve equal which has only one interval size.
But if we used pure fifths - then when you get to twelve equal it has two interval sizes, the chromatic and the diatonic semitone (those are the two sizes of semitone in Pythagorean twelve tone).
The thing to notice here is - that the systems that humans like tend to be ones with only two interval sizes.
Erv Wilson calls these "moments of symmetry".
So - then 17 is another moment of symmetry for pure fifths, as is 53, and you keep getting more moments of symmetry as you keep stacking more and more pure fifths on top of each other.
This is perfectly general - you get them not only with the conventional pure fifths, or tempered fifths, in various temperaments, but with any musical interval if you choose to stack it over and over.
The result is then called a "linear temperament" and it will normally have three interval sizes, but when you hit a "moment of symmetry" it has just two interval sizes.
If it eventually hits the octave (some multiple of) then it's an equal temperament and has only the one interval size, of course, at that point.
For an example of linear temperaments that are novel to the ear and not at all like twelve tone systems, but pleasing and great for making interesting music, have a look at the "Golden horograms"
This uses a diagram a bit lke the rings of a tree to show successive repeats of the interval you chose to repeat, whatever it is.
Golden Horagrams Introduction and Explanation
"CIRCLES" OF INTERVALS
Take any musical interval - doesn't matter if it is a fifth, or a third, or a fourth or whatever - and now repeat the same interval over and over.
So if you do that with major thirds you get
C E Ab C'
which is a kind of "circle" of major thirds.
Now that's with equal temperament. But if you do the same with pure harmonic series major thirds - the most harmonious ones you can get - the interval you will naturally sing if you sing over a long drone for instance - then each of the major thirds is a smidgen flat compared with twelve equal.
So, then, E is a bit flat, Ab is a bit flatter, and when you get to the final C', you end up with a note that is a long way flat of the octave above your original C.
Mathematically that's because the pure major third is the ratio of the fifth to the fourth harmonic (is the interval between them) so a frequency ratio of 5/4 - and our ears hear identical frequency ratios, rather than identical frequency differences, as the "same interval".
So when you add musical intervals, you multiply ratios.
So if you try your circle of major thirds it's (5/4) * (5/4) * (5/4) or 125/64.
That's way flat of the octave which would be 128/64. To get the major thirds of twelve equal, you need to tune all your thirds quite a bit sharp and then they can hit the octave.
The same thing also happens with fifths 3/2, but - after going up two fifths you are already well over the octave so you aren't going to hit the octave, but instead some multiple of an octave. As it turns out, you hit the nineteenth octave, after stacking twelve fifths.
And, just as we found before, for the 5/4s, you can't possibly hit the octave exactly. No matter how many times you multiply 3/2s together, the result will never be a power of 2 (when you go up an octave you double the frequency, so to hit the octave you have to hit a power of 2).
Turns out it ends up a bit sharp.
So - to get it to work, you have to make all the fifths of twelve equal a smidgen flat.
That's the circle of fifths as usually understood.
CIRCLES OF 53 OR MORE FIFTHS
But you can keep going, because it didn't hit the octave exactly after all. Keep going and you find the next close miss is after 17 notes - but that's only a little bit closer.
The next really close near miss is at 53 notes, which gives you the cycle of 53 fifths used in a lot of Turkish and Arabic music theory.
And you could go even further.
MOMENTS OF SYMMETRY
Now there are intermediate moments as well when you get interesting tunings. These are known as "moments of symmetry".
So if we look at our circle of tempered twelve equal fifths - and start at C and keep stacking fifths, you get
C G
C G D
C G D A
C G D A E
Now some of these are of special interest. Look at the intervals when arranged in ascending order
C G C' (fifth, fourth)
C D G C' (whole tone, fourth, fourth)
C D G A C' (whole tone, fourth, whole tone, minor third)
C D E G A C' (whole tone, whole tone, minor third, whole tone, minor third)
C D E G A B C' (whole tone, whole tone, minor third, whole tone, whole tone, whole tone, semitone)
Then - most so far have only two interval sized, but
C D G A C' has three interval sizes for the successive steps, as does C G D E G A B C'.
Turns out - this gets harder to achieve from now on in. Most of these fragments have three interval sizes.
The next one to have only two interval sizes is
C D E F G A B C
our diatonic system
Then finally you get to twelve equal which has only one interval size.
But if we used pure fifths - then when you get to twelve equal it has two interval sizes, the chromatic and the diatonic semitone (those are the two sizes of semitone in Pythagorean twelve tone).
The thing to notice here is - that the systems that humans like tend to be ones with only two interval sizes.
Erv Wilson calls these "moments of symmetry".
So - then 17 is another moment of symmetry for pure fifths, as is 53, and you keep getting more moments of symmetry as you keep stacking more and more pure fifths on top of each other.
LINEAR TEMPERAMENTS
This is perfectly general - you get them not only with the conventional pure fifths, or tempered fifths, in various temperaments, but with any musical interval if you choose to stack it over and over.
The result is then called a "linear temperament" and it will normally have three interval sizes, but when you hit a "moment of symmetry" it has just two interval sizes.
If it eventually hits the octave (some multiple of) then it's an equal temperament and has only the one interval size, of course, at that point.
For an example of linear temperaments that are novel to the ear and not at all like twelve tone systems, but pleasing and great for making interesting music, have a look at the "Golden horograms"
This uses a diagram a bit lke the rings of a tree to show successive repeats of the interval you chose to repeat, whatever it is.
Golden Horagrams Introduction and Explanation
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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