18 December 2012

I was emailing back and forth with a student research assistant about some algebraic properties of adding, subtracting, multiplying, and dividing intervals, and I thought that I might share the email (in slightly revised form) with the Net.  I would be very surprised if this hasn't been written up already by someone else, but the only things I could find quickly about interval multiplication was all in the 12-tone/serial usage, which is less interesting to me today.

An interval is just a ratio (fraction), so the octave is 2:1.  That ratio could apply to frequency (a note an octave up has a frequency twice that of the lower note), or the reciprocal gives the string length (so a string half the length of another is an octave higher than the longer string) or length from mouthpiece to first open tone hole, etc.

The octave is always 2:1.  The other interval ratios depend on what temperament we’re talking about.  In equal temperament, all other ascending intervals are integer exponents of L , where L is a the 12th root of 2 (I’m using L because it’s the 12th letter).  So a major third (4 semitones) is L^4 or the cube-root-of 2.  In other temperaments, the ratios are integer ratios, but they’re not standardized.  So a major third can be either 5:4 or 81:64 (I like the latter, also called a ditone, for reasons below).  You just have to memorize them all.  But once you have P5 as 3:2 and major second as 9:8, you can basically derive the rest.

When we add intervals, what we’re actually doing is multiplying ratios.  So P5 + P4 = 3/2 * 4/3 = 2/1 = P8. When we subtract intervals we’re dividing ratios, which is multiplying by the reciprocal, so P5 – P4 = 3/2 * 3/4 = 9/8 = M2.  And P8 – P5 = 2/1 * 2/3 = 4/3 = P4.  Since multiplication of ratios is commutative, then addition of intervals is also commutative.  The generic interval (the "5" in "P5") of the sum of ascending intervals is always the sum of the two generic intervals minus 1.  For descending intervals or a mix or if adding multiple intervals at once, take each generic interval and subtract 1 from the absolute value and then restore the sign.  In the end add one to the absolute value of the result and then restore the sign.  If it was 0 before, make it positive, since we only have P1 for a unison, never P-1.  (Though how we should actually designate the sign for d1 or dd1, is unclear, maybe it should be d-1? but then it's not analogous to dd2, such as between E# and Fb, where the generic interval is definitely ascending -- in programming this possibility is always one of those cases that bites you in the ass later later, where you have an ascending diatonic interval that is a descending chromatic interval, so if you mix ASCENDING tests for generic intervals with > 0 tests for chromatic intervals, you'll get inconsistent results depending on which form of the interval you're looking at.)

Getting back to the main subject.  So what does it mean to multiply an interval by an integer?  M2 * 2 = ?  Well, what’s M2 + M2 = 9/8 * 9/8 = 81/64 = M3.  So M2 * 3 = M2 + M2 + M2 = 9/8 * 9/8 * 9/8 = A4/Tritone (729/512), and so M2 * 3 = (9/8)^3, so when we multiply an interval by a number it’s like taking the ratio to a power.

Since exponentiation is not commutative, 2 * P5 would be different than P5 * 2;  2 * P5 = 2^(3/2) = 2 radical 2, while P5 * 2 = (3/2)^2 = 9/4 which is a M9 (P8 + M2 = 9/8 * 2/1).  However, we’re defining our own algebraic system, so we could define * as always placing the integer in the exponent and thus make this commutative.  HS and early college math doesn’t talk much about defining our own algebras, but we do it all the time.  (otherwise we couldn’t define 11:00 + 3hours = 2:00, etc.)

So, does it make any sense to multiply intervals?  What would M3 * P5 be?  Well, if we convert it to ratios, then it’d be (81/64)^(3/2), or (9/8)^3, or 729/512, which we defined as an Augmented 4th.  Most ratios * other ratios though will create irrational ratios, which we don’t like unless we’re in Equal Temperament (“irrational ratio” is an oxymoron if you think about it).  In equal temperament though we’d end up with irrational numbers raised to irrational exponents.  Your calculator will calculate these things, by substituting in the nearest rational number, and in fact to take a number to an irrational ratio, you need to find the limit of the ratio of the base to the closest smaller rational number exponent with the base of the closest larger rational number exponent.  (btw – did you ever notice that any negative number to an irrational power is undefined? because it depends on whether the irrational number can be expressed as a ratio with an even or odd denominator, and irrational numbers are not ratios.  Fortunately, we don’t need to deal with negative ratios in music).

A nice property of defining multiplication of intervals as a form of exponentiation is that descending intervals (whose ratios are positive but < 1) can also be used.  I like M3 * P-8, or major third times descending perfect octave; or (81/64)^(1/2) power, or 9/8, or M2.

Consider what multiplying by an interval by an interval might be used for.  M2 * P8 = (9/8)^(2/1) = 81/64 = M3.  So a M2 occupies the same proportion of the harmonic space of one octave as a M3 does for two octaves.  This process (multiplying an interval by P5) could be used to convert intervals in standard, octave repeating, space into Bohlen-Pierce space which is based on the P12.  Or it can translate the ratios produced by fingering patterns in the lower vs. upper register of the flute (based on P8) into the ratio you’d get on the clarinet (based on P12).

Also notice that multiplying any interval (ascending or descending) by a descending perfect infinity (P-∞) (or the limit as the number of descending octaves increases without bound) condenses the available interval space to nothing. So every interval becomes a unison.  E.g., P4 * P-∞ = (4/3)^(1/(2^∞)) = (4/3)^(1/∞) = (4/3)^(0) = 1 (since any non-zero number to the zeroth power = 1) and 1 = 1:1 = P1.

The question of what diatonic intervals result from any addition or multiplication isn’t something I’ve touched on here.  It’s easy to figure out what the generic interval under addition will be as I described above.  The specifier (major, minor, augmented, diminished, perfect, etc.) is harder to determine.  I’ll leave that as an exercise – it’s messy and I solved it a while back, but I can’t remember the exact solution right now.  Under multiplication of an interval with an integer, it’ll be easy to solve what the diatonic interval will be, without converting to ratios, once you’ve solved the previous problem.  But for multiplication of an interval by another interval the math becomes harder.  The first question to solve there is, is the answer dependent on the temperament system chosen, or can it be generalized for any temperament?

Btw, raising intervals to the power of other intervals is just silly.  So say I. :-)

Michael Scott Cuthbert said...

Edited to change 6:5 to 5:4 for a ratio of the major third. Thanks NE @ Harvard. Anonymous said...

Fourier Uncertainty Principle or time-frequency uncertainty is also non-commutative.

Equal-tempered tuning converts the non-commutative of the C - F 2/3 subharmonic of C to G 3/2 harmonic overtone into C to F 4/3 for Archytas' commutative equation creating equal-tempered tuning.

There is a great secret of sound lost by using commutative algebra as the foundation for logarithmic equations. Check out math professor Luigi Borzacchini's research on this -
Borzacchini, Luigi
Incommensurability, music and continuum: a cognitive approach. (English)
Arch. Hist. Exact Sci. 61, No. 3, 273-302 (2007)

Timothy Aveni said...

You've got a quote in there that doesn't sound quite right: "So a major third (4 semitones) is L^4/2 or the cube-root-of 2."

Wouldn't a major third be L^4 or 2^(4/12), but not L^4/2 (whether you're interpreting it as (L^4)/2 or L^(4/2))? That's what would make it a cube-root-of 2. Even if this was meant to read "L^(4/12)", that doesn't sound right: you've already taken the twelfth root in defining L.

The previous sentence also has a spare word "a" in it, but I'm not so concerned with English flaws ;)

Michael Scott Cuthbert said...

Aha! That should be L^4 or 2^(4/12) as you note! I'll change that. Thanks for noticing! Best, Myke

Timothy Aveni said...

No problem. I definitely enjoyed reading the post!