When we left off of stacking threes, we’d arrived at a power of three (531441) that was close enough to some power of 2 (524288) such that dividing them gave us a value that was approximately equal to 1.00 (531441/524288 =~ 1.0136). We then took that approximation and treated it as if it actually was 1.00 (or treated 531441 as if it was 524288) which meant that our stacking had effectively circled around and returned to our original pitch class, which was our goal.
Unfortunately, it’s not that easy.
In reality, we haven’t really escaped the fact that a power of three can never really be equal to a power of two. 531441 may be close to 524288, but we know that they’re not equal, and so the circle that we’ve created by stacking threes has an error. It’s supposed to be a circle where each successive note is reached by multiplying the current note’s frequency by 3, but right at the end we have an inconsistency. If we multiply 177147 by 3, we get to 531441, but if we do that, we end up with a spiral instead of a circle. In order to flatten the circle, we need to get from 177147 to 524288. But then, that’s not multiplying by 3, that’s multiplying by 2.9596. This is an example of what’s known as a ‘wolf’ interval. We have an almost uniform sequence of our desired interval building up our pitch palette, but then we have this unintended interval in there as a result of some limitation in our constraints.
How do we solve this? There are several approaches. One is to leave it as is and work around it. Historically, another approach was to “temper” the tuning, i.e., distributing the error in some way throughout the circle. For a time, one popular method was to distribute the error in a non-uniform way, producing purer intervals between some notes, and intentionally less pure intervals between others. In that way, different keys would have different “color” (in terms of feeling). Ultimately, the solution that would go on to become the standard that we still use today, was equal temperament. This meant taking the error and distributing it throughout the circle in such a way that every successive step would have the same amount of error.
In our case that would mean that we’re no longer stacking 3s on the harmonic series to create our circle, but instead stacking 2.9966. Why 2.9966? It has to do with the fact that multiplying by 2, or some power of 2, gives us the same pitch class (440 = A, 440*2 = 880 = A), and that we want our circle (or chromatic pitch palette) to be made up of 12 unique pitch classes. In other words, we want to start at 1.00, jump 12 times, each time multiplying by the same constant, and then arrive at 2.00 at the end. What decimal constant can we multiply 12 times, starting from 1.00, and get to 2.00? Let’s do some math:
1.00 * x^12 = 2.00
x^12 = 2.00 / 1.00 = 2.00
x = 2.00^(1/12)
x =~ 1.059463
This gives us the interval that we can use to build up our chromatic scale in the order of successive pitches.
2^(0/12): 1.000000
2^(1/12): 1.059463
2^(2/12): 1.122462
2^(3/12): 1.189207
2^(4/12): 1.259921
2^(5/12): 1.334840
2^(6/12): 1.414214
2^(7/12): 1.498307
2^(8/12): 1.587401
2^(9/12): 1.681793
2^(10/12): 1.781797
2^(11/12): 1.887749
2^(12/12): 2.000000
From this group, the interval that’s analogous to our harmonic 3s is the 1.498307 (2^(7/12)) interval. In this state, it’s reduced so that it sits between 1.00 and 2.00, but multiplying it by 2 (which would produce the same pitch class as the 1.498307 interval does) gives us our 3 (or third harmonic) approximation, 2.996614 (2^(19/12)). We can now use this interval (or its reduced form), in the same way that we used our 3s, stacking until we return to the original pitch class, in a closed circle. And this time, we accomplish that exactly.
As you can see, under the system of 12-TET, none of the intervals, other than the octaves, are “perfect” by the definition that we’re accustomed to, i.e., small integer fractions. But, to me, 12-TET raises some new questions. Do we actually want pure intervals? Th slight impurities of 12-TET bring with them a little more movement and sizzle than the pure intervals do without actually sacrificing our ability to understand the musical intention. Is 12-TET an approximation, or is 12-TET itself the true form of the chromatic scale? This second question stems from the natural fact of pitch-class equivalence under multiplication by 2, the presence of sqrt(2) as the tritone, and the property of 12-TET where it doesn’t seem to be just an approximation of 3-limit Pythagorean tuning, but also seems to be able to approximate several tunings simultaneously.
And yes, there are other TETs, but many are odd, so certain interval combinations don’t add the way you’d want them to, and they grow in the number of pitch classes very quickly, such that you have multiple similarly sounding intervals, and have to employ an extra step of filtering to select a chromatic scale from within the larger set.
It’s almost strange how clean a solution 12-TET is, and I think it’s worthy of inspection as another path toward understanding the fundamental properties of musical pitch. Maybe it leads nowhere, or maybe it leads to something profound.