Blumeyer Comma Pump
11.09.2015 § 2 Comments
Four years ago, I noticed that 11 * 13 * 17 was almost equal to 19 * 27. With 2432 on the left and 2431 on the right, the numbers are off by just one, and at that scale we are talking less than five hundredths of a percent difference.
Who cares? Well, in music, then, a movement by an 11th harmonic followed by a 13th harmonic followed by a 17th harmonic is nearly exactly canceled out by a movement by a 19th harmonic in the opposite direction (those seven 2’s are taken care of by assuming an octave reduced environment, that is, a musical system where notes separated by any factor of 2 are considered to be of the same pitch class and thus get named with the same letter).
At that time I developed a just intonation tuning based around these four harmonics and made a blog post about it. In this post I also claimed the 2432:2431 interval as the “Blumeyer comma”. If I had fully understood what a comma was back then, though, I wouldn’t have stopped there, leaving just these two things in the same place together. And now that I do better understand what a comma is, I can see how silly I was to do so!
I was at least correct about a comma being a ratio that is very close to 1:1. But the purpose of a comma is to be “tempered out”, erased, nudged into becoming exactly 1:1. Calling my ratio a comma implied that my intention was to come up with a tuning system in which moving by an 11th, a 13th, a 17th, and then down a 19th would not land you just really really close to where you started, but in fact exactly where you started, and that it would do so by distributing that comma in some way among those four intervals, making each of them ever so slightly inaccurate until the comma vanished. Most musicians approach commas as problems to be solved, tiny little undesired disagreements in pitch that arise when working with pure ratios that can never be exactly in terms of each other. The fact that in the very center of my JI lattice I left a two pitches that were 2432:2431 apart together flies in the face of the concept of a comma. I went in the exact opposite direction, going so far as to embrace that tiny difference between 1:1 and 2432:2431 as part of my tuning system, rather than figuring out a way to make it disappear. Ah, the ignorance of youth! 🙂
Anyway, since I recently had this “aha” moment about commas, I thought I’d go back and finish the job: come up with a tuning system designed to temper out the Blumeyer comma. Not only did I do that, I composed this brief piece to showcase it.
And I may look back on this one day and scoff, too, but as far as I understand things today, I believe what I should call this particular type of piece I’ve composed is a “pump”. This is because it prominently figures a repeating progression of intervals which in its just form would result in drifting in pitch by exactly one comma each iteration, but though the magic of my temperament manages to stay in the same place. In my case this is a move downwards by a tempered 17th, upward by a tempered 19th, downward by a tempered 13th but simultaneously upward by an octave, then finally downward by a tempered 11th. These are the four opening notes of the piece, and while complexity gradually layers on, they remain the constant bass line to the end.
To be clear, “vanishing” a comma does not remove its wonder from your music, it transforms it. In my JI tuning, the wonder was expressed like this: move in dramatically different harmonic ways, but magically get back to nearly the same place, then feel that nearness as an energy in itself. In my tempered tuning, the wonder is expressed like this: move in dramatically different harmonic ways, but magically get back to exactly the same place, then retroactively feel the seeming impossibility of it.
Monzo for the Blumeyer comma, for those interested: |7 0 0 0 -1 -1 -1 1>
The fundamental thing one does when creating a tempered tuning is settle on an atomic interval of pitch. This atomic interval becomes the single, central, structural foundation of your harmony. The trick is to find an interval small enough that multiples of it can approximate all of the pure ratios you care about. Of course you can always choose a smaller atomic interval to approximate more pure ratios and approximate them better, but if you make your atomic interval so small that you can’t tell steps of it apart, then you’re essentially playing in free pitch already and have lost any of the advantageous practicality its structure may have provided.
Using Graham Breed’s nifty temperament finder, I was quickly able to pinpoint the sweet spot in accuracy and practicality: 57 equal divisions of 2:1. In other words, my atomic interval is the fifty-seventh root of 2, 57√2, or approximately 1.0122347160 (for comparison, the most common tuning system in modern practice is 12 equal divisions of 2; each successive key on a standard piano is one 12√2 times the pitch of the previous). This is the lowest equal division of the pure octave which approximates all four of my higher primes closely enough that the average musician cannot discern that they’re out of tune at all — the inaccuracies are each less than 5 cents (where that standard 12√2 interval is 100¢, naturally).
57 is 3 * 19, and thus 57ed2 “extends” 19ed2. One could think of it like taking a guitar fretted for 19ed2 and dividing each of its frets into three equal smaller parts. 19 is a historically popular tuning, so that is to my advantage.
The patent val for 126.96.36.199.19 prime limit 57ed2 is < 57 197 211 233 242 |. What this means is that 57 of my atomic steps will best approximate the ratio 2:1, 197 of them will best approximate 11:1, 211 of them 13:1, and so on. Only the 2 is perfectly approximated, and this is common. In my tempered tuning, everywhere one finds an 11 in the basis for a pitch ratio, however, the number 10.9749339402 ((57√2)197) has been substituted. Similarly, 13, 17, and 19 have been substituted with 13.011851757, 17.0030222301, and 18.9695563769, respectively.
One can see that the Blumeyer comma is tempered out by 57ed2 using one of two methods. The first uses multiplication; if one replaces the numbers in the expression 11 * 13 * 17 / 19 with the particular approximations above:
10.9749339402 * 13.011851757 * 17.0030222301 / 18.9695563769
one gets exactly 128, which is 27. The second method uses addition; one can see that 197 + 211 + 233 – 242 = 399, which analogously is 57 * 7. As you can see, the temperament has given us a way to reason about multiplicative ratios using addition, shifting us down a gear on operational complexity.
In practice, we do not plan to use these harmonics without octave reducing them. My tuning will repeat at each octave, so the version of the 11th harmonic that I care about is the one that is between 1 and 2, that is, 11:8. Similarly, I care about 13:8, 17:16, and 19:16; I divide by the next power of 2 until I’m less than 2. To get the number of steps of 57ed2 that represent each of these intervals, I just have to perform the analogous operation on each of the numbers in the val: modulus by 57. For example, 197 % 57 = 26; the three 57’s divided out correspond to the three 2’s in the 8 of 11:8. Thus we arrive at a final, usable conversion of pure JI harmonics into tempered steps:
- 11 becomes 26 steps.
- 13 becomes 40 steps.
- 17 becomes 5 steps.
- 19 becomes 14 steps.
Everything observed above about the Blumeyer comma holds for these octave reduced values: 26 + 40 + 5 – 14 = 57.
And now that the numbers are all nice and small, several other tricks reveal themselves.
- This temperament vanishes 209/208, as 26 + 14 = 40; moving by an 11 and 19 is the same as moving by a 13.
- Moving up by two 11’s and then an 17 gets you nowhere: 26 + 26 + 5 = 57.
In more detail:
- 11:8 -> 547.368¢ (pure would be 551.317942¢, error of -3.950¢)
- 13:8 -> 842.105¢ (pure would be 840.527¢, error of 1.578¢)
- 17:16 -> 105.263¢ (pure would be 104.955¢, error of 0.308¢)
- 19:16 -> 294.737¢ (pure would be 297.513¢, error of -2.776¢)
Remember what I said about atomic intervals getting too small? A 57th of an octave is a little on the small side. So I set out to find a mode of 57 — a subset of its 57 pitches — one which still housed the tempered versions of all four of these higher harmonic intervals.
One thing I knew about modes was that the better sounding ones tend to have exactly two scale step sizes, and distribute two scale step sizes as evenly as possible. Look at traditional modes in 12ed2: all of them are rotations of 2212221. You jump 2 atomic intervals, then 2, then 1, then 2, 2, 2 and then 1 for a total of 12. You never jump any number other than 2 or 1, and the ordering is not imbalanced such that, say, all the 1’s are in a row and all the 2’s are together. So I wanted to find something like this, but for my 57ed2.
That part would be easy, actually. The tricky part was going to be finding one where you could still express all of the intervals 26, 40, 5, and 14. Quick counterexample: say I divided 57 into 3’s and 7’s. Sure, looks like I can make 26 (3 + 3 + 3 + 3 + 7 + 7), and 40 (3 + 3 + 3 + 3 + 7 + 7 + 7 + 7), but wait, how am I supposed to make 5?
Where to start? The first thing that was clear was that the smaller of my two intervals would be no larger than 5. But I also somehow needed to capture 14, and if I had only 5’s to work with, I couldn’t do that; I could only get 10 or 15. So I tried using a 9, the remaining difference being the next logical thing to do. Could I build a 26 using only 5’s and 9’s, though? Unfortunately not, so I tried the next logical thing: making the 14 out of two of my original 5’s and treating that remainder, 4, as my potential second scale step size. And indeed I could build a 26 out of 4’s and 5’s: 5 + 5 + 4 + 4 + 4 + 4. I could build a 40 out of 4’s and 5’s, too, in several different ways, even: ten 4’s, eight 5’s, or five 4’s plus four 5’s. Finally, 57 itself could be made out of 4’s and 5’s: either one 5 and thirteen 4’s, or (continuing to exercise this handy four 5’s for five 4’s exchange rate) five 5’s and eight 4’s, or finally nine 5’s and 3 4’s.
The question then became which of these three modes of 57 supported sufficient counts of 4’s and 5’s for each of the intervals 5, 14, 26, and 40. It was immediately obvious that the first extreme with only a single 5 could not suffice, as the 14 requires two 5’s. And it was also apparent that the other extreme — the one with only three 4’s — could not support 26 since it called for four 4’s. So I was forced to go with the most balanced set: five 5’s and eight 4’s.
So now the question became: while I knew by the total counts of 5’s and 4’s that the possibility at least existed that the intervals I was interested in could be found, I couldn’t be sure yet that when these 5’s and 4’s were positioned as they needed to be — as evenly distributed as possible — that I would be able to find each of the particular sets of 5’s and 4’s that I needed to capture 5, 14, 26, and 40.
Well, the most even distribution of eight 4’s and five 5’s looks like this:
and I was in luck:
- The 5 exists five times, of course, once for each .
- The 14 exists twice, once for each .
- The 26 exists ten times! For each , you can take the six numbers to the left or the six numbers to the right (looping back around if you reach an edge; this is a cyclical set), and you’ll get 26. This is an 11th-harmonic-heavy mode.
- The 40 exists six times! Like the 26, you can easily find each instance by looking to the ’s; for each one, you can take the nine numbers to the left or the nine numbers to the right and you’ll get 40. You only get six 40’s because unlike with the 26’s, some of these overlap (connecting two 5’s together).
While the 11th harmonic is the most prevalent sound, the 19th is the rarest. Since the 19th is the key to the comma, being the one of the four higher harmonics that sits by its lonesome on one side of the ratio while the other three party together, I figured that I’d have to pay special attention to the moments in the scale with .
Some of the more xenharmonic amongst my readers may have noticed that my mode is otherwise known as a Moment of Symmetry scale — I just arrived at it in a totally ass backwards fashion! Most MOS scales begin with a window (usually an octave) and a generator, then iterate that generator in a loop around the window, stopping at any point where the generator has divided the window into sections of exactly two different lengths (the maximally even distribution is a natural result of this process). So, for those MOS-curious of you out there, had I begun with my generator, it would have been the 35th step of 57, which is 736.8421¢, associated with the 21st harmonic. And my set of five 5’s and eight 4’s could be generalized as a 5L 8s scale, in the tridecatonic family.
Here is the Scala file for the tuning:
Blumeyer comma scale, 5L8s MOS of 57ed2
Amusingly, this 4454454544545 turns out to have nothing to do with 19ed2, despite as mentioned previously the fact that 57 is 19 * 3. There are just no threes anywhere in there, only 4’s and 5’s!
However, my tuning did turn out to have similarities with a different smaller ed2. When using Breed’s temperament finder initially, I had noticed that 13ed2 was another point of interest in the accuracy & simplicity space. Certainly less accurate (errors of 2.528¢, -9.758¢, -12.648¢, and -20.590¢ respectively for 11, 13, 17, 19; musicians can usually sense when a pitch is off by 10), but also certainly simpler! So it’s interesting to notice that the mode of 57 that I came up with is remarkably similar to 13ed2. Being a tridecatonic scale it also has 13 steps, and since the long and short steps of my mode are only ever so slightly different from each other (compare to traditional modes where the long steps are twice as long) it takes on a chromatic character. When you really think about it, what I ended up with is kind of a whacked out 13ed2: pinched and pulled in a particular way to more perfectly capture my four precious primes. It’s no surprise that 13 * 4 = 52 and 13 * 5 = 65, since 57 is exactly 5/8ths of the way between 52 and 65, corresponding to the ratio of 4’s to 5’s in the MOS.
I rendered a version of my comma pump in 13, and I was disappointed to find that it doesn’t sound all too different to my ears, and that I’d perhaps gone through a ton of work for nothing. Perhaps I’m just not a trained enough listener. And I don’t know the exact math behind this type of thing, but it seems intuitive that the higher a harmonic you are dealing with, the more accurate a tempering has to be for the harmonic to be perceivable as such. For example, consider the 19th harmonic (297.5¢): it is much closer to the minor third in standard tuning (300¢) than the ratio 6:5 (315¢), yet since 6:5 is made out of more fundamental primes, it takes precedence. Some listeners can literally feel the polyrhythmic vibration of six against five in peaks and troughs of a waveform, while most can’t count nineteen there. To really drive home a 19, I felt, I couldn’t be 20¢ off like I’d be in 13ed2. But turns out, what do I know.
And that all said, the constraints of my 57ed2 mode bred creativity. Compare what 13ed2 looks like in the same terms as my 57ed2 mode. There’s a ton more freedom in 13 — I can go anywhere anywhen anyway I want — but that also makes things more boring. Using the bizarre restrictions of my 57ed2 mode, I came up with rules about which pitches I would allow myself to move between, and that helped me come up with the “melodies” that enter later my pump. Basically, I disallowed myself from moving between pitches not related by any of my tempered higher harmonics, and I disallowed myself from moving to any pitch that was part of the comma pump itself when moving in between them.
One thing I didn’t look into yet is a mode of 13 equal that would capture the steps of the comma pump (0, 1, 3, and 8). This could be done with an MOS with steps 12122122.
I found a tuning on the Xenharmonic Wiki called hilim13 (High Limit 13?) by Gene Ward Smith, which also has thirteen steps and is built out of ratios using primes 11, 13, 17, and 19. This is what his tuning looks like, more asymmetrical:
And this is what my original JI tuning looks like in the same terms (pretty cool if you ask me):
I really would like to go back soon and write some music for my original JI tuning, which I never actually did. When I drew this out I noticed how balanced/symmetrical it was, which reminded me of one of the theoretical advantages of Combination Product Sets (CPS), and going back on its construction I realized my JI tuning was actually a combination of four common CPSs: (4 choose 1), (4 choose 2), and (4 choose 3), a tetrany, hexany, and tetrany again, respectively. I also threw in (4 choose 0), the unison, and (4 choose 4), for a total of 1 + 4 + 6 + 4 + 1 = 16 pitches (3 pairs of which are close enough that they more or less collapse into each other, resulting in a 13 note scale — interesting how all four of these higher harmonic limit tunings end up as 13 notes!) I believe this makes it an Euler genus. I have remained quite fond of this musical pun shared among the consecutive primes 11, 13, 17, and 19, and excited to realize it in further work. Maybe a using these four primes would kick butt too.