Subconscious Mario Paint Constraints

In 1992, my family got Mario Paint. You know, the “video game” that came with its own controller (a mouse) which was basically a souped up Microsoft Paint. You could also make simple animations, and you could even compose music.

These tools were limited. You could only have one animated element at a time, and it could only endlessly loop around either 4, 6, or 9 frames you’d drawn (the more frames, the smaller cells you got to draw in), and this looping element could be further animated in a path around the screen but only in any path you could record contiguously with your mouse movement.

And as for the music, you could only have three notes at once, choosing from a palette of only fifteen instruments (including the Super Mushroom, the Baby Yoshi, and the Game Boy), within a pitch range of less than two octaves, without accidentals (thus forcibly diatonic), without dynamics, and you could only use staccato quarter notes or rests, up to 96 beats in either 3/4 or 4/4, choosing from a predefined set of tempos.

I recently visited my parents’ house in Georgia and found the SNES cartridge on a shelf in a basement closet. I brought it back with me to California and kept it near me as a talisman while preparing a composition to be performed and recorded at the Untwelve Microtonal Summer Camp — to remind me where I got my start composing.

The work I’ve been doing this month is to interpret my composition for humans, not just computers. These days I have better digital audio workstations than Mario Paint, so it’s no big deal to tell a computer to play whatever notes I damn well please, such as one with a frequency of 453.884586154785 Hz. But to get a human to play that note it’s better to tell them something more like that they’re shooting for a note 44% percent between the Eb5 and E5 familiar to them, and also that they’re shooting for a note 16/13ths the pitch they just played and 11/8ths the pitch another performer is playing simultaneously.

To avoid manually drawing all 161 of the diagrams peppered throughout the sheet music for this purpose, I scripted up a webpage through which I could feed a specially formatted spreadsheet version of my song, that would then draw the diagram with HTML5 Canvas and spit out each diagram as an image file. I had hoped to add a fifth part to my piece in time for camp, but the overhead to prepare these pitch diagrams (and write tests to confirm they came out correctly) precluded that.

Clearly the pitch system I developed for this piece is too complex to have been written in Mario Paint. It’s not even twelve tone equal temperament, let alone diatonic. But given that I originally composed this piece a year ago — long before Mario Paint was fresh in my head again — I am amused to notice that other than pitch, its other musical qualities could be rendered within the constraints of Mario Paint!

While nowadays I normally write music in like, 11/3 + 9/4 + 7/5 time, this one is straightforward 4/4. And while it’s over seven minutes long, it’s so slow that it’s only 23 bars, or 92 beats, just within MP’s limit. That molasses-slow tempo it proceeds in is precisely the slowest tempo MP makes available. It boasts only a single type of instrument, almost always only up to 3 different pitches at once, and a pitch range within MP’s limits. Up until adapting it for humans, I called for no dynamics either.

Part of me knows that I intentionally made the piece as simple as possible other than the pitch system, in order that the performers and listeners could focus on the complexities of its pitch system (trickery deriving from the fact that movement by the 19th harmonic is almost exactly equal to a movement by the 11th, 13th, and the 17th) (which I originally developed five years ago at the Xenharmonic Praxis Summer Camp, an earlier iteration of this annual alt-tuning retreat). But part of me playfully wonders if some of the Mario Paint near the core of my musical self subconsciously worked its way to the surface here.

(No, sadly, I cannot find any trace of the first song I remember composing on Mario Paint, “Flying”, which was a duet for two Geese.)

Blumeyer Comma JI Unpump

Four years ago I invented a tuning system based on the pun whereby movement by a 19th harmonic almost exactly cancels out movement by an 11th, 13th, and 17th (2432:2431). Finally, I’ve written a piece in it:

I call it an “unpump” because rather than spiraling out in looping increments of the comma as JI pumps tend to do, I instead modulate by the tiny intervals in the tuning in order to return back to the home pitch as tempered pumps tend to do.

Here are two different maps of the bass to the track:

Basically I found the three instances of the pump I used for the tempered version (here) and noticed that they each shared one pitch with each other, so I decided to give each one at least one full loop around itself before switching tracks to the next, looping back around the full three-pump cycle in a total of 23 bars. The exact pattern is:

ABCDABCDEFGDEFGHBEGHBCD

As you can see, ABCD, DEFG, and GHBE are the pumps. The first two links are seamless, but the link from the pump starting with G back to the one starting with A requires a little more connective tissue. It’s fine, though, I think, since it results in a nearly even distribution of presence of chords per pump (that is, while individual chords appear between 2 and 4 times each – a factor of 2x difference – sets of chords appear in more similar amounts: A’s, B’s, C’s, and D’s together appear 13 times; D’s, E’s, F’s, and G’s together appear 12 times; G’s, H’s, B’s, and E’s together also appear 12 times).

The chords are built from the pitches directly connected to each of these bass notes but which were not captured along this path. I guess you could call these the “left-out” notes.

The harmony comes from choosing from left-out notes, too. But instead of building a stack of all such notes for each bass note, what I did was pick a single note, then connected them by moving step-wise through the left-out notes. How long I hold each harmony note just depends on how many steps it takes to get between one bass note’s harmony note and its next’s using the shortest possible path. Only one note has no neighbor left-out notes: the unison.

I did some other tricky things that I don’t exactly remember and won’t attempt to explain. Enjoy!

 

Blumeyer Comma Pump

Background

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.

The prime exponent vector for the Blumeyer comma, for those interested: |7 0 0 0 -1 -1 -1 1>

 

 

The temperament

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 2.11.13.17.19 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¢)

 

The mode

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.

The tuning I used for the Blumeyer Comma Pump.

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:

4454454544545

and I was in luck:

• The 5 exists five times, of course, once for each [5].
• The 14 exists twice, once for each [545].
• The 26 exists ten times! For each [5], 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 [5]’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 [545].

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_tempered.scl
!
Blumeyer comma scale, 5L8s MOS of 57ed2
13
!
84.21053
168.42105
273.68421
357.89474
442.10526
547.36842
631.57895
736.84211
821.05263
905.26316
1010.52632
1094.73684
2/1

 

Versus 13ed2

 

13ed2, with tempered 11th, 13th, 17th, and 19th harmonics

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.

 

 

 

 

 

 

 

And beyond

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.

Othoundsto: rotating musical houndstooth

Yaw

Othoundsto is a musical work, and an illustrative example of the power of one of my Fun Musical Ideas: musical yaw.

You can experience a demo of othoundsto on my personal site here: othoundsto.douglasblumeyer.com.

The basic idea is that the red bar is the play cursor. Just like the play cursor of any other music sequencing program, it combs across the music, playing notes as it crosses them. Notes the cursor strikes toward its top are higher in pitch, and notes it strikes towards its bottom are lower in pitch. The bar is the “now”; notes to the right are to be played in the future, and notes to the left have already been played. Since it is useful for “now” to stay on the screen, typically sheet music is split into segments which the play cursor reads left to read in rows (like you read words on this page), but another common implementation is for the cursor to hold in place and the music to comb under it, somewhat like a player piano’s reader is fixed in place and the roll winds through it; othoundsto works more like the latter. And of course while othoundsto doesn’t look much like sheet music or piano roll, the same type of information is encoded in it: borders between black area and white area are the notes, with vertices getting the stress and the line segments connecting them treated like glissando ties.

Musical yaw is when the sheet music fluxing across the cursor membrane rotates as it does so. Once it has rotated 90 degrees, all distances which used to affect note durations now affect pitch durations instead, and vice versa. In between 0 and 90 degrees you get some funky effects.

 

Teeters

In othoundsto, I chose to link the composition to moments of vertical and/or horizontal alignment. It “teeters”, I call it, rocking back and forth, counter-clockwise and clockwise, with each turn swinging a little further that direction than it did last time, each time stopping at the next furthest angle of alignment. At these alignments, at least two notes either share the same pitch or occur sameltimeously, and there tend accordingly to be a minimum of interval sizes (both pitch and duration).

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Since pitch and duration are not actually related (well, of course they are, but at a scale too distant for this effect) the choice of 300 cents as the equivalent pitch unit to the 1 durational unit (the length of one horizontal line segment) was arbitrary. I could have picked any amount to get a different sound (and you are welcome to experiment). I chose 300 cents because that results in the octave (1200 cents, or the simplest 2:1 frequency ratio) repeating with the 2×2 supertile that houndstooth uses to tile the plane. You can really see how in musical yaw there are two types of duration: the micro-duration, the material of the pattern that can become another property in rotation, and actual time in which such rotation may occur.

An important note is that the play cursor is not a line but just a line segment. If it extended infinitely up and down, there would simply be too much crossing it at a time (even if we account for pitches going out of human hearing range); except at the moments of alignment, every pitch class would get heard. But it also would sound weird if pitches sliding off the top or bottom of the bar just disappeared. Thus, pitch circularity is applied. The pitches that strike in the center of the bar are loudest, and fall off in volume at a rate shaped like a bell curve toward the extreme top and bottom. This way, I can simply infinitely tile the plane with houndsteeth in all directions, and rotate the entire world about a pivot centered on the middle of the play cursor.

 

WIP

Now the thing is that when you’ve rotated this thing 45 degrees one way or the other, the former diagonal lines take precedence as your verticals and horizontals, and they are the square root of 2 the length of the original horizontal and vertical lines. While the square root of two is a beautiful transcendent number, unfortunately as a frequency ratio is not particularly impressive (as aren’t many important transcendent numbers like e and pi).

I admit that I am somewhat disappointed by the harmonic variety one gets out of othoundsto. The moments of alignment do not sound significantly less chromatic to me than the stretches of chaos in between them, even with the ratio between the stress of the vertices upped (de-emphasizing the glissandoing segments joining them).

What I think would make othoundsto more interesting would be a greater variety of timbres (than the one square wave, yes, I know). But seriously, this is a big part of what made houndstooth appeal to me for the purpose of illustrating yaw in the first place: it has just the right amounts of simplicity and complexity when dealing with rotations of itself. In particular, if you set a rule where a voice/timbre may be continuous whenever the line it’s following does not move backwards, you already begin to see a lot of interesting effects:

Here is the repo for othoundsto: http://github.com/DougBlumeyer/othoundsto

The implementation is unfortunately extremely specific to houndstooth at this point. I would love to generalize the yaw tool so that any midi file could be given to it, with instructions for rotations at given times. I would also love for it to be able to generate sheet music that could be used by live performers.

In terms of the existing repo, it would be great if I could have a stronger visual indication of the connection between the collision of the play cursor and lines in the houndstooth pattern – perhaps some glowing yellow points of contact – to help parse the sound. Though ideally I imagined the power of yaw would be being able to perceive the rotation between pitch and duration without needing it to be visualized – I’ll have to keep hacking on that.

harmonic circle

I have created a novel aural illusion whereby the end of a given octave of the harmonic series is made to seem like it reconnects seamlessly with its own beginning. I propose calling this illusion “harmonic circularity”, modeling its proposed name after the widely accepted names for key components of the effect: pitch circularity and tempo circularity. An instance of my new illusion could then be called a harmonic circle, and I have included one such harmonic circle for your listening pleasure below.

This particular harmonic circle is made from the 8th octave of the harmonic series (harmonics 128 through 255). I’ve found the 8th octave to be the ideal for appreciating the effect: it has enough steps to concealingly spread the pitch and tempo circularities across, while not too many steps such that the harmonic relationships between the intervals become imperceptible.

Here is a high level summary of how I achieve this effect (more detail to come):

1. Establish a pitch circular environment.
2. Choose an octave of the harmonic series and loop it, so that pitch seems to climb forever. The ending connects to the beginning because in pitch circularity one octave higher is back where you started.
3. Recognize that the logarithmic nature of the harmonic series poses a problem to the seamlessness of the connection between the end and beginning of the looped octave: intervals at the beginning are twice as wide as those at the end. Correct for this with a proportional tempo acceleration, so that by the end of each octave two notes pass in the space that two pass at the beginning, thereby linearizing the rate of pitch climb.
4. But now there is an obvious seam between the ending and beginning in that suddenly the tempo drops by half on each loopback. Correct for this by gradually fading out every other note, so that by the time the tempo is 2x, only half the notes are actually being heard.
5. Finally, a subtle tweak to complete the illusion and impart stylistic consistency.

 

 

Step One: Pitch Circularity

If you already understand pitch circularity, you can safely skip this section. However, I recommend refreshing yourself on it, or at least seeing it through the lens I see it, since themes to its structure will recur in later steps.

Pitch circularity is the property of a sonic space in which pitch can be made to seem to continually rise (or fall) while getting nowhere — over time seeming to stay in about the same place. This aural illusion is achieved by fading out frequencies as they get too high (or low), while fading in frequencies to take their place from the opposite end of the pitch continuum. Listen below to a Shepard scale, a simple showcase of a pitch circular environment.

Most people perceive frequencies related by the simplest possible harmonic ratio, 2 (also known as an octave), to be so similar that in some senses they are essentially the same note, or “pitch class equivalent,” and in traditional Western musical notation we even refer to these frequencies with the same letter (A, B, C, D, E, F, or G). This is the interval, therefore, used between frequencies that take each other’s place in pitch circular environments. When a pitch begins to get too high, we begin replacing it with another pitch having half the frequency; when a pitch begins to get too low, we begin replacing it with another pitch having twice the frequency.

frequencies related by the ratio 2:1, each an octave apart

 

The fading in and out of frequencies as they get higher and lower is handled by an “amplitude curve”, which is simply a graph relating pitch to loudness. In the middle of this graph, where pitch is moderate, loudness reaches its maximum; at the extreme edges of this graph, where pitch is very high or very low, loudness approaches zero. Such an amplitude curve could be drawn as a simple triangle connecting with line segments the pitch-central loudness-maximum to its two pitch-extreme loudness-zeroes. However, the most effective amplitude curves smooth out the process into a shape like a normal distribution’s bell curve. This way, there are no sudden changes in the derivative of the graph, also known as its slope.

The derivative of a bell curve is smooth and continuous; the rate of change has no sudden changes…

 

…whereas the derivative of a triangle-shaped “curve” is shaped like a square wave; frequent discontinuities where the rate of change flips suddenly from positive to negative.

 

In other words, the rate of change in amplitude of a voice in a pitch circularity environment as it changed frequency would never itself suddenly change; the rate of change should change gradually. Bell-shaped amplitude curves are more effective for pitch circularity because the illusion relies upon concealing its underlying mechanisms, and gradual changes are more difficult to perceive than sudden ones. Here’s what that Shepherd-Risset Glissando would have sounded like with a triangular, non-curvy amplitude “curve”:

The simplest implementation of pitch circularity is to begin with a single pure frequency and center an amplitude curve on it. As the pure frequency begins to rise in pitch it also begins to descend in amplitude down the right half of the bell toward a point where it simultaneously reaches one octave higher in pitch and silence in amplitude. In the meantime a new frequency having appeared at a point one octave lower in pitch and silent in amplitude has been climbing the left half of the bell toward the point the first frequency originally occupied at the top of the bell where it is central in pitch and maximal in amplitude.

However, if our intent is to conceal the mechanism by which the pitch circularity is being achieved, this simplest implementation will be unsuccessful, for a couple reasons.

1. Texture. The sonic texture when the sound is mostly consisted by a single pure frequency at the top and center of the bell will be easily distinguished from the rest of the time when the sound consists of two quieter frequencies an octave apart on opposite slopes of the bell.
2. Isolation. With only two total frequencies being heard, it is readily apparent that one is fading in while the other fading out.

Here’s what this simplistic version sounds like:

In order to effectively conceal the octave-stacking mechanism that enables pitch circularity it is necessary for the sound at all times to consist of a larger number of stacked octaves, with the amplitude bell curve spanning not only two octaves as was described before, but all the way across the many-octave range.

1. Texture. This corrects for the textural issue because it is far more difficult to pick out the moment of convergence (when one frequency is dead-center on the amplitude curve and two others are silent at its extreme ends) when the net minus one frequency it results in is out of many frequencies rather than only a total two. Said another way, superparticular ratios start at infinity and approach 1 as they increase in size ( x + 1 / x -> 1 as x goes to infinity), meaning that the numerator and denominator become more alike.
2. Isolation. This also corrects for the issue with being able to perceive individual fading frequencies because the process of fading is now distributed across many laps around the pitch circle, which are all overlaid with each other (the stretch from halfway up the bell to the tip-top is concurrent with the stretch from the very bottom to halfway up).

Here’s what this slightly denser sounding but more obscure version looks like:

While it is still possible to listen closely and figure out what is going on and feel where the moment of convergence occurs, in general the technique described here is sufficient to produce an environment where pitches can seem to climb or descend infinitely while staying in place. That said, if you’re interested, I have described a technique whereby the pitch circular effect and its artifacts could be even further obscured: recursive Shepard-Risset glissandoing.

 

 

Step Two: Enter the Harmonic Series

I have been fascinated by pitch circularity for many years, but it only recently occurred to me to wonder how it could be applied to the harmonic series.

As you may know, a musical harmonic series simply maps a sequence of consecutive integers to multiples of a base frequency. Here’s what that looks like:

Because pitch space is logarithmic (remember, each octave is related not by a constant amount, but by a factor of 2), each successive step up a harmonic series results in a smaller perceived difference in pitch. This shrinking of interval size can be readily perceived by listening to the first 16 notes of a harmonic series (the first four octaves):.

And here’s what the first part of that looks like graphed on a non-logarithmic graph:

Looking at it the other way, each successive octave in the harmonic series holds twice as many pitches as the previous octave.

Diagram courtesy of the amazing Andrew Heathwaite.

 

As can be seen, the pitches in each octave “bunch up” toward the top, the intervals between them approaching the smallness of the first interval of the next octave (which in turn is the largest interval within that octave). In other words, it is not at all the case that the interval size instantly halves at a punctuated moment of octave ascension; the interval size is always gradually changing and irrespective to octaves.

Now we finally come to the crux of why I was interested in what happens to the harmonic series in pitch circularity. Let’s listen to the 5th octave of the harmonic series, with pitch circularity naively applied.

Something sounds a bit off, right? It’s very easy to pick out the point where it loops back around, is it not? Well, we’re using our grade-A pitch circularity, here, nothing is amiss with pitch circularity itself. The problem resides in the nature of the harmonic series. When we snap back to an earlier place in a logarithmic dimension, resolution changes. The pitch interval sizes suddenly change.

What we have at this point is the illusion of infinitely rising pitch, but rather than a smooth infinite rise, a really jerky rise with one hump after another. The pitches suddenly begin to climb rapidly from step to step, gradually climbing less and less with each step, and then suddenly starting again to climb rapidly again, and repeating that over and over.

For this illusion to work, we need that change to happen gradually. But how?

 

 

Step Three: Vary Tempo

To flatten out those logarithmic humps into a linear climb it will be necessary to proportionally increase the tempo as we proceed through each octave. This way, while the pitch interval between the penultimate and final harmonic is smaller than the pitch interval between the first and second harmonic, the time interval is equivalently smaller. In other words, by the time a harmonic step covers half the pitch as an earlier harmonic step, we’ll have it whizzing by in half the time, neutralizing the logarithmicity. Pitch climb is proceeding linearly.

As pitch intervals decrease, duration intervals are decreased proportionally, linearizing the pitch climb (though the notes grow denser… more on that later)

 

This, however, introduces yet another problem! Now we’ve concealed the pitch circularity and the logarithmic rate of pitch climb inherent to the harmonic series, however at the moment we switch back around to the beginning of the harmonic series, there is an extreme change in tempo: it drops by half. This is what that sounds like:

And this is what that looks like in my audio software:

 

 

 

Step Four: Rudimentary Tempo Circularity

In order to correct for this, it is necessary to introduce another process whereby every other note is fading out toward silence (this fading out is independent from and additional to any amplitude attenuation caused by the pitch circularity).

With tempo doubled, twice as many notes are occurring at the end of the process, yes, but with half of those notes descending into nothingness, the process cancels out. Here’s what that looks like in my audio software:

Finally the two ends of our harmonic series octave sound the same and can be connected. This makes sense, since as can be clearly seen in an earlier diagram, one of the beautiful aspects of the harmonic series, in that each successive octave essentially recovers the previous one while filling in another set of notes in-between each previous note; we’re fading out that new set of notes.

But we’re not quite done yet. You may have noticed that this most recent sample still wasn’t quite as impressive as the final result we listened to at the outset of this discussion (and not just because of lack of reverb finesse). Our tempo circularity isn’t concealed enough at this point.

 

 

Step Five: Philosophical Coherence

The final subtlety to make this effect really come together is to realize that to leave the every-other-note-fading-out-effect in this state would be to commit the same offense that we avoided when implementing pitch circularity: not burying the fading out within enough similar simultaneous fadings, leaving it exposed and easy to pick out.

Analogously to how we stacked many octaves of pitch and drew out the fade out process over them, then, we will actually take the every-other fade-out effect and extend it over multiple iterations of the loop. Notes which are multiples of 4 stay at 100% amplitude throughout, while notes that are multiples of 2 fade to half strength by the end, connecting up with those at the beginning which are odd harmonics, which begin at 50% and fade to zero. And we do this along the shape of a bell curve, again, of course:

 

 

Conclusion

Which brings us back to where we started:

Thank you for listening and reading! Please feel free to use harmonic circles in your own compositions. Here is a link to my GitHub entry with the final product: https://github.com/DougBlumeyer/Harmonic-Circle

I’ll be back someday soon hopefully with some fun spatial circlings to complement these.

Virtual Reality Dynamic Positional Audio Experimental Sound Sculpture

This is a little something I whipped up in one day. You will need the Oculus Rift DK2 to experience it.

Windows | Mac

You will find yourself floating invisibly inside an icosahedron, listening to some strange music. Look at different triangular faces to change the tuning of the instruments. Each of the twenty faces is associated with a different Hexany, one for each combination of three of the first six odd numbers past one, including one as the fourth number ([1,3,5,7], [1,3,5,9], [1,3,5,11], [1,3,5,13], [1,3,7,9], … [1,9,11,13]). Every hexany consists of six pitches, and these are mapped to the six different instrument timbres, which are mapped to the six cardinal directions (above, below, left, right, before, behind).

This project was inspired in part by a demo Total Cinema 360 gave at the first VR Cinema meetup. In this demo, you found yourself inside three virtual realities at once (arranged as trienspheres): look ahead to be flying over a volcano, look behind and to the right to be in a living room hanging out with some folks, and look behind and to the left to be on stage of a rock concert. The kicker is that when you’re looking into any one of these three worlds, the sound surrounding you is of that world. In other words, look into the volcano and you can hear the volcano behind you, even though you know that if you look back there you will see a room or a concert and the sound of the volcano will disappear. Thus an element of sight has been lent to sound: you can only hear things in a line extending from the front of your head, rather than the normal state of being able to hear everything around you no matter where you look. This is what I refer to as Dynamic Positional Audio.

There’s a lot left to improve on this, and hopefully I will be able to get to it soon. I would use more distinct tunings, maybe have a continuum between them, maybe have the sound sources in motion (in response to your motion even maybe), and just make it more beautiful to look at and hear. But hey, I’m a beginner. Ship.

I used Different Methods’s SpatialAudio Unity plugin after much hunting and attempts to do it myself using a MaxMSP+Unity integration.

Jwimwheul

This song switches between cycling 3 and 5 against 4.

ROBOT HEAD, ROBOT SHOULDERS, ROBOT KNEES, AND ROBOT TOES

ROBOT HEAD, ROBOT SHOULDERS, ROBOT KNEES, AND ROBOT TOES.

ROBOT HEAD, ROBOT SHOULDERS, ROBOT KNEES, AND ROBOT TOES.

AND CAMERAS, AND MICROPHONES, AND SPEAKERS, AND ROBOT NOSE.

ROBOT HEAD, ROBOT SHOULDERS, ROBOT KNEES, AND ROBOT TOES.

Oumaotcouwao’ist

Oumaotcouwao’ist is in 5/3 time. I discussed this with a few friends and we decided that since the “note value that constitutes a beat” element of the time signature, that is, the bottom number, is arbitrary to the “number of beats in a measure” element, the former’s greatest contribution to the meaning of the time signature is to define the resolution of the subdivisions of each beat. That is, were I to have instead said that Umowchuwowiest was in 5/4, I’d still be saying that each measure had 5 beats; it makes no difference whether I call each of these beats “quarter” notes or “one third” notes. However, a “quarter” note implies that it further divides into eighths and sixteenths and thirty-secondths, etc.; the subdivisions are all 1/4 multiplied by 1/2^x. By calling it a “one third” note I hope to imply that each beat divides into sixths and twelfths and twenty-fourths, etc.; the subdivisions are all 1/3 multiplied by 1/2^x. Implemented, it’s just like each measure has 15 beats: snare drum hits every 3, kick drum hits every 5.

At least, that’s what I had decided about Oumaotcouwao’ist several years ago. The truth is, its beats do not divide into 3 parts then 6 then 12 then 24, that is, 1/3 does not get multiplied by 1/2^x. It gets multiplied further by 1/3 in some places, and 1/5 in others. I suppose the best way to describe its rhythm would be 1/(3^x)(5^y), because it operates by dividing every beat by either 3 or 5. This is a rhythmic style I am seeking to explore further in my songs that will feature infinitely decreasing tempos, diving deeper and deeper into recursively defined patterns.

If you’d like to listen to the older version, which is a bit slower, more robotic, and not tuned to Bohlen-Pierce, try this:

And here’s a yet older song that I had nearly forgotten about, that uses the same 5 against 3 rhythm without taking it anywhere deeper than that, but is still pretty nuts:

Souwasinorites

Souwasinorites has 33 beats per measure which subdivide into both 3 sets of 11 and 11 sets of 3 (0:00-0:32), then 36 beats per measure which subdivide into both 4 sets of 9 and 9 sets of 4 (0:33-1:18), then 35 beats per measure which subdivide into both 5 sets of 7 and 7 sets of 5 (1:18-1:53).