Beware any number larger than 5 .
The number of digits on one hand
is a fundamental limit in our
thinking ( feeling )
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The same product (for a ratio reduced to lowest terms) gives us the period if I’m not mistaken
Yes I think that’s right. I think the mathematical term is “least common multiple” - so for a 5:4 rhythm it will have a fundamental period of 20 before it repeats.
I should also add that a signal doesn’t have to be 100% periodic to still have a fundamental period - I think that’s where this becomes interesting for rhythm because you can accommodate some amount of randomness or variation while still referring to something with a definite cycle. The question is, how random can you make it before it changes the character of the rhythm? Also, does it make a difference where the randomness occurs e.g. off beats vs on beats? (intuition says it probably does)
If we jump back to the audio waveform domain for a sec, it’s fairly obvious from looking at the waveform of a guitar playing a single note that there’s a lot of chaos and variation in there but in terms of what we hear (ie the pitch of the note) there’s something periodic and repetitive about that which our ears can latch onto. A simple way to extract that is to look at how the zero crossings of the waveform are based where it should be easier to see a pattern.
My thoughts on all this aren’t fully formed btw, I’m really just throwing stuff out there to stimulate a discussion 
Regarding Gamelan, culture etc. it is indeed a massive component of what we as individuals are drawn to for harmony and rhythm. That said, I’m of the belief that we are all “wired for music” on a basic level (there’s a decent amount of studies to support this idea) and so there are some fundamental or universal building blocks of music appreciation that we should try to explore. Trying to come up with computational models or algorithms is a good test of this hypothesis - if we can make something that produces widely appealing rhythms then there’s probably something there. Sonic Pi is as good a tool as any to explore these things with!
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I’ve always gone by the terminology “irrational rhythm” as a lingua franca to communicate with other musicians regarding tuplets that don’t divide by 2. If “polyrhythm” emphasizes stacking different tuplets vertically as an analog to pitch harmony, then playing tuplets sequentially in different arrangements would be the analog of melody. For the sake of communication, I’ve always gone by the terminology “temporal modulation” here. Two rhythmic “melodies” happening at the same time then have both a harmonic (polyrhythmic) and a contrapuntal (melody vs. melody) relationship; it’s just a matter of how you want to color things, want to bring things out in your music, etc.
For example, we call Bach contrapuntal (four simultaneous beefy melodic voices) even though his harmonic syntax always makes good sense, and we call Beethoven homophonic for greatly reducing the emphasis on melody to one lead voice and 2-3 “lighter,” accompanying ones for the sake of gestural elements that would obliterate too many melodies. Regardless, you have harmony and melody going on all the time (even with a solo instrument’s overtones), and it’s just a question of what you want to do with them, what sounds good to you.
There are lots of examples of particularly good music emphasizing temporal modulation, so I’ll just recommend these classics and this wonderful operatic piece. Also, a temporal modulation module recently came out for Eurorack, and it’s pretty cool to interface its CV with MIDI gear. This guy does some nice demos. I was feeding Flux’s rhythms in MIDI format into SPI, which works quite nicely. The Flux module excels in CV gates (rhythm), but is rather lacking in melodic capacity. What’s more, I think temporally modulated music really wants to go with microtunings other than 12 equal divisions of the octave, so I thought it would be nice to implement my own temporal modulation + microtuning tool in SPI that I can use to drive my beautiful-sounding MIDI synths in live performance. I’m just getting that together now, as my rustiness at programming slows down development.
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Ah, rhythmic melody! Yes, I’ve been wondering about the sequential permutations of rhythmic units, and it may be the missing piece for a gap I’m mulling over. At the beginning of this discussion, I brought up how 16 ticks (each corresponding to a 16th note) could be grouped and how the groupings could be permuted, e.g. [6,6,2,2], [8,4,4]. It’s a constricted rhythmic melody which doesn’t delve into “irrational rhythms”, but I think it’s the same idea. Most of the resources shared up to this point have been about steady rhythms. I want to explore “melodic” rhythms next.
I think the “counterpoint” of two rhythmic melodies would involve their syncopation. The mathematical procedure I’ve stumbled upon for determining syncopation is to compare the partial sum series of the two rhythms. E.g. [6,6,2,2] has a partial sum series of (6,12,14,16) while [8,4,4] has a series of (8,12,16). Because the smaller series is not a subset of the larger series, the two rhythms can be said to syncopate. (Also, their composite rhythm has partial sums of (6,8,12,14,16), corresponding to a rhythm of [6,2,4,2,2].) I think the independence of two rhythms depends on the degree to which they syncopate.
I think the bridge between melodic and harmonic rhythm has to do with periodicity. That’s what I’m mulling over at the moment.
An update by the way (addressing the thread):
After reading Stockhausen and exploring these resources, my horizons have broadened. What I previously called a “note” I’ve generalised as a “phoneme”, and what I called an abstract “motif” is now an abstract “chronomorph”, literally a shape in time. I think the theory suggests something far more ambitious than the code. After all, the implementation is merely a 4/4 tonal music generator 
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This thread has gone way beyond my understanding, but I’m enjoying the music links. I’ll just say that playing polyrhythms on a drum kit are a lot of fun, especially when you’re chugging away and your own sense flips between one frame and the other. Like one of those optical iilusions where you flip between seeing a vase and two faces.
In that Stockhausen Samstag aus Licht piece at least he thoughtfully puts in an anvil (or clanging bit of metal whatever it is) to give a clue to the time. Is that the fundemental period you are on about?
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That piece sounded good to me. I paid attention to the composer’s manipulation of time at different scales to get form and rhythm, as he discussed in the first of his four criteria.
The real-life performance aspect of polyrhythm bewilders me. I think Neely mentioned a performer gets a feeling for the composite rhythm, because otherwise I have no clue how one gets to multi-task like that. I guess it’s easier for steady rhythms. Anyways, your perspective on things is much appreciated! 
With regards to the fundamental period, I think it refers to the period of a composite rhythm. Thus, a 3:2 composite rhythm would have a fundamental period of 6 beats. I didn’t really get a feeling for polyrhythm in that Stockhausen piece, except to notice the agitated passages.
It’s true, so actually you feel the three things - the two individual rhythms and the composite. Try the first movement of for size https://youtu.be/wcMYxDMuM4c I couldn’t play it but did enjoy unpicking some of the phrases. If you thought that Neely clicking his fingers in polyrhythm was tricky…
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For rhythms like Stockhausen’s and others, players do learn it by counting it out, then endlessly practice until they get a good enough feel for it so that it’s practically memorized, and they don’t have to depend on the score except for quick reminders. Remember, those orchestra folks essentially dedicate their entire lives to playing one instrument and don’t get time to write music with cool things like SPI . Then they rehearse endlessly in small groups before the whole ensemble combines for full rehearsals. Add in those tempo change markings, and some stuff simply can’t be played without the players having earphone click tracks. That takes some of the aesthetic impact out of it, I guess, but pop bands also depend on in-ear monitors to stay in sync. Here, we’re depending on SPI to help us play stuff that exceeds our skills or number of hands, but the more you find out how stuff works and sounds, the more easily you can get to or discover the things you want to hear and still have time left over to go to work and make money.
Without a score to look at, I don’t think it’s possible to identify just what’s going on temporally or otherwise in Lucifer’s Dance, let alone identify a fundamental period. Like any music, it either speaks to you to one degree or another, or not. You listen a couple of times, and then you’re listening for that part you particularly like (I’ve always loved from around 17:50 to 21:20—the drum kit is Lucifer’s nose), then more of it grows on you, or perhaps never does. Pop music consumption works pretty much the same if you ask me. With SPI, we can try stuff out quickly to see what we like without having to have concert pianists’ skills or players dedicated to performing for us.
Oh yeah, Stockhausen usually has a visual or theatrical element going on, and I think it would be cool to coordinate SPI with something like Resolume.
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Yep, it’s not that our routines can’t be done without the aid of technology, but that they are unfeasible at scale. At some point however, I’m interested in how as Stockhausen puts it:
New means change the method; new methods change the experience, and new experiences change man.
Digital algorithmic art is a new means. My 2400 lines could be performed manually, but at that speed and precision?
On another note, I’m a fan of Schoenberg not because I can analyse dodecaphony but because it sounds good to me.
Lastly, someone mentioned IanniX to me some time back. I haven’t figured out how to work with visuals.
Currently, I’m toying with rhythmic melody and harmony alongside pitch melody and harmony, using a tick (the 16th) as the finest resolvable unit.
What the above comprises is a selection of melodic “chronomorphic” ideas (the pattern of each beat) defined within a harmonic space-time system (the overarching pitch relationships and rhythm relationships).
I’m not sure what’s going on with Iannix, but Hydra is live coding visuals that works straight out of the box. The dev, though, hasn’t provided a way to interface it with MIDI, but maybe that will come someday. It does look really good, not so much as a work of art, but decorative, and a good accompaniment to music. I guess a skilled VJ could make more of a show with it.
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Eli.exe has stopped responding. Please reboot Eliverse.
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LOL, all this theory! At the end of the day, it must sound good, otherwise it’s all for naught.
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I can recommend hydra, I’ve been using it for a while. Mostly to add some visuals to audio tracks so as to put them on the internet, but also with a view to doing some live visuals, when we get out of here. There is some MIDI interaction (CC’s only I think) labelled as experimental. I did try it breifly but it didn’t work right of the bat so I didn’t persue it.
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I have an idea to implement in the next rewrite of my program. It stems from this comparison: if I’m not mistaken, the root of a chord is a sort of fundamental frequency (I realise we’re not talking about the harmonic series). Anyways, if we slow down that “fundamental” frequency to the realm of rhythm, could that frequency be taken as a basis for polyrhythm? Instead of a chord root, I call this a “rhyme”. In my program, all rhythms will be composite with this rhyme.
Here’s what that looks like:
In the above, the rhyme has a beat count of 4. The rhythm at the top is its composite with a beat count of 3. The rhythm at the bottom “rhymes with” 5. I find the result to be more unified than onefold polyrhythm.
Just as there are chord progressions, I’m going to implement “rhyme progressions” so that the musical system evolves.
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This looks great - no idea what’s going on - but I’ll eagerly await the results.
Not sure this is helping with @Eli 's reboot though 
I know a bit of theory, and I was attending a jazz class before lockdown. It was fine up to a point, and I loved the playing, but at some stage I realised I could know longer understand what the teacher’s analysis was on about. Rather similar to a recurring dream I have where I’m taking a maths exam and the questions make no sense at all 
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Yes, this is totally possible - all chords are made up of notes and they have their own patterns of peaks interacting with one another. What you might find though is that things change a lot with phase (ie which subdivision of the bar each component starts on). The rhythms it generates might also be a little too complex for most listeners - that was my feeling from some of the examples that Neely used.
My understanding is that rhythm perception, while sharing some underlying similarities with pitch and harmony perception, is still different in terms of how we hear and understand it.
I couldn’t resist adding this cool example of polyrhythms making an awesome groove:
Listen to that first (at least the first minute, although stick around for piano solo if you have time) and try to figure out whats going on.
When you’ve given up, there’s an awesome explainer video here to help you
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I think I’ve found a listenable compromise with Euclidean rhythms, though I haven’t tried it with more extreme values.
rotate adds a whole other dimension to things that I haven’t explored in my theory 
To say nothing of rhythm offset! Another dimension.
Mhm! Stockhausen observed how pitch, rhythm, and form are three distinct time scales of perception. Continuously changing between those time scales is mind-bending!
I pieced this together after watching the explainer! (I try to use a language-neutral syntax.)
use_bpm(12);
live_loop(:pinzin_kinzin_beat) do
sample(:drum_bass_soft) if spread(7, 32, rotate: 1).tick;
sample(:drum_cymbal_closed) if spread(8, 32).look;
sample(:drum_snare_soft) if spread(4, 32)[look + 4];
wait(1/32.to_f);
end
That was really fun actually! Great song 
I can only ever understand theory on paper (if at all) 
Listening, it just sounds good and different or it doesn’t 
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Wow that explanation video is great! Cool piece.
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Oh, Avishai Cohen! You’re a man of taste!
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