I am reading Alphabet, a book of poetry by Inger Christensen where she used the Fibonacci series to generate the structure of the verses. I wonder if this could somehow be done with sounds/rhythms.
I love the visuals that you generated from the polyrhythms. And you raise some interesting questions around the musical capability of machines in an effort toward “perfection.” I work with sound and so often am trying to get the digital stuff to perform less “perfectly,” more analog, but for sure you have a point that (so) many things are beyond human performance capability, and in this respect we have a lot to learn. Anyway, thanks for the words and great images.
Thanks for your kind words Trees & Booms. I am glad you like the visuals! It's not really so much that I consider digital images/music perfect, as that they open the door to new possibilities. I find attempting to explore the previously 'unattainable' much more interesting than trying to replicate the already attained. Unfortunately, there is an attempt to blur the two at the moment, whereas I think music thrives when they are more distinct.
You're welcome. And thank you for clarifying - I understand your point about exploring the previously unattainable. That's a noble effort. The visuals are terrific and work great as your album art as well.
1. I'm not sure I understand how these images represent polyrhythms exactly. Sure, you have an algorithm that maps a set of geometrical relationships. What does that have to do with "polyrhythms"? I have some background on this subject and the specifics are of interest to me. A better explanation would be helpful.
2. I'm curious about the program you developed for this visualization and whether or not it could run a similar simulation. Let's say you had three "arms" of a clock running at different rates in the same direction, but eventually converging across some period. You could think about this way: what if in a period of time, one arm made (5) full rotations, another (3) and the last one (2). Can you represent: a) where they coincide; and b) how many coincidences there are for each arm?
Hi Jeffrey, thanks for the questions. It's good to know what to clarify.
In a sense, you answered your first question. What else is a polyrhythm but a set of geometrical relationships? We might not be familiar with representing them that way, but that is an accurate way of depicting them.
If we use your clock hands: if the hands of the clock go around at an even speed (clockwise or anticlockwise - it doesn't matter as the patterns are mirror symmetrical) then as they hit each line they will make a sound. If you have divided the circle by 1/1, 1/2s, 1/3s, 1/4s etc then you will have a polyrhythm of 1:2:3:4.
The convergence of polyrhythms can be worked out quite simply by using the lowest common multiple calculations (there are calculators online that do this). So for 1:2:3:4:6, it would be 12 (perhaps why it's so popular as a cycle). With more numbers, the convergence takes much longer e.g., 1:2:3:4:5:6:7 = 420.
In terms of where the polyrhythms within polyrhythms coincide, it's the same maths. Many small convergences exist within a pattern working towards a total convergence. These convergences are represented in the circular images by darker lines. You can also detect them vertically by looking through the original image.
In general, all of this is moving towards animation for me. I want to work on a big system that will put all this into motion with sound. I hope that it will make sense at a very intuitive level,
I would push back on your claim that rhythms are geometrical by nature. Geometry is *spatial*, whereas rhythm is *temporal*. In space, everything is related to everything else simultaneously and commutatively, whereas with time, convergence is only ever partial, and relationships are generally asymmetrical and transformative. My background on the subject of rhythm comes from Lerdahl and Jackendoff's system for scanning rhythms, which is componential -- meter, grouping and prolongation are organized under different parameters and can be scanned both separately and in conjunction. The representations you're suggesting don't lend themselves very well to the detail captured in L&J's representations, much less something more complex, like polymeter, so I have to express some skepticism. I think too often, as versed as we are, culturally, in spatial thinking, we make the mistake of reducing temporal thinking to spatial metaphors, rather than dealing with time on its own terms.
My question on the "clock hands" is: what if the arms are moving at different speeds? Can you track the convergences for each hand in its own tally?
To be clear, I am not suggesting polyrhythms have an essential geometric nature, but rather that they can be accurately depicted geometrically, just as they can be accurately depicted using standard notation or numbers.
You make a distinction between the spatial and the temporal, but where does time exist if not in the spatial? If you have two sine waves travelling through space at different frequencies and you mark their zero points it will create a pattern. If you have 150 sine waves of integer frequencies from 1-150 and mark their 0 points you would get something that looks like the 'Visualisation of 150 polyrhythms' that I shared in the post.
Lerdahl and Jackendoff's system is quite different to what I am trying to achieve, though you could certainly use the methods I used to depict polymeters. The images I created are not intended as tools for standard musical analysis but to explore highly complex polyrhythms, far beyond the playing capabilities of musicians. They provide a general map of these rhythms which would be impossible to accurately understand or compute mentally. That is a very important point. They are impossible to make sense of at the level of detail that playing requires.
In terms of tracking convergences in the example you give. The 2&3 would converge at 0°. The 3&5 would converge at 0° and 180°. The 2&5 would converge at 0° 240° and 120°.
In general, I do not believe in a universal, objective way of depicting rhythm as every system has strengths and limitations. I aim to create visualisations that are beautiful and intriguing objects in themselves, as well as musically interesting patterns that provide a large overview of highly complex polyrhythms that can only be realised by machines.
These are complementary modes of physical organization. My point is that, "scientifically", we have had difficulty intellectualizing time as organized in terms of its own principles, and rely too much on spatial *metaphors* that don't withstand scrutiny as first principles.
I'm not sure I buy your assumption here that a sine wave is tantamount to a "rhythm". A sine wave could map onto a process that is rhythmic under the right conditions, but that only gets you part of the way to a more general claim about what rhythms are and how they work. I'd like to hear a better account of why that would be the case.
L&J's work is useful not just because it's a tool of analysis, but because, explicitly or implicitly, it works from a theory of what a rhythm is, or what a rhythm consists in. Your representations are interesting, but they're not really descriptive. Any mapping from what you've created to a piece of music seems arbitrary and somewhat limited in scope to me; good descriptive representations are both formally apposite and phenomenologically extensive. For example, how would one identify syncopation -- where a prominent voiced event is locally juxtaposed to a weak gestural event, or vice versa? Syncopation is a VERY important rhythmic phenomenon in music (albeit moreso in some traditions than others). I'm less concerned here with "objective" or "universal" representations, and more with representations that cover more observational ground and are not philosophically misleading.
I think your following two statements get to the heart of why we see things differently.
'Any mapping from what you've created to a piece of music seems arbitrary and somewhat limited in scope to me;'
I would agree with the limited scope for traditional music, however, it presumes that I intended to map these onto traditional music, which has never been the case.
'good descriptive representations are both formally apposite and phenomenologically extensive.'
I don't share this value judgement. It's not that I object to systems that do this - I don't - but I don't believe it is some universal gold standard. For starters, such a system roots music excessively in past practices. What I am describing are ideas and symbols that help us direct machines in music-making. Ideas such as complexity theory and system theory are more helpful to me here than a system that maps neatly onto a madrigal. As the computer is new in music, I believe it will need some new symbolic representations.
For example, you place a stress on the importance of syncopation. If you have 750 rhythms simultaneously what does syncopation even mean? There is no beat. There are no bars. There's no '1'... Or if you want to think in terms of 1 there are 750 '1s'. The first track The Code 'Discovering Quasar' demonstrates this (with more than 750 voices).
I disagree with the idea that this approach is somehow philosophically misleading. The images are accurate representations of complex polyrhythms. If you object to spatial metaphors used in symbolic representations of music then there is a lot to object to. There are no high or low notes, does that make staves misleading? Despite objecting to spatial metaphors you used the metaphor of clock hands to describe a rhythm. What method truly avoids this? And given that the world's best physicists can't agree on what time is, how can one represent it 'in terms of its own principles'?
The images I presented here help me navigate composing music with massive polyrhythms and have direct experience of their utility. I'm sure there are many things they offer that I have not explored. I am planning to build more advanced systems for this task, as there are limitations on what is possible with still images.
Lastly, keep in mind this post is part of a series called 'open studio'. The idea is to show certain processes and ideas I'm exploring and provide a more open approach to creativity and art. What I'm not doing, however, is trying to present some Chomskian universal grammar of rhythm.
so fun to read these explorations, I feel I’m playing around with many related raw materials (generative, polyrhythmic, human-made machine, the possible imageries of time-bound work) but ending up touring vastly different landscapes
I am reading Alphabet, a book of poetry by Inger Christensen where she used the Fibonacci series to generate the structure of the verses. I wonder if this could somehow be done with sounds/rhythms.
Interesting sounding book. It could definitely be done. If I get a moment I’ll try. Good idea Su!
I want to hear what you do with this idea Dom!
No sound, yet, but I've updated the post with an image using the Fibonacci series.
I love the visuals that you generated from the polyrhythms. And you raise some interesting questions around the musical capability of machines in an effort toward “perfection.” I work with sound and so often am trying to get the digital stuff to perform less “perfectly,” more analog, but for sure you have a point that (so) many things are beyond human performance capability, and in this respect we have a lot to learn. Anyway, thanks for the words and great images.
Thanks for your kind words Trees & Booms. I am glad you like the visuals! It's not really so much that I consider digital images/music perfect, as that they open the door to new possibilities. I find attempting to explore the previously 'unattainable' much more interesting than trying to replicate the already attained. Unfortunately, there is an attempt to blur the two at the moment, whereas I think music thrives when they are more distinct.
You're welcome. And thank you for clarifying - I understand your point about exploring the previously unattainable. That's a noble effort. The visuals are terrific and work great as your album art as well.
Thank you so much. Comments like yours make the effort all worth it.
I have two questions.
1. I'm not sure I understand how these images represent polyrhythms exactly. Sure, you have an algorithm that maps a set of geometrical relationships. What does that have to do with "polyrhythms"? I have some background on this subject and the specifics are of interest to me. A better explanation would be helpful.
2. I'm curious about the program you developed for this visualization and whether or not it could run a similar simulation. Let's say you had three "arms" of a clock running at different rates in the same direction, but eventually converging across some period. You could think about this way: what if in a period of time, one arm made (5) full rotations, another (3) and the last one (2). Can you represent: a) where they coincide; and b) how many coincidences there are for each arm?
Hi Jeffrey, thanks for the questions. It's good to know what to clarify.
In a sense, you answered your first question. What else is a polyrhythm but a set of geometrical relationships? We might not be familiar with representing them that way, but that is an accurate way of depicting them.
If we use your clock hands: if the hands of the clock go around at an even speed (clockwise or anticlockwise - it doesn't matter as the patterns are mirror symmetrical) then as they hit each line they will make a sound. If you have divided the circle by 1/1, 1/2s, 1/3s, 1/4s etc then you will have a polyrhythm of 1:2:3:4.
The convergence of polyrhythms can be worked out quite simply by using the lowest common multiple calculations (there are calculators online that do this). So for 1:2:3:4:6, it would be 12 (perhaps why it's so popular as a cycle). With more numbers, the convergence takes much longer e.g., 1:2:3:4:5:6:7 = 420.
In terms of where the polyrhythms within polyrhythms coincide, it's the same maths. Many small convergences exist within a pattern working towards a total convergence. These convergences are represented in the circular images by darker lines. You can also detect them vertically by looking through the original image.
In general, all of this is moving towards animation for me. I want to work on a big system that will put all this into motion with sound. I hope that it will make sense at a very intuitive level,
Does that clarify things?
I would push back on your claim that rhythms are geometrical by nature. Geometry is *spatial*, whereas rhythm is *temporal*. In space, everything is related to everything else simultaneously and commutatively, whereas with time, convergence is only ever partial, and relationships are generally asymmetrical and transformative. My background on the subject of rhythm comes from Lerdahl and Jackendoff's system for scanning rhythms, which is componential -- meter, grouping and prolongation are organized under different parameters and can be scanned both separately and in conjunction. The representations you're suggesting don't lend themselves very well to the detail captured in L&J's representations, much less something more complex, like polymeter, so I have to express some skepticism. I think too often, as versed as we are, culturally, in spatial thinking, we make the mistake of reducing temporal thinking to spatial metaphors, rather than dealing with time on its own terms.
My question on the "clock hands" is: what if the arms are moving at different speeds? Can you track the convergences for each hand in its own tally?
Hi Jeffrey,
Thanks for sharing your thoughts.
To be clear, I am not suggesting polyrhythms have an essential geometric nature, but rather that they can be accurately depicted geometrically, just as they can be accurately depicted using standard notation or numbers.
You make a distinction between the spatial and the temporal, but where does time exist if not in the spatial? If you have two sine waves travelling through space at different frequencies and you mark their zero points it will create a pattern. If you have 150 sine waves of integer frequencies from 1-150 and mark their 0 points you would get something that looks like the 'Visualisation of 150 polyrhythms' that I shared in the post.
Lerdahl and Jackendoff's system is quite different to what I am trying to achieve, though you could certainly use the methods I used to depict polymeters. The images I created are not intended as tools for standard musical analysis but to explore highly complex polyrhythms, far beyond the playing capabilities of musicians. They provide a general map of these rhythms which would be impossible to accurately understand or compute mentally. That is a very important point. They are impossible to make sense of at the level of detail that playing requires.
In terms of tracking convergences in the example you give. The 2&3 would converge at 0°. The 3&5 would converge at 0° and 180°. The 2&5 would converge at 0° 240° and 120°.
In general, I do not believe in a universal, objective way of depicting rhythm as every system has strengths and limitations. I aim to create visualisations that are beautiful and intriguing objects in themselves, as well as musically interesting patterns that provide a large overview of highly complex polyrhythms that can only be realised by machines.
How can space exist if it exists in no time?
These are complementary modes of physical organization. My point is that, "scientifically", we have had difficulty intellectualizing time as organized in terms of its own principles, and rely too much on spatial *metaphors* that don't withstand scrutiny as first principles.
I'm not sure I buy your assumption here that a sine wave is tantamount to a "rhythm". A sine wave could map onto a process that is rhythmic under the right conditions, but that only gets you part of the way to a more general claim about what rhythms are and how they work. I'd like to hear a better account of why that would be the case.
L&J's work is useful not just because it's a tool of analysis, but because, explicitly or implicitly, it works from a theory of what a rhythm is, or what a rhythm consists in. Your representations are interesting, but they're not really descriptive. Any mapping from what you've created to a piece of music seems arbitrary and somewhat limited in scope to me; good descriptive representations are both formally apposite and phenomenologically extensive. For example, how would one identify syncopation -- where a prominent voiced event is locally juxtaposed to a weak gestural event, or vice versa? Syncopation is a VERY important rhythmic phenomenon in music (albeit moreso in some traditions than others). I'm less concerned here with "objective" or "universal" representations, and more with representations that cover more observational ground and are not philosophically misleading.
Hi Jeffery,
I think your following two statements get to the heart of why we see things differently.
'Any mapping from what you've created to a piece of music seems arbitrary and somewhat limited in scope to me;'
I would agree with the limited scope for traditional music, however, it presumes that I intended to map these onto traditional music, which has never been the case.
'good descriptive representations are both formally apposite and phenomenologically extensive.'
I don't share this value judgement. It's not that I object to systems that do this - I don't - but I don't believe it is some universal gold standard. For starters, such a system roots music excessively in past practices. What I am describing are ideas and symbols that help us direct machines in music-making. Ideas such as complexity theory and system theory are more helpful to me here than a system that maps neatly onto a madrigal. As the computer is new in music, I believe it will need some new symbolic representations.
For example, you place a stress on the importance of syncopation. If you have 750 rhythms simultaneously what does syncopation even mean? There is no beat. There are no bars. There's no '1'... Or if you want to think in terms of 1 there are 750 '1s'. The first track The Code 'Discovering Quasar' demonstrates this (with more than 750 voices).
I disagree with the idea that this approach is somehow philosophically misleading. The images are accurate representations of complex polyrhythms. If you object to spatial metaphors used in symbolic representations of music then there is a lot to object to. There are no high or low notes, does that make staves misleading? Despite objecting to spatial metaphors you used the metaphor of clock hands to describe a rhythm. What method truly avoids this? And given that the world's best physicists can't agree on what time is, how can one represent it 'in terms of its own principles'?
The images I presented here help me navigate composing music with massive polyrhythms and have direct experience of their utility. I'm sure there are many things they offer that I have not explored. I am planning to build more advanced systems for this task, as there are limitations on what is possible with still images.
Lastly, keep in mind this post is part of a series called 'open studio'. The idea is to show certain processes and ideas I'm exploring and provide a more open approach to creativity and art. What I'm not doing, however, is trying to present some Chomskian universal grammar of rhythm.
so fun to read these explorations, I feel I’m playing around with many related raw materials (generative, polyrhythmic, human-made machine, the possible imageries of time-bound work) but ending up touring vastly different landscapes
Sounds interesting Eric. It’s great that through such methods people can arrive at quite different places.
Hola , Fascinante Ensayo. La Portada De Tú Álbum Es Magnífica , Al Igual Que La Música. Un Saludo.
Gracias Una, muchas gracias por las amables palabras sobre el álbum! Un saludo!