> " In other words, the Fibonacci numbers mod 7 are cyclic after 16 numbers and that cycle is used for the bottom notes."
To my naive view, that's really interesting. Whereas duodecimal number systems fell out of cultural fashion around the world, we have settled on one in the chromatic scale of western music and the way we often develop an intuition for the proportions and symmetries in music is using a mod/base-12 system. I have no real insight to add other than to just appreciate the coincidence that the periodicity of this cycle sounds nice. The idea that music can be discovered is just appealing.
The magic of having 12 equally spaced notes in the chromatic scale is that specific important intervals are closely approximated by ratios of small integers, and you can transpose a piece to any key (this doesn't work if you tune to exact ratios). Small, integer ratios supposedly mean the frequencies will sound similar or related because the peaks and troughs sync up every 2-5 cycles.
I say supposedly because studies have found remote tribes, played the, the same note, but in different octaves, and asked if they're the same of different. A lot of people couldn't associate that they're the same, which would imply these ratios aren't actually important. I have serious doubts about these studies, though; from a signal processing perspective, 440 Hz and 880 Hz are related at a physical level, so it's surprising that recognizing this similarity isn't almost innate.
Peaks don't have to necessary align in frequency multiples (harmonics). You could tweak the phase offset between them so that peaks don't align at all and it will still sound same. Your ear it is sensing and analyzing in frequency domain and not in time domain.
I think there is another reason we can identify when frequency ratios are integers in complex sound. This ability gives brain power to "group together" frequencies so it can assign them to the same sound source. This seems like great optimization for computation and bandwidth. It would make sense for hardwired part of brain to make preliminary analysis of sound.
For example, you hear an animal sound in a forest. You want to know the direction, want to estimate the size of animal that made the sound, and you'd also want a small 'fingerprint' of a sound, so you can recall from memories if you heard it before. This has to happen as quickly because you might be bombarded with multiple sounds per second. All of them are sounding over each other, so your ear & brain had to develop ability to group them into separate sources and analyze them independently.
Most sound sources are coherent, which means they will contain integer multiples of some base frequencies. This goes for picked string, struck anvil, howling wind, but also for sounds made by animal throat. When a bird sings it can be approximated with handful of sine waves whose pitches slide up and down in sync. Therefore it makes sense for ear to communicate only the spectral analysis to brain, and discard other sensed data. So integer ratios are good indication that heard frequency components can be grouped together and assigned to single source of sound. Music exploits this mechanism by playing same or related pitches across different instruments.
This has been done many times before. Composers who explicitly based their rhythms on the Fibonacci sequence include Per Nørgård and Sofia Gubaidulina. (While Bartók is sometimes said to have done so, evidence is lacking.) Brian Ferneyhough makes use of prime numbers in one scene from his opera Shadowtime.
In the popular-music world, BT wrote a Fibonacci-sequence tune that was a club hit in 1999.
I would say the main part of the audio file is the seeming regularity of the prime numbers. I just added the Fibonacci numbers to have a "leading" base line.
The song Lateralus by the band Tool, is all based on the Fibonacci sequence, and it's a masterpiece. Here goes the link if someone is curious:
https://www.youtube.com/watch?v=Y7JG63IuaWs
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[ 1170 ms ] story [ 1082 ms ] threadTo my naive view, that's really interesting. Whereas duodecimal number systems fell out of cultural fashion around the world, we have settled on one in the chromatic scale of western music and the way we often develop an intuition for the proportions and symmetries in music is using a mod/base-12 system. I have no real insight to add other than to just appreciate the coincidence that the periodicity of this cycle sounds nice. The idea that music can be discovered is just appealing.
I say supposedly because studies have found remote tribes, played the, the same note, but in different octaves, and asked if they're the same of different. A lot of people couldn't associate that they're the same, which would imply these ratios aren't actually important. I have serious doubts about these studies, though; from a signal processing perspective, 440 Hz and 880 Hz are related at a physical level, so it's surprising that recognizing this similarity isn't almost innate.
I think there is another reason we can identify when frequency ratios are integers in complex sound. This ability gives brain power to "group together" frequencies so it can assign them to the same sound source. This seems like great optimization for computation and bandwidth. It would make sense for hardwired part of brain to make preliminary analysis of sound.
For example, you hear an animal sound in a forest. You want to know the direction, want to estimate the size of animal that made the sound, and you'd also want a small 'fingerprint' of a sound, so you can recall from memories if you heard it before. This has to happen as quickly because you might be bombarded with multiple sounds per second. All of them are sounding over each other, so your ear & brain had to develop ability to group them into separate sources and analyze them independently.
Most sound sources are coherent, which means they will contain integer multiples of some base frequencies. This goes for picked string, struck anvil, howling wind, but also for sounds made by animal throat. When a bird sings it can be approximated with handful of sine waves whose pitches slide up and down in sync. Therefore it makes sense for ear to communicate only the spectral analysis to brain, and discard other sensed data. So integer ratios are good indication that heard frequency components can be grouped together and assigned to single source of sound. Music exploits this mechanism by playing same or related pitches across different instruments.
In the popular-music world, BT wrote a Fibonacci-sequence tune that was a club hit in 1999.
https://www.youtube.com/watch?v=ybgsUo5kcyM
Enjoy :)