2022-11-25

model

instrument

cluster ...

envelope ...

- sine(amp, frq, phs) / filtered_noise(amp, frq_start, frq_end) ...

notes

partials summed make complex wave shapes

using clusters to build instruments works, because even instruments are made of some distinguishable objects

partials in a cluster should be similar enough to each other so they are not recognised as separate sounds, but different enough that they arent recognized as mere amplification

- "The relative amplitudes (strengths) of the various harmonics primarily determine the timbre of different instruments and sounds, though onset transients, formants, noises, and inharmonicities also play a role."
- harmonics are multiples of the fundamental (pitch) frequency. the fundamental is the frequency at which the entire wave vibrates
- multiples and added divisions can be used
- variables to select a partial: fundamental + divisions + shift
- irrational frequencies have their period never restart at the same time as the fundamental again
- mixing odd and even divisions is less harmonic
examples of subdivisions, times frequency:

- 1/1, 1/1 + 1/2, 2/1
- 1/1, 1/1 + 1/3, 1/1 + 2/3, 2/1
- 1/1, 1/1 + 1/4, 1/1 + 2/4, 1/1 + 3/4, 2/1

relations to music theory

- an octave is a doubling in frequency
- a common scale divides octaves into twelve tones. without just intonation, these intervals are historically not equally spaced
- a perfect fifth is a combination that is periodic on two periods of the lower note, a 2:3 frequency relation. the beginning of the second repetition of the fundamental is at the center of the second repetition of the other tone

some possible partial frequency relations

equidistant

increasing distance

decreasing distance

varying distance

sparse

dense

- statistics are a digest/abstraction of the actual details of a pattern
- they can be used to compare or generate patterns (monte carlo method)
- statistics can be a target for a generator
- it can also be used as an analysis step to make parts of patterns more similar or to blend them together
- note that pointwise interpolation is another method to create similar patterns
- statistics on the differences between values can also be useful
some interesting statistical methods

- standard deviation: the variation from the mean (variance) with reduced bias to extremes
- arithmetic mean: one share of the total sum equally distributed. the point where the sum of smaller values matches the sum of larger values
- median: the center between higher and lower values. the point where the count of smaller values matches the count of larger values
- range: the difference between the maximum and minimum
- minimum: the minimum value in the dataset
- maximum: the maximum value in the dataset
kurtosis

- describes the distribution of numbers. for example, if values are spread out and more equally likely, or if there are peaks in the distribution, which would mean that a range of values appears in the dataset more often
- mean((x - mean(data)) ** 4) / (mean((x - mean(data)) ** 2) ** 2)

skewness

- describes the distribution of numbers. gives an indication about if there are more low or more high numbers
- mean((x - mean(data)) ** 3) / (mean((x - mean(data)) ** 2) ** 3/2)

center of mass

- the distribution of mass is balanced around the center of mass, and the average of the weighted position coordinates of the distributed mass defines its coordinates.
- sum(n x(n)) / sum(x(n))

the number of unique subsequences as a potential statistic

the count of repetitions of overlapping subsequences of length 0..n

can find the subpattern lengths with the highest proportion of unique subpatterns relative to the possible unique subpatterns

examples

low: 11111 112112

high: 12345 112212

- autocorrelation gives a degree of difference with large local difference being pronounced to differentiate it from many small differences.
- correlation is -1 to 1
- correlation only tests linear dependence. (1 2 3) and (4 5 6) are 1
- mean absolute difference goes to infinity

the choice and availability of all sine configuration details before synthesis allows perfect analysis/knowledge unachivable by sound analyis with the fast fourier transform and other methods

- this freedom is not necessarily lost even when samples are used as a basis to choose parameters, unless the goal is an exact recreation of the source material

even if sines can therotically represent triangles (with infinite sines), a triangle is the ideal (even if impossible) shape

- summing sines can match the limits of reproducibility that exist anyway
- using the ideal shapes might lead to aliasing

data types

- continuous: sample -1..1; amplitude: 0..1
- discrete: sampling-time

- one can imagine a spectrum between harmonicity and noise, where harmonic sounds dissolve into noise
- real human-like complexity in patterns is a variety of, and layers of, modifications over time. it is not the number of unique subsequences
- panning effects can be left to right then again left to right, or left to right then back from right to left
- to share parameter values, sound generators do not need to somehow trigger other sound generators (inefficient, possibly cylic). instead the values can be shared from a parent context
- partials can be more causative/dependent/similar or more correlative/independent/dissimilar