Giants Steps : Fractal Structure in Coltrane’s Solo | Harlan Brothers
Music as Data Series
The distribution of interval-related data in Coltrane’s famous solo is a wonderful example of a power-law in action
“When you begin to see the possibilities of music, you desire to do something good for people, to help humanity free itself from its hang-ups.”
— John Coltrane (1926–1967)
John Coltrane’s improvisation on his composition Giant Steps, first recorded in 1959, stands out as one of the seminal recordings in jazz history. There are 26 chord changes that occur over the course of a 16-bar song form. During that relatively short time, the tonal center changes between 3 keys (B, G, and E) a total of 10 times. Combined with a blistering tempo of approximately 291 beats per minute, the ability to smoothly improvise over it is widely considered a benchmark that separates the top jazz players from everyone else.
The fact is, Coltrane’s solo is beautifully melodic and fluidly phrased. It is also a wonderful manifestation of a power-law relation with respect to its distribution of interval-related data. Below is an animation that offers a dynamic window into the structure of the solo. The data for each instrument is represented as a time series with a height corresponding the note value and a length corresponding to note duration. The baselines of the ranges for each instrument, relative to each other, are adjusted for visual clarity.
Watching the animation, one can get an intuitive sense of the structure we want to measure by looking at the distribution of the vertical spacings between the orange saxophone notes. Overall, there are lots of small jumps and fewer medium-sized jumps. These are all punctuated by relatively few large jumps. This relationship between count and size can be accurately captured by a power-law.
Power-laws
Power-laws are a fundamental property in the study of fractal geometry. They describe a relationship between two variables such that one is proportional to a power of the other. For example, the allometry of animal metabolic rates can be determined by the relation metabolic rate = k⋅(weight)^(3/4). Power-laws underly a wide variety of phenomena in the natural and social sciences from the distribution of galaxies to the distribution of wealth.
They can also arise in music with respect to many characteristics like pitch, interval, duration, rhythmic variation, and a form of fractal structure called motivic scaling wherein a melodic or rhythmic motif is repeated simultaneously at different time scales [1].
Fractal Music
A quick search for the term fractal music offers a hodgepodge of descriptions. In most cases, the results refer simply to music that was generated by mapping a fractal image to some combination of pitches, durations, dynamics, and instrument sounds. This type of music can be compelling, but just mapping pixels to musical sounds does not ensure that the result possess fractal characteristics. Any such assumption would be akin expecting a poem by Henry Wadsworth Longfellow to rhyme when translated into Greek.
Benoit Mandelbrot had a strong intuition that music could be fractal in a measurable sense. Given that fractal geometry is, first and foremost, geometry, it makes sense for us to reserve the term fractal music for music that possesses some clearly measurable scaling property.
It’s important to note that fractal music is not a modern creation. Motivic scaling made an appearance at least five centuries ago with the work of Franco-Flemish composer, Josquin des Prez. We can now recognize that work as being among the earliest known examples of fractal music.
As in the case of Giant Steps, the fractal characteristics of music often appear as a statistically self-similar distribution. In such cases, we are looking for a power-law in the context of a heterogeneous collection of elements. This means that wherever we zoom into to the collection, we find roughly the same distribution of sizes.
How to Detect a Power-law
For a heterogeneous series of melodic intervals of different sizes, {s₁ , s₂ , s₃ , …}, we can count the number of intervals of each size {N₁ , N₂ , N₃ , …}, where Nₖ corresponds to size sₖ. The sequence possess melodic interval scaling if, when we graph a log-log plot of the Nₖ against sₖ, the points fall along a straight line. Provided the fit is reasonably good, the slope of that line can be interpreted as the dimension d of the set of elements. This is because taking the log of a power-law relation transforms it into a linear equation:
In Equation 1, if we let y=Log(N) and x=Log(1/s), then what we have is the equation of a line, y=mx+b, where d=m and b=0. For details regarding this type of analysis and its application to music, see [1, 2].
The Data
For the saxophone solo, I used pitch data from The Jazzomat Research Project [3]. Here is the distribution of intervals where the size of each interval is equal to 2^(k/12) for an interval of k semitones:
The value 2^(k/12) comes from the physics of the equal temperament scale commonly used in Western music. Below is a plot of the interval data. It yields a dimensions of d≈8.3:
We can use a data smoothing technique called binning to, in a sense, loosen our focus and, in this case, obtain a slightly better fit. We’ll place the intervals into equal-size bins where each bin contains intervals of roughly the same size. The representative size of a bin is typically assigned using either the lowest, highest or mean values with the choice applied consistently across all bins. To be clear, while binning can help to reveal the presence of a power-law, it cannot produce a power-law where none exists.
Here is the distribution of intervals where each bin is three semitones wide, along with a plot of the data which yields a dimension of d≈8:
The R² value is marginally better and, by general scientific standards, is certainly respectable. However, we can see that the strength of the relationship between size and count is still not as consistent as we’d like as evidenced by the counts in bins of representative size 8 and 11. Those counts are roughy equal and, in fact, increase rather than decrease.
On the other hand, when we look at the way the intervals change, the picture is clearer. We refer to the differences between intervals as melodic moments. If the interval is the first difference of pitch, then the melodic moment is the second difference. Here is the distribution of raw moment data:
Note that the smallest moment size is 0 semitones which corresponds the case when two consecutive intervals have the same size. Here are plots of the raw data and the binned data with a bin width of 3 semitones:
The raw and binned data show excellent agreement with respective fractal dimensions of d≈8.1 and d≈8.3. With R² values of .930 and .998, the melodic moment distribution shows unequivocal evidence of fractal structure.
Closing Thoughts
When analyzing music in this fashion it’s important to keep in mind that, generally speaking, the datasets we’re exploring are small by scientific standards. We therefore work under the assumption that there is a strong relation between the character of a melody and its underlying intervallic distribution. For instance, a scale-wise melody sounds very different from one containing large leaps. In this sense, where interval-based scaling exists, we can assert that a piece that continued in the same melodic fashion would also continue to exhibit a self-similar distribution of elements across several scales of measurement.
Over the centuries, it seems that the greatest music has a sense of transparency and timelessness about it, whether it be Bach or the Beatles. I am certainly not alone in the perception that John Coltrane’s music possesses those same qualities. To the extent that his music also reflected the world around him, it should not surprise us that, in some sense, it expresses the fractal nature of the world as he experienced it.
Music as Data Series
The distribution of interval-related data in Coltrane’s famous solo is a wonderful example of a power-law in action
“When you begin to see the possibilities of music, you desire to do something good for people, to help humanity free itself from its hang-ups.”
— John Coltrane (1926–1967)
John Coltrane’s improvisation on his composition Giant Steps, first recorded in 1959, stands out as one of the seminal recordings in jazz history. There are 26 chord changes that occur over the course of a 16-bar song form. During that relatively short time, the tonal center changes between 3 keys (B, G, and E) a total of 10 times. Combined with a blistering tempo of approximately 291 beats per minute, the ability to smoothly improvise over it is widely considered a benchmark that separates the top jazz players from everyone else.
The fact is, Coltrane’s solo is beautifully melodic and fluidly phrased. It is also a wonderful manifestation of a power-law relation with respect to its distribution of interval-related data. Below is an animation that offers a dynamic window into the structure of the solo. The data for each instrument is represented as a time series with a height corresponding the note value and a length corresponding to note duration. The baselines of the ranges for each instrument, relative to each other, are adjusted for visual clarity.
Watching the animation, one can get an intuitive sense of the structure we want to measure by looking at the distribution of the vertical spacings between the orange saxophone notes. Overall, there are lots of small jumps and fewer medium-sized jumps. These are all punctuated by relatively few large jumps. This relationship between count and size can be accurately captured by a power-law.
Power-laws
Power-laws are a fundamental property in the study of fractal geometry. They describe a relationship between two variables such that one is proportional to a power of the other. For example, the allometry of animal metabolic rates can be determined by the relation metabolic rate = k⋅(weight)^(3/4). Power-laws underly a wide variety of phenomena in the natural and social sciences from the distribution of galaxies to the distribution of wealth.
They can also arise in music with respect to many characteristics like pitch, interval, duration, rhythmic variation, and a form of fractal structure called motivic scaling wherein a melodic or rhythmic motif is repeated simultaneously at different time scales [1].
Fractal Music
A quick search for the term fractal music offers a hodgepodge of descriptions. In most cases, the results refer simply to music that was generated by mapping a fractal image to some combination of pitches, durations, dynamics, and instrument sounds. This type of music can be compelling, but just mapping pixels to musical sounds does not ensure that the result possess fractal characteristics. Any such assumption would be akin expecting a poem by Henry Wadsworth Longfellow to rhyme when translated into Greek.
Benoit Mandelbrot had a strong intuition that music could be fractal in a measurable sense. Given that fractal geometry is, first and foremost, geometry, it makes sense for us to reserve the term fractal music for music that possesses some clearly measurable scaling property.
It’s important to note that fractal music is not a modern creation. Motivic scaling made an appearance at least five centuries ago with the work of Franco-Flemish composer, Josquin des Prez. We can now recognize that work as being among the earliest known examples of fractal music.
As in the case of Giant Steps, the fractal characteristics of music often appear as a statistically self-similar distribution. In such cases, we are looking for a power-law in the context of a heterogeneous collection of elements. This means that wherever we zoom into to the collection, we find roughly the same distribution of sizes.
How to Detect a Power-law
For a heterogeneous series of melodic intervals of different sizes, {s₁ , s₂ , s₃ , …}, we can count the number of intervals of each size {N₁ , N₂ , N₃ , …}, where Nₖ corresponds to size sₖ. The sequence possess melodic interval scaling if, when we graph a log-log plot of the Nₖ against sₖ, the points fall along a straight line. Provided the fit is reasonably good, the slope of that line can be interpreted as the dimension d of the set of elements. This is because taking the log of a power-law relation transforms it into a linear equation:
In Equation 1, if we let y=Log(N) and x=Log(1/s), then what we have is the equation of a line, y=mx+b, where d=m and b=0. For details regarding this type of analysis and its application to music, see [1, 2].
The Data
For the saxophone solo, I used pitch data from The Jazzomat Research Project [3]. Here is the distribution of intervals where the size of each interval is equal to 2^(k/12) for an interval of k semitones:
The value 2^(k/12) comes from the physics of the equal temperament scale commonly used in Western music. Below is a plot of the interval data. It yields a dimensions of d≈8.3:
We can use a data smoothing technique called binning to, in a sense, loosen our focus and, in this case, obtain a slightly better fit. We’ll place the intervals into equal-size bins where each bin contains intervals of roughly the same size. The representative size of a bin is typically assigned using either the lowest, highest or mean values with the choice applied consistently across all bins. To be clear, while binning can help to reveal the presence of a power-law, it cannot produce a power-law where none exists.
Here is the distribution of intervals where each bin is three semitones wide, along with a plot of the data which yields a dimension of d≈8:
The R² value is marginally better and, by general scientific standards, is certainly respectable. However, we can see that the strength of the relationship between size and count is still not as consistent as we’d like as evidenced by the counts in bins of representative size 8 and 11. Those counts are roughy equal and, in fact, increase rather than decrease.
On the other hand, when we look at the way the intervals change, the picture is clearer. We refer to the differences between intervals as melodic moments. If the interval is the first difference of pitch, then the melodic moment is the second difference. Here is the distribution of raw moment data:
Note that the smallest moment size is 0 semitones which corresponds the case when two consecutive intervals have the same size. Here are plots of the raw data and the binned data with a bin width of 3 semitones:
The raw and binned data show excellent agreement with respective fractal dimensions of d≈8.1 and d≈8.3. With R² values of .930 and .998, the melodic moment distribution shows unequivocal evidence of fractal structure.
Closing Thoughts
When analyzing music in this fashion it’s important to keep in mind that, generally speaking, the datasets we’re exploring are small by scientific standards. We therefore work under the assumption that there is a strong relation between the character of a melody and its underlying intervallic distribution. For instance, a scale-wise melody sounds very different from one containing large leaps. In this sense, where interval-based scaling exists, we can assert that a piece that continued in the same melodic fashion would also continue to exhibit a self-similar distribution of elements across several scales of measurement.
Over the centuries, it seems that the greatest music has a sense of transparency and timelessness about it, whether it be Bach or the Beatles. I am certainly not alone in the perception that John Coltrane’s music possesses those same qualities. To the extent that his music also reflected the world around him, it should not surprise us that, in some sense, it expresses the fractal nature of the world as he experienced it.
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