jwilzewske: statistics

Showing posts with label statistics. Show all posts

Sunday, January 1, 2012

Jan Beran’s ‘Winter711’, fractional Brownian motifs, long-range interdependence

D reams pass into the reality of action. From the actions stems the dream again. And this interdependence produces the highest form of living.”
— Anaïs Nin.

A new year; an occasion for us to meditate on how our mutual feelings and actions—our own; those of friends and family members—are meaningfully intertwined over long stretches of time.

H ere’s an MP3 of Jan Beran (piano) and Christopher Raphael (oboe) performing Beran’s ‘Winter 711’. Beautiful and evocative of long-range interdependence/entanglement.

I f you’re interested, some of the mathematical underpinnings of the compositional method used to create ‘Winter 711’ can be found in the links below. Happy New Year!

I believe it is not possible to coordinate musical parts in a purely responsive manner. That is, a system that simply triggers its events on the detection of notes in the solo part will perform badly in most musical contexts. This is due, in part, to the inherent detection latency that is built into the problem which makes the responsive system perpetually late. Instead, Music++ schedules the accompaniment’s note times by continually predicting into the future based on what it has currently observed.”
— Chris Raphael, Indiana Univ.

J an says of ‘Winter711’ that he tried to avoid any familiar sense of musical flow. In other words Jan’s music doesn’t give the performer (and listener) the usual cues needed to rhythmically organize the music. While the music is sometimes highly rhythmic, many sections contain no recognizable steady pulse, nor clear points of emphasis whose times differ in simple ways, as in much mixed meter music. While I think these pieces are engaging on their own terms, they work as a wonderful showcase for the [sequencer] accompaniment system, since I doubt they could be played by an all-human ensemble.”
— Chris Raphael, Indiana Univ.

Tuesday, June 14, 2011

New Quantitative Ways to Look at Small-Scale Variability in Meter and Rhythm

A pianist friend of mine says that the 2-part Inventions are merely ‘steep hills’ but that the 3-part Sinfonias are ‘craggy mountains whose paths have lots of twists and turns’. Having lived with these pieces most of my life, I have to agree.”
— Lee Ridgway, harpsichordist, BEMF 2011 Fringe Recital.

H arpsichordist Lee Ridgway performed all 15 of J.S. Bach’s 2-part Inventions and all 15 of the 3-part Sinfonias yesterday, in one of the ‘fringe’ recitals at the Boston Early Music Festival. He performed them in the sequence in which they originally appeared in W.F. Bach’s clavier-büchlein. His masterful and sensitive treatment of each provoked a number of reveries for me, for which I am grateful.

A mong these was a reflection on how the moment-to-moment agogic metric accents might be analyzed with mathematical and statistical signal processing and spectrum-analytic methods similar to those that have been applied over the past 20 years in cardiology, to understand heart rate variability (HRV) and develop predictive models of cardiac disorders. The pulse quickens; the pulse also slows! Quantitative details of small-scale dispersion of meter—and the abnormal, motoric diminution of HRV that occurs in stress and in disease states—can tell us so much! Just to see what interesting things may turn up, I begin with one of the simplest ‘variability’ measures—the root-mean-square standard deviation, RMSSD:

I look in the music theory literature and recent doctoral dissertations, and so far I am unable to find anybody who has pursued this sort of thing. If I am wrong, please email me or add a comment below.

O vernight, I download some MP3s of others’ performances of the 2-part Inventions and begin to run some analyses using the MATLAB signal processing toolbox software—the same modules that I use for analyzing digital electrocardiograms. I pick off beat-by-beat interbeat intervals in the left-hand part and the right-hand part. I exclude marked ritardando/accelerando bars; also exclude fermatas.

T here is more or less constant variation in articulation and timing... successive notes are crafted with varying durations—this is especially true on harpsichord more than other keyboard instruments, but it would be true as well for winds, voice, any instrument. The plot above is a composite across 2-part variations—a quick exploratory ‘look’ only; not what we would do for a proper analysis.

D etached releases ranging from ‘sharp’ ones to ‘lingering’ ones… this all shows up in the RMSSD and other time-domain variability measures. It also shows up in frequency-domain power-spectrum and other FFT- and wavelet-transform-based measures.

A nd, naturally, the variability and rhythmic power-spectrum measures change throughout the course of each piece. The power spectrum has ‘shoulders’ on it, on both the left (slower, longer IBI) and right (faster, shorter IBI) side of the median interbeat interval or rhythmic frequency. In principle, we would watch and measure the time-evolution of the power-spectrum from the beginning of a piece to the end of a piece—look at these evolutions or trajectories for different performers whose interpretations and styles differ. Then we would be better able to account objectively and quantitatively for how they do what they do!

L inkages across the bar-line are frequent, often involving small melodic voice-leading in one part (hand). These were particularly prominent in Ridgway’s accounts of the C-major, F-major, G-major, and C-minor 2-part variations. Beautiful, truly.

T hrough thoughtful, passionate application of agogic variations in small-scale timing, we get an acute sense of tension between (1) propulsion from ordinary and extraordinary accents in the metric frame and (2) propulsion from motivic and other phrase units. It is a major source of musical drama and beauty—not just in Bach but, I think, in all music. It was simply Ridgway’s fantastic playing yesterday that makes me think of these mathematical ways of explaining and understanding how it works.

R idgway is a solo organ and harpsichord performer, has performed for more than 40 years throughout North America and Europe. A native of Oklahoma, he received his Bachelor of Arts from the University of Oklahoma and Master of Arts from New England Conservatory of Music. Presently the organist and choir director at St. Chrysostom[s Episcopal Church in Quincy, MA, he is also Dean of the Boston Chapter of the American Guild of Organists. The instrument on which he performed yesterday at Emmanuel Church is one made by Boston’s Allan Winkler, a 2-manual adaptation of a 1716 design by Carl Conrad Fleischer… gorgeous sound.

B eyond their pedagogical applications, the Inventions and Sinfonias offer us wonderful, individual pieces of music… The similarities or contrasts between pieces in the same key; the [affective] characteristics of different keys; the characteristics of individual themes; … how the voices engage in dialogue or games of chase...”
— Lee Ridgway, harpsichordist, BEMF 2011 Fringe Recital.

Sunday, April 12, 2009

Beglarian’s Chamber Music: Magic through Risk-Sharing, Intimacy through Obligate FM

J esus said, ‘I have cast fire upon the world and, see, I am guarding it until it blazes.’ ”
— Gospel of Thomas, v. 10, quoted by Eve Beglarian, ‘Until It Blazes’.

I am (you are) the light that shines over all things. I am (you are) everywhere. From me (you) all came forth, and to me (you) all return. Split a piece of wood, and I am there. Lift a stone, and you will find me there. Tend a sick sheep, and you will find me there.”
— Gospel of Thomas, v. 77.

E very atom belonging to me as good belongs to you.”
— Walt Whitman, ‘Leaves of Grass’.

L istening for the magic, as if our lives depended on it.”
— Eve Beglarian, quoted in Hinkle, p. 148.

T he music and writings of Eve Beglarian are terse, capable of having radical impact for those who are amenable to attending to them.

[50-sec clip, Eve Beglarian, ‘Until It Blazes’, Segment 1, 1.6MB MP3]

[50-sec clip, Eve Beglarian, ‘Until It Blazes’, Segment 2, 1.6MB MP3]

[50-sec clip, Eve Beglarian, Messiah Remix, ‘Be/Hold’, Segment 1, 1.6MB MP3]

[50-sec clip, Eve Beglarian, Messiah Remix, ‘Be/Hold’, Segment 2, 1.6MB MP3]

E ve’s iconoclastic tendencies are endearing. Only phonies/fogies would consider them seriously threatening. And her compositional methods are wonderfully diverse—pushing envelopes that we didn’t know were there. I listen to some of the synth ‘chirps’ and tone-pips in several of her pieces...

[50-sec clip, Ray Lynch, Deep Breakfast, ‘Tiny Geometries’, 1.6MB MP3]

E ve’s use of ambient chirps and tone-pips is not like, say, Ray Lynch’s use of them. Many communication sounds, including those of primates such as New World monkey twitter calls, contain frequency-modulated (FM) swept-spectrum signals or tone-pips. Others have rhythmic pulses that are FM swept click-trains. Complex natural sounds (e.g., vocalizations or speech) can be characterized by specific spectro-temporal patterns the components of which change in both frequency (FM) and amplitude (AM).

B ut the timing and properties of Eve’s tone-pips are measurably, statistically different from Lynch’s—Eve’s are more like natural primates and birds—and therein lies, I think, a source of the radical, profound effects that Eve’s compositions have. She is ‘wired’ into a part of our innate neurophysiology—a part that is engaged in our reflexively apprehending patterns in our surroundings, and disambiguating and prioritizing the profusion of patterns. Upon a first hearing Eve’s music confronts us with and makes us aware of details of our own inner workings that we may not have known existed; upon subsequent hearings, we revisit what we learned about those details, and reflect upon what it means and how it matters.

I t’s interesting to find what the probability density function and statistical kurtosis of a music signal are, as a means to figure out why the music works or has the effects on us that it does. In a mixture of a Gaussian signal with a sub-Gaussian signal, the Gaussian signal will be significantly attenuated or suppressed, in terms of the pulse-train intensity and auditory evoked potentials in the brain [see links below].

T he probability densities of Beglarian’s music that I’ve examined with my signal-processing set-up tend to be super-Gaussian (kurtosis > 3.0). This generally occurs because there is a substantial amount of low-amplitude time in her compositions (tracks). By concentrating the energy near zero during these times, the probability density function (PD) becomes more peaked, and therefore becomes more super-Gaussian.

Super-Gaussian passage, Eve Beglarian, ‘Until It Blazes’

I f the raw, super-Gaussian music signals are admixed with others, they will have less ‘gain’ (in human auditory processing) than, say, a competing sub-Gaussian signal. One might want, therefore, to alter the probability density function of the music signals from super-Gaussian to sub-Gaussian. A frequency-modulated (FM) signal is always sub-Gaussian.

A n FM signal, f_c(t), is mathematically defined by the function [see Dunlop & Smith]:

where A is the carrier-frequency amplitude, ω_c is the carrier frequency, f_m(t) is the modulating signal, and B is proportional to the modulation depth. This equation shows that the amplitude of the carrier never is changed, but the frequency varies according to the modulation signal. As a result, the probability density function is exactly the same as the probability density function of the carrier itself:

T his particular probability density function is the probability density function of a sinusoidal, and has a kurtosis of 1.5 (or an excess kurtosis of -1.5).

B ut there is no ‘law’ that says that the carrier has to be of a single pitch or frequency, that it has to be sinusoidal. Likewise, there is no ‘law’ that says that the carrier has to be many orders of magnitude different in frequency from the modulating signal. In FM radio it is true that the frequencies in the audio signal are many tens of thousands of times lower than the FM radio carrier frequency, but that is because of the radio broadcast use-case, which is not applicable in Eve’s musical-soundscape use-case.

S o all signals, whether super-Gaussian, Gaussian, or sub-Gaussian, are necessarily, obligatorily converted to a sub-Gaussian signal if they are frequency-modulated—and that is part of the physics that Eve’s compositions take advantage of, to achieve the effects and impact that they do. Gaussian and super-Gaussian signals do not give way to themes that are conveyed by sub-Gaussian signals; they are ‘taken over’ by the sub-Gaussians. Quasi-vivisectionist/neuroanatomist, Eve takes us apart: her pieces are fascinating lessons, revealing to us how we work, deep down inside. Different interpretation of chamber music as ‘intimacy’.

I n ‘Until It Blazes’, the latter part of the piece is overwhelmed by a high-frequency noise spectrum that, essentially, becomes an FM ‘carrier’ for the other musical signals with which it’s admixed and which it takes over. It’s not that the other signals are buried; no, the amplitudes stay the same. Rather, the admixture/modulation causes new processing by our hearing and auditory cortex—processing by which are able to detect and respond to FM acoustic signals. The musical result—and the gnostic text that inspired it—compel us to reprioritize what we believe, what we believe we know; re-think the evidence that is the basis for our knowing... This isn’t merely ‘edginess’ or gratuitous excitement. It’s politically interventional, potentially life-changing—features that are characteristic of Eve’s work. It’s an invitation to a self-styled, radically self-examining intimate community.

T he ‘gnostic’ gospels reference the Greek word ‘gnosis’, meaning ‘knowledge’. Arguably, Gnostic philosophy—finding answers to spiritual questions within oneself, without reliance on a church or priests; individuals learning to free themselves from the material world by way of meditation and enlightenment—did begin in pre-Christian times; either that, or it was a Jewish reaction to Judaism reacting to early Christianity. Some introductory comments about the apocrypha, gnostic Gospel of Thomas and the 114 ‘sayings’ it contains are here. A fine provocation for meditation, I think, on the occasion of Easter.

E ve started out as an Uptowner with a Princeton-Columbia education, but then jumped ship and began composing music her professors couldn’t countenance. Her approach to musical sources is omnivorous. She’s made collages of disco music, updated the 14th-century composer Guillaume de Machaut, made theater pieces out of Kurt Schwitters’ nonsensical ‘Ur-Sonata’, and quoted Gregorian chants along with the sounds of a couple having orgasms. One of her best pieces is ‘No Man’s Land’, a gritty poem to New York whose thoughtful text is accompanied by sampled sounds as grating as the city itself. Beglarian is an amazingly high-tech sampling artist, and much of her music—even orchestra pieces like ‘FlamingO’—uses brilliantly manipulated prerecorded noises.”
— Kyle Gann.