The unfathomable strangeness of American music Part 3. Stockhausen in London

I have one short encounter with Karlheinz Stockhuasen. In 1985, I attended his festival at the Barbican in London. During one of the concerts, my girlfriend asked me "does he get along eith Boulez?" I said that they have a public rift, but that I thought it was just manufactured for the sake of publicity. I hadn't realized that Stockhausen was sitting behind us in the audience. Someone tapped me on the shoulder, I turned to see Karlheinz in his faux nineteenth century collarless jacket. He pointed his finger at his face, smiled and nodded.

Last night, my composer friend Ezequiel Viñao and I heard Stockhausen's instrumental opera "Michaels Reise um die Erde". I'm not always his fan, but the beautiful writing floored me, as did the production.  Lots of brass and very few strings in a large amplified chamber group. The first two thirds are sort of doom metal images of Michael's destruction of Earth with brass, percussion, keyboards and some harp. One surprise that kept it exciting was a lot of jazz, mostly from Michael's  trumpet - he must have worked out these solos with his son Markus, who is a very good jazz trumpeter. I guess that some sections are improvised, including a pizzicato bass duo that would be exactly at home at the Village Gate.

Occasional spots of dark humor: Ezequiel pointed out that when Michael destroys Cologne, you hear a bunch of academic serial music, while the destruction of New York brought back the jazz.

Towards the end, after our version of the world is destroyed, sex, beauty and procreation reappear with extended and complex bird call courting duos on clarinets, trumpets and basset horns. Along with long memorized parts that include a lot of breathing into their instruments, the two bird couples, one of whom are Michael and Eve, are choreographed in intricate mating behavior dance rituals. The ending section sounded ike Olivier Messiaen's bird calls and recreations of the outdoors.

To me, Messiaen's spirit hovers over the whole piece - but what a great spirit to summon when considering the end of man together with thoughtful considerations of nature.

See this link for some examples of how Messiaen transcribed bird songs for human musicians.

Music Mathematics Part IV. Timeless Music Theory: composing with fractals on purpose

Read Parts 1 through 3 first  

I think pure noise, as wonderful as it is, provides too easy an answer to our challenge to develop timeless music. While it is infinitely variable, it always sounds the same.

Let’s come up with timeless music that has discernable patterns that can’t be fully described in the two dimensions of amplitude and time, even if we use them to reproduce it.

We’ll use a means by which music was stored and reproduced prior to audio recording. Classical musical scores also have two dimensions, with time again running from left to right, at least, but rather than showing the amplitude of air pressure in the vertical dimension, we indicate frequencies to be played or sung: for example a dot on the second space in a treble clef means a note with that changes the air pressure 440 times a second.

Classical composed music works if these notes, which are packets of frequencies, follow one another fast enough so that there is a perception of a flowing pattern, the way a cartoon flipbook looks continuous if your thumb is fast enough.

Since classical compositions use an up / down dimension with notes, and a left / right dimension of time, let’s produce music with the same pattern at multiple scales, similar to using different size rulers to measure the coast or duration of the fractal Sinatra in Part I.

I’ve selected a five note pattern that Arnold Schoenberg uses as a basis of his second string quartet. The notes are A Bb A B D, or on the piano, intervals of up one note, back down one note, up two notes, and finally up three notes.

We can transpose these intervals around the piano. It’s not only easy to hear the relationship, but moving up three notes from a low note is the same sonic distance as moving up three notes from a high note (this is based on math associated with Pythagorus, and may be a future blog topic). For now, adding or subtracting notes to a pattern is like climbing or descending steps near the middle of a long staircase, and moving up one step, back down one, up two and up three changes your relative position similarly.

Here’s an example with Schoenberg’s pattern played on low notes on the piano one time, while the high notes play the pattern five times as fast, so that there are five repeats, and we hear the same melody at two time scales. The fast phrase adds the frequencies to the slow notes, so the melody changes to higher or lower notes, in the way someone appears taller or shorter depending on where they stand on the staircase.

Let’s expand on this and listen to the PubliQuartet play this five note phrase at four different time scales: this is the Fractal Variation from my String Quartet #3, The Essential, which is based on math manipulations (Download the score). The quartet plays the phrase in four voices, one in which the five note phrase repeats 125 times, another slower scale where it is played 25 times, another 5 times, and finally one only time. Despite the fact that there are only four time scales, and every note uses the same steps in that five note phrase, the notes add to each other so that the music constantly changes and nothing precisely repeats.

Music Mathematics Part III. Timeless Music Theory: Musical dimensions and why Noise is the most advanced style

Read Parts 1 and 2 first  

So much for geometry of nation states, but what dimensions describe music? The straightforward answer might surprise you: the vast majority of all music we listen to is recorded and reproduced from a two dimensional grid, like a coastline. The two dimensions are 1) amplitude running up and down and 2) time running left to right. It is just like drawing a pencil line on a roll of chart paper.

These two dimensions record and replay all sound, ranging from a sine wave, which is easily described from a simple formula: this is smooth wave with 440 peaks and troughs per second. When we play this through a speaker, it moves the air in the same pattern, and you'll hear a sin wave tone.

 The waves are far more complex with an orchestra of many instruments and can't be described by any simple formula, but can still be drawn and reproduced extremely well with the pencil on chart paper. For example, here is 1 second from an orchestra playing Thung Kwian Sunrise.
and here is the chart (in mono) of the sound wave of that 1 second: if you play it into a stereo speaker or a telephone, or press it into vinyl or recording tape and then play it, it will play back what you heard the orchestra play

Here is just  10 milliseconds of that sound: see how undulating it is?

Sound is produced by changes in amplitude of air pressure, so for the sin wave above, an air pressure wave similar to waves in the ocean moves from peak to trough 440 times a second. This change in air pressure encounters your ear and your brain interprets it as music.

Sound recording measures the peaks and troughs of that air pressure wave with a pencil line on a chart moving from left to right. Our two dimensional sound recording chart can be stored as changes in the height in the grooves in LP records, or as metal particles on a sound tape, or as bits in a sound file on a computer chip or compact disc. If we replay the chart thorough a speaker, the speaker regenerates the air pressure and sound.

So how do we measure the dimensions of music? The length of the pencil line could be described in only a time dimension with a ruler or stopwatch, which is like using a long ruler for the British coastline (see Part II). But while we measure a line in one dimension, the shape of our music line is like the coastline of Britain: it's not quite 2 dimensional as it doesn't have area, and it obviously isn't a one dimensional line, so should be able to determine it's length more accurately using a partial dimensiona between 1 and 2. A periodic wave like our 440 Hz sine wave has a length that is easily described using a sine wave function. The louder and more irregular the sounds our chart is recording, that is the nosier the sound, the longer our pencil line will be, and the closer to filling an area.

So we can jump now to an easy answer to how to create timeless music: pure noise has infinitely long music line that would fill up our chart paper. In contrast to a sin wave, we can’t accurately describe it, although we can find a dimension that can approximate it's the length of the wave. Now questions of fast or slow are irrelevant, as is the duration We can measure its duration with a stopwatch or ruler, but there is no pattern and no duration of a musical pattern, so the piece doesn't have an ending, you can listen as long as you want.

White noise is this pattern in sound, and is the sonic analog to a coastline of Britain so complex that it is almost two dimensional (this is one second of white noise)

and here is 10 milliseconds of white noise: see how non-regular and complicated it is, compared to either the sin wave (which is still a sin wave no matter how much you stretch it out) or the orchestra sample?

so the country might be described as land, sea or a frothy muck, and so would this extremely complex music, which encompasses virtually every frequency you can hear.
Whenever you are between stations on your radio dial, you are hearing our new timeless music composition, White Noise for Air Columns.

Orchestra of humans plays music by elephants (Thai Elephant Orchestra)

Following the post a couple of days ago of Steve Beck playing my arrangement (and Wade Ripka's transcription) of a piece composed by Phoong the elephant from the Thai Elephant Orchestra, here's the orchestra from the group Composer's Concordance playing our transcription of Thung Kwian Sunrise, which is on the elephant's first CD.



You can download the score for Thung Kwian Sunrise.

Music Mathematics Part II. Timeless Music Theory: Intuitive Fractal British Coastline

Read Part I first!

The repetition at multiple scales we just constructed with Sinatra's musical phrase occurs throughout nature – many examples are found in Benoit Mandelbrot’s book, The Fractal Geometry of Nature.

The best known example is the problem of how to measure the length of the British coastline. There is a straightforward approach, which is like using a stopwatch to measure the duration of music: you buy four very long rulers, you and three friends stand at the four corners of Britain, and voila, you add the 4 lengths together, and calculate say 2,000 kilometers.

But rulers don’t account for the fact that the British coastline is craggy at different scales, ranging from a promontory seen from a space satellite to tiny pools in the sand seen with a magnifying glass. If you measure the coast with a 200 kilometer ruler, you need to place the ruler in 12 straight lines that approximate this irregular coastline end to end, and calculate the coastline is 2400 kilometers.

Or with a 50 km ruler, we place it end to end 68 times, and come up with a length of 3400 kilometers instead of 2400.

The reason the coastline length is so hard to measure is that a straight line measures one dimension, length alone, but not the pattern of the coastline, which is better described in at least two dimensions, length and width.

The problem  is similar to the question of how to measure the circumference of a circle: it’s hard to measure the edge of a circle with a ruler, but you can measure it’s diameter with a ruler in one dimension, and then use a trick: multiply the diameter by the number pi. This  yields a precise measurement of the circumference in one dimension, even though the circumference itself has a two dimensional shape with length and width. If we measured the surface area of a ball instead of a circle, we can use the same kind of trick, measuring the radius with a ruler, squaring it, and multiplying it by 4 times pi. The ball is in three dimensions, but the surface area in two dimensions.

Mandelbrot’s insight was that the coastline doesn’t expand infinitely, but converges on a length. He introduced a math trick, which is to add a little bit of an extra dimension, called Hausdorff or fractal dimensions: the description of the length of the British coast turns out to be accurate if you estimate it using a about 1 and one quarter dimensions. So closer to a line than an area, but getting towards an area. Countries with straighter boundaries like Germany have a dimension of about 1.15.

While imagining fractional dimensions is frustrating, you can imagine that the repeated smaller and smaller rulers placed along the coastline fill up more and more of the country’s area. If the smaller eddies and promontories continued forever, they would eventually cover the entire country. Then Britain would be all coastline, and either all land or all sea, in which case our linear coastline would be completely two dimensional by occupying all of the length and width. Britain would either be all land or all sea, or some kind of muck between: since it isn't, the length converges somewhere.

In a couple of more entries on this blog, you'll learn to make music that is analogus: without a time dimension, and neither music nor not-music, but some kind of muck that is in between.

Steve Beck performs music by Phoong the elephant

Phoong, an elephant living at the Thai Elephant Conservation Center and member of the Thai Elephant Orchestra, was for years an avid improviser on our giant xylophone, which we call a renaat after a traditional Thai instrument.

Thousands of people have heard him perform at the daily shows at the Center. We recorded one of his solo improvisations and released it on the CD Elephonic Rhapsodies.

Wade Ripka transcribed it, and I arranged it for solo piano (download the score) . Here's a new recording of the piece by a great classical pianist, Steve Beck.


Or listen to Phoog's solo in a new window

Music Mathematics Part I. Timeless Music Theory: Timeless Sinatra

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Originally for WFMU dj Vicki Bennett’s People Like Us Radio

Let’s explore music that is neither slow nor fast and for which time is arbitrary. To do so, we travel into the basics of what music is made from.

Let’s start with a building block of 19 seconds from Frank Sinatra’s “It’s a Lonesome Old Town” with a bit of background by arranger Nelson Riddle.

I selected this bit because it’s simple, only voice and a bit of instrumentation, and because a great performer like Sinatra uses subtle shading and attacks that develop features at different time scales. A sine wave is still a sine wave when you change speed: with Sinatra, however, if we slow the recording 8 times so that the word “around” lasts nearly 10 seconds,  you hear a beautiful glissando, while the vocal vibrato at the end of the syllable of “round” nearly becomes a melody itself.

Here’s an easy way to make “fast” and “slow” irrelevant, by playing the phrase at different speeds simultaneously. This is the phrase played at 7 speeds, at each multiple of 2 from the 8 times slower to 8 times faster.

So we have to call that tempo slow, fast, and many shades in between.

Here’s another playback at multiple time scales that overlap to produce new features. I hear something like a church bell that tolls five times.

That was Sinatra’s phrase at 7 speeds, in multiples of three:  9 times slower than normal, 3 times slower, original speed, and 3, 9, 81, and 243 times faster than normal. You might have discerned new patterns in the 10 seconds we just heard. These result, for example, from the peak of one soundwave superimposed on the peak of another to produce a sound that wasn’t in the original. Less noticeably, some of the soundwaves are out of phase and cancel out some of the original pattern.

A stopwatch says that little piece was 10 seconds long. But I think the length of the music, in contrast to the length of the original which is genuinely 19 seconds, is ambiguous. While you heard Sinatra at seven different time scales, we used 2 seconds of overlapping speeds repeated 5 times. The original phrase was ~20 seconds, the kernels that makes up the piece range from one 81^st of a second to 2 seconds, and ALL  the information ise heard in 2 seconds. So any length greater than 2 seconds, like the 10 second length we just heard, is arbitrary.

More provocatively, you may have heard patterns that last longer than the 2 second loop duration! For instance, I hear a slow melody repeated two and a half times: this is like a Middle Eastern geometric art form in which tiles in a mosaic produce patterns that are welcomed but not intended by the artist.

Here’s another time scale overlap of the same material, which somehow contains the sound of a train.

In that case, multiple one second loops of each of the different speeds from the previous piece contained all of the information of the entire piece. So the piece we just heard could be performed for any length of time longer than a second, unless you perceive patterns that are longer than a second.

The above was an intuitive introduction to fractal music composition: much more to come.

"The Revolutionary Mr. Jenkins" performed by Daniel Reyes Llinas and Luis Iianes

Following up on  the previous post on a story about the great Leroy Jenkins, here's a piece, The Revolutionary Mr. Jenkins, I wrote in commemoration in 2007. William Hooker, Sabir Mateen and I had a trio concert at Roulette,  released live CD by Engine Records, called Yearn for Certainty by the William Hooker Trio, including a version of this piece.

Last week, our friends at Composer's Concordance produced  guitar concert last week and asked  for a piece:  I scored it one out for two electric guitars, and here it is played with a hard rock improvised verse by Daniel Reyes Llinas and Luis Ianes.