Archive for September, 2010

House of Mirrors

Saturday, September 11th, 2010

House of Mirrors

This is a short piece I made as a music theory project in 2004, during my undergraduate at USC. I’ve never liked the title but it’s fairly apt and I haven’t found a better one. The music theory class in question, incidentally, was taught by the talented and inspiring Veronika Krausas.

The piece consists of a single set of pitches (all realized as sine waves), presented in two ways and in three different layers.

The foundation of the piece is a cantus firmus, a simple just-intonation melody in a low register. The cantus lasts for the entire duration of the piece. It does not repeat, but it is composed of two similar sections. The duration of each note in the cantus is inversely proportional to its pitch, such that high notes are shorter than low notes in a proportional way. The lowest pitch of the cantus (and of the entire piece) is a concert A at 55Hz, which is high enough to hear but low enough that small speakers won’t be able to produce it. You’ll want a woofer of some sort to get the full spectrum of this piece.

A diagram of the cantus follows (click it to embiggen). The x-axis is time, labeled in minutes and seconds. The y-axis is pitch frequency in Hz, which for some reason I represent linearly instead of logarithmically. Horizontal gridlines show the locations of concert A’s.

The cantus is divided into groups of pitches labeled A–I on the above diagram. Each of these groups is presented as a simultaneous chord in the “Primary Theme” (PT). The PT occurs twice, once at the beginning of the piece and once at approximately 1:10. The “Secondary Theme” (ST) consists simply of the cantus melody, presented in higher registers and at faster durations than the original cantus. The ST also occurs twice, once at about 0:50 and once at the end of the piece.

The following diagram indicates how the PT’s and ST’s are arranged in relation to the cantus firmus. This diagram can be thought of as a “score” for the piece, or perhaps better as a graphical analysis of it.

Each instance of the PT and ST is arranged such that its lowest note has the same pitch as the cooccurring note in the cantus. All other pitches are modulated accordingly. Thus when the ST occurs over an E (as it does around time 0:50), this entire occurrence of the ST is a fifth above the ST at the end of the piece, which occurs over an A.

Moreover, each entire instance of PT or ST lasts exactly as long as the cooccurring note in the cantus. All relative durations are preserved, but absolute durations are modified such that each theme fits exactly onto one note of the cantus.

People who have taken a music theory course including the analysis of Western musical forms should be suspicious that I am talking in terms of primary and secondary themes. Perhaps they are shaking their heads in shame, or trying to forget bad memories. Yes, I did engineer this piece to be in a (simplified) sonata form. The secondary theme’s occurring first in the dominant and then finally in the tonic is typical of simple sonata form. Also there are two “transitions”—in this case, sections of higher pitches in the cantus. Of these, the first transition wanders afield from simple overtones of A; whereas the second has the same melody shape as the first, but all of its pitches are strictly overtones of A (55Hz). This is also intended to be reminiscent of sonata form. The only thing lacking from canonical sonata form is a development (although I have optimistically labeled the first occurrence of the secondary theme as “ST & Dev” in the diagram above).

The version of House of Mirrors presented here is an exact recreation of the version I made in college, the only difference being in the process of production. The original sound file was produced laboriously by generating individual sine waves in CoolEdit and mix-pasting them on top of one another manually. A month ago I reopened the project and automated the production process. Now a perl script takes a set of numerals representing the cantus melody and generates a csound score file. The rest of the work is done by csound, bless its little binary heart. So far I have only used this process to reproduce the original piece, but potentially I can also make pieces of the same concept but with different melodies. Or I could make a longer version of the original piece which includes a proper development. Stay tuned!

scale phases

Thursday, September 9th, 2010

Recently I made several very simple pieces which I call “scale phases.” Each of these pieces is a minimalist presentation of an individual scale, in which the pitches from a one-octave range of that scale fade in and out at regular intervals. The idea was to be able to let the sonorities of the scales speak for themselves, so to speak—to hear each scale in itself, rather than to hear a scale embedded in a more structured piece of music.

To this end, I made my scale phases very simple. Each pitch-class is represented by one sine wave at a comfortable frequency. Each scale is presented in “root position,” with the root being the lowest pitch. The loudness of each pitch is determined also by a simple sine wave, every pitch having a unique periodicity. With this simple technique, various combinations of pitches will be heard as the process evolves. Different aspects of the scales’ textures will arise and fall back again into the background.

To concretely illustrate the technique, the following graphic indicates the times at which each of the five pitches in the “5-limit pentatonic phases” piece are at their loudest points. Each horizontal line represents a different pitch (labeled on the y-axis as a ratio from the root pitch), and time is measured on the x-axis, each minute indicated by a vertical line. You can play the 5-limit pentatonic phases piece and follow along to this score. (Click to open the image and it will look less scrunched!)

There is nothing else to these pieces, except for presentational details. I tweaked the overall volume of each pitch to obtain a good balance, while also making the more simple intervals a little more prominent than the more extended intervals. I also had to cut off the pieces at some point, since the process of pitches fading in and out could in principle last forever. Thus I cut off every scale phase arbitrarily at 1000 seconds. This means that the pieces are very long, but also it is in no way necessary to listen to the whole thing, or to start at the beginning. I think the best way to compare the pieces is to listen to little snippets of each one side by side. I also enjoy listening to an entire one of these pieces all the way through — I find them quite relaxing.

The audio files are here, and more specific information about each piece can be found below. The files were realized using csound.
major phases
chromatic phases
3-limit pentatonic phases
offset 5-limit pentatonic phases
7-limit pentatonic phases
Grainger tri-pentatonic phases
Grady tri-pentatonic 14-tone #1: phases
Grady tri-pentatonic 14-tone #2
Grady tri-pentatonic 14-tone #3

“Major phases” and “chromatic phases” both represent scales playable on standard Western equal-tempered instruments. “Major phases” is the 7 pitches of the equal-tempered A major scale, with the root pitch at 220Hz. “Chromatic phases” is all 12 pitches of the chromatic scale, again with the “root” (lowest) pitch at 220Hz. These two scales are very familiar, but presented here in an unusual way.

The rest of the scales are all just-intonation scales of some variety. In other words, all of the intervals in these scales can be represented as ratios of integers.

“3-limit pentatonic phases” is a very simple scale. It is built in the Pythagorean fashion, being entirely composed of adjacent perfect fifths (i.e. the ratio 3:2). These are then compressed into one octave. The complete set of ratios in this scale is {1:1, 9:8, 81:64, 3:2, 27:16}; Pythagorean just-intonation equivalents of root, major second, major third, perfect fifth, and major sixth, respectively.

The 5-limit pentatonic is similar to the 3-limit pentatonic, but instead of the Pythagorean major third (81:64) and major sixth (27:16), it has the 5-limit major third (5:4) and major sixth (5:3). Although the resulting frequencies are not so distant from one another, I think the scales end up “feeling” different. I find that the 5-limit scale sounds more soft and sonorous. The scale-phase pieces representing these two scales above have been realized so as to be directly comparable to one another. Both have a concert C root, and identical temporal patterns of loudness. I generated these two scale phases at the request of Steve Grainger.

The 7-limit pentatonic is somewhat different from the other two pentatonics. Instead of a major third, it has a 7-limit minor third (7:6); and instead of a major sixth it has a 7-limit minor seventh (7:4). It has no major second (9:8), but has a perfect fourth (4:3). I find that this scale sounds very sonorous and also mournful. It has been realized with its root at 200Hz, and the full set of intervals from the root present in this scale is {1:1, 7:6, 4:3, 3:2, 7:6}. Note that the number 5 is absent from all of these ratios.

The rest of the scales presented here are based on this 7-limit pentatonic.

The Grainger tri-pentatonic is a single scale comprising three 7-limit pentatonics spaced at perfect fifths from one another, their constituent pitches then octave-corrected into the space of a single octave. A nice diagram of this scale can be found in this post from Anaphoria. The simple 7-limit pentatonic is represented on the top left of the diagram, and the dark lines in the central part of the diagram show the three conjoined pentatonics comprising the tri-pentatonic. The resulting scale has nine tones, at the following intervals from the root: {1:1, 9:8, 7:6, 21:16, 4:3, 3:2, 14:9, 7:4, 16:9}. This scale was developed by Steve Grainger.

The other three scales are 14-tone extensions of the Grainger tri-pentatonic. All of these were developed by Kraig Grady, and are also discussed in the Anaphoria post mentioned above. See this post for information about intervals present in the scales, etc. #1 is the “left-most” scale in the diagram (that including the ratio 112:81), #2 is the other scale represented on the flat plane, and #3 is the scale represented as including the diagram’s vertical dimension. The scales are similar to one another, and to the simple tri-pentatonic, but have differing flavors of dissonances. I have generated these three scales with their roots at 200Hz and with identical time courses of loudness so that they can be directly compared to one another, and to the tri-pentatonic. However each of them, like all of the other scales in this post, can stand on its own — either for concentrated listening or as “furniture music” for setting a textural mood.