## What Sequences Sound Like

Is there a way to put a sequence of numbers to music? The system that comes immediately to mind is to match a number to a particular pitch. The difference between any two neighboring integers is the same, so it is logical to assume that the same tone interval should correspond to the same difference in integers. After we decide which tone interval corresponds to the difference of 1, we need to find our starting point. That is, we need to choose the pitch that corresponds to the number 1. After that, all numbers can be automatically matched to pitches.

After we know the pitches for our numbers, to make it into music we need to decide on the time interval between the notes. The music should be uniquely defined by the sequence, hence the only logical way would be to have a fixed time interval between two consecutive notes.

We see that there are several parameters here: the starting point, the pitch difference corresponding to 1, and the time interval between notes. The Online Encyclopedia of Integer Sequences offers the conversion to music for any sequence. It gives you freedom to set the parameters yourself. The sequences do not sound melodic because mathematical sequences will not necessarily follow rules that comply with a nice melody. Moreover, there are no interesting rhythms because the time interval between the notes is always the same.

One day I received an email from a stranger named Michael Blake. He sent me a link to his video on YouTube called “What Pi Sounds Like.” He converted the digits of Pi to music. My stomach hurt while I was listening to his music. My stomach hurts now while I am writing this. He just numbered white keys on the piano from 1 to 9 starting from C. Then he played the digits of Pi. Clearly, Michael is not a mathematician, as he does not seem to know what to do with 0. Luckily for him the first 32 digits of Pi do not contain zero, so Michael played the first several digits over and over. My stomach hurts because he lost the basic math property of digits: the difference between the neighboring digits is the same. In his interpretation the digits that differ by one can have a tone interval of minor or major second in a random order corresponding to his random starting point.

I am not writing this to trash Michael. He is a free man in a free country and can do whatever he wants with the digits of Pi. Oops, I am sorry, he can’t do whatever he wants. Michael’s video was removed from YouTube due to an odd copyright infringement claim by Lars Erickson, who wrote a symphony using the digits of Pi.

Luckily for my readers Michael’s video appears in some other places, for example at the New Scientist channel. As Michael didn’t follow the symmetry of numbers and instead replaced the math rules with some music rules, his interpretation of Pi is one of the most melodic I’ve heard. The more randomly and non-mathematically you interpret digits, the more freedom you have to make a nice piece of music. I will say, however, that Michael’s video is nicely done, and I am glad that musicians are promoting Pi.

Other musicians do other strange things. For example, Steven Rochen composed a violin solo based on the digits of Pi. Unlike Michael, he used the same tone interval for progressing from one number to the next, like a mathematician would do. He started with A representing 1 and each subsequent number corresponded to an increase of half a tone. That is, A# is 2 and so on. Like Michael Blake he didn’t know what to do with 0 and used it for rest. In addition, when he encountered 10, 11, and 12 as part of the decimal expansion he didn’t use them as two digits, but combined them, and used them for F#, G, G# respectively. To him this was the way to cover all possible notes within one octave, but for me, it unfortunately caused another twinge in my stomach.

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Have you ever played with the “listen” feature of the Online Encyclopedia of Integer Sequences?

http://oeis.org/play?seq=A000796

2. #### Andrew MW:

If they used a base 7 version of pi it would better correspond to the typical diatonic scale. Another option would be a base 12 pi – which would correspond to a chromatic scale (a purer, more “mathematical” scale, following your lead). Whole tone scale is another option – requiring base 6.

3. #### Lyudmila German:

In Arnold Schoenberg’s so-called 12-tone system, 0 corresponds to C, 1 to C#, etc, up to 12 obviously. This system has nothing to do with pi and is not designed to represent any mathematical formula, moreover, it doesn’t contain any rules as to the rhythmic value of notes, but it was quite a while ago that connecting digits to notes was systematized (have to look it up, but I think some time in 1920s). The next generation of composers, such as Stockhausen, in the 1950s, tried to “serialize” other aspects of music, such as duration (how long) and dynamics (how loud/soft). That’s a step further into ‘mechanizing’ music, although I can’t say that the result was bad.

4. #### Peter Gerdes:

I’m sorry this is just dumb!

You can no insight from Pi by hearing this and you won’t get any deeply good music you couldn’t have gotten out of a random number generator (the regularities in Pi are too subtle to show up at the level of our musical appreciation).

Ultimately the whole thing is silly since if you wanted to produce good music you would look for an algorithm to do that and ignore pi and any regularities pi happens to share in common with good music could also be straightforwardly produced using that algorithm.

Sorry but as a mathematician who is not into classical music or particularly musically talented I get really fed up with this constant, empirically unfounded, attempt to suggest math and music are deeply related or similar. It’s true that for cultural reasons there is often overlap between people who study/practice music as children and those who go on to do math but that’s just cultural coincidence.

Even the studies suggesting that listenting to music while young makes you smarter have been largely discredited or never suggested the effect lasted more than 15m

5. #### Kivett Bednar:

Love the blog so far! Im a Berklee College of Music grad, and I got my degree in audio engineering. So I know there an awful lot of people who have successfully applied mathematical principles and patterns to music. To name a few: Johann Pachalbel (Canon in D), John Coltrane (Giant Steps), Bill Schoenberg (serial music). Some have been less successful (Schoenberg again :P). But I don’t think it’s out and out wrong to state that the subtlety of pi could be captured musically, albeit to some finite decimal place (just like in math).

I mean, I find it conceivable that a computer program could be created to find the most dulcet, for instance, combination of
Given that we provide the program with a definition of what “dulcet” is. I believe this definition, of “dulcet,” could be easily agreed upon since western music already defines the consonance (which in this context could be thought of as “dulcet-ness”) of the intervallic relationships found between the 12 notes it has at it’s disposal.

I happen to know classes are taught at North Texas that deal with this exact issue. The students write algorithms that in turn create music, is my understanding.

Interesting questions though.

6. #### John:

I agree with Peter Geres, there is no relationship between math and music!

7. #### ot:

There is a deep relationship between math and music. But it does not consist in such trivialities as converting a sequence in tones and playing it. Check, for instance, the work by David Lewin, Tom Fiore, Dmitri Tymoczko or Guerino Mazzola (to name just a few) to see the very deep theories that have been developed to explain music in mathematical terms.