Brief Explanation of Differences Between Pythagorean and Non-Pythagorean Scales

On their latest CD, New Magnetic Wonder, The Apples in Stereo feature two songs composed in a new type of scale called a Non-Pythagorean scale. Apples frontman Robert Schneider’s dual fascinations with music and math led him to create the scale as a way to explore new musical avenues. But what’s so different about it?

WARNING: Music Theory and Mathematics Content
Music scales are basically defined by the intervals they contain. Standard scales are arrangements of whole steps and half steps. These steps are defined by the ratio of the frequencies of the pitches, measured in Hertz (Hz). An octave of a typical scale is made up of the notes that fall between the frequency of the first note and the next occurrence of that note, which is at precisely double the frequency. For example, a C Major scale can start on Middle C (264 Hz) and end on the next C (528 Hz). The eleven notes between are arranged at equal intervals between the two notes (at least they are when you use equal temperament…but that’s another story…). Using all of these notes gives us a chromatic scale. Most modern music is made using scales that leave out many of the notes. The result is the familiar series of tensions and resolution we are accustomed to. This system was first written down by Greek mathematician Pythagoras, so these can be considered Pythagorean Scales.

Pythagorean Scale Frequencies

So, what’s so different about Robert Schneider’s Non-Pythagorean Scale? The ratio of the intervals’ frequencies is not equal. It’s based on the natural logarithm of the scale tone, leading to a non-linear distribution of intervals. The space between notes at the bottom of the scale is much larger than the space at the top of it. Most of the pitches defined this way do not correspond to notes found in traditional scales because their pitches end up being irrational numbers (numbers that that cannot be derived by n/m, where n and m are integers), and all of the pitches in the Pythagorean system are rational. This results in melodies and harmonies that are at once completely foreign and oddly satisfying.

nonpythagorean

“Non-Pythagorean Composition 1″

Audio clip: Adobe Flash Player (version 9 or above) is required to play this audio clip. Download the latest version here. You also need to have JavaScript enabled in your browser.

For more information on the Non-Pythagorean Scale check out the bonus materials on the New Magnetic Wonder CD.

UPDATE: The scale explanation from the CD is now available on The Apples in Stereo website

ANOTHER UPDATE: They’ve redesigned their site and the explanation isn’t there anymore.

9 comments

Interesting… I think the composition sounds like a Pythagorean piece that’s being played underwater, thus distorting pitch. I did not find it satisfying.

Other scales are not Pythagorean and sound much better, like Gamelan music.

I’ll stick to ratio-driven intervals.

“Non-Pythagorean scales” are emphatically not /new/. Even equal temperament is based on logarithmic relationships between notes, chosen to approximate some of the most fundamental Pythagorean relationships. Non-Western scales using non-Pythagorean relationships have existed for millennia.

True, I probably worded that poorly. There are various non-Pythagorean scales in use all over the world. Cut me a little slack…it’s been almost 15 years since I took music theory, and even then I slept through most of it.

That hurts the ears. Such ugly sounds…

Yuck! I can’t stand when people think they are being creative by putting notes together that are clearly clashing frequencies.

Jimmy Jam you have no idea what your are talking about. If you have ever listen or played any pieces of Eric Whitacre then you will become to respect clashing. Music is all about the tension and release. I highly recommenced “Sleep” by Eric Whitacre. I’ve played the trans-composed piece to band and it’s nothing but gorgeous.

Justin,

While I will not get into the merits (or demerits, as it were) of Whitacre’s compositions, he employs dissonance that is readily accepted by Western culture. Non Pythagorean chords do not create the same effect at all. Think about it this way: what makes a tritone more dissonant than a major 7th?

There are two causes:

First: Overtones/harmonics are structured in such a way that some notes are more closely related to the fundamental than others. For example, a perfect fifth can be found in the second harmonic, so it’s practically the same note. A major 7th above the fundamental is more closely related in the overtones than a tritone is. The same goes for these new scale tones, because they do not exist in the early harmonics.

Second: A tritone is more dissonant because two notes must move to resolve. In these new chords, a simple half step in one direction will not create consonance.

To sum it up, lay off Jimmy. Dude knows what he likes.