# Research of Scale

Jan 7th, 2011

Kelly Foss

Research of Scale

Even the most musically un-inclined are familiar with the musical scales. You might recall the major scale: do, re, mi, fa, sol, la, ti, do, from an elementary or high school music class, or perhaps C, D, E, F, G, A, B and C.

Dr. George Miminis, a professor in the Department of Computer Science, has spent a considerable amount of time thinking about the scales. In particular, asking: “Just how many possible scales are out there and what kind of structural rules do they obey?”

A musician by hobby, Dr. Miminis has founded two Greek bands in the city, The Forgotten Bouzouki and Acousmata. The bands have played at concerts, festivals, benefits and events with the local Greek community.

While he isn’t a music theorist, over the years he played traditional Greek as well as old folk Greek (Rebetika) music and began to realize that those melodies followed patterns or scales that were a lot more complicated than western melodies or even contemporary Greek music. More specifically, most western melodies (with some exceptions) follow either the major scale or one of three minor scales: natural, melodic or harmonic. In contrast, Rebetika and traditional Greek songs follow about 20 different scales.

The existence of so many ways to write a song made him wonder about the nature of scales. For example, what may be a limit on the number of them or are there structural relations amongst them?

He discovered that musical scales have an intriguing structure which offers itself to rigorous mathematical study. He then set foundations to such a study that he hoped to enhance their theoretical understanding.

A scale is defined as any sequence of consecutive notes or equivalently, intervals. “Taking the major scale, instead of saying it as a sequence of notes, for example C, D, E, F, G, A, B, C, you can also say it as a sequence of intervals,” he explained. “In this case, it is: [tone-tone-semitone]-tone-[tone-tone-semitone].”

Since going from C to D we need a tone, D to E another tone, from E to F a semitone, etc.

This very general definition, however, does not allow much room for analysis. Dr. Miminis restricted the definition to that of a natural scale. He defined a natural scale to be a sequence of seven consecutive intervals (or equivalently eight consecutive notes) made up of semitones, tones and trisemitones (three semitones) to a total of 12 semitones. A natural scale spans an octave.

“All the Greek songs that I have encountered, and they are many, as well as most western songs fall into this category,” he said.

“If you change the order of the intervals of a scale, for example from the major scale to [semitone-tone-tone]-tone-[semitone-tone-tone] you get a completely different structure and if you play it on any instrument it has a different mood. As a mathematician and computer scientist, I started wondering what the possibilities were.

“Is there a limit to the different ways we can group these intervals? Can we create families of scales that have similarities? And what would those similarities be?”

Dr. Miminis decided to put his background to use and ultimately created a mathematical model that could describe all the natural scales and how to group them in a way that could reveal their structural relationships. This could also give an answer to how many there were. He found that there were 38 families of natural scales where each family consisted of seven different scales, giving therefore a total of 266 natural scales. This has been documented in a paper entitled Musical Scales: A Structural Point of View.

“Did that help me in playing music? No, but it helped my curiosity,” he said. “It was pretty unsettling when I didn’t know what restrictions scales had and I think that’s how scientists work. They want to find the space to which the subjects they study belong and then determine its limits, properties, etc.”

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