We all know the C-major scale: do-re-mi-fa-sol-la-ti-do. But what’s behind it? And how many other scales there are? It’s complicated. Let me do a brief introduction into the theory first, without trying to be precise or complete.

In use are more than a dozen scales, and the popular one in the western world are the major, minor (natural, harmonic), and the “old modes”: Dorian, Lydian, Locrian, etc. All these are heptatonic (7-tone) scales. There are also pentatonic (5-tone) scales, and also other scales like Turkish, Indian, Arabic. All of them share a common purpose: to constraint melodies in order to make them sound pleasant. The notes in each scale trigger a different level of consonance with each other, which in turn provides different “feel”. The predominant scales all fall within the so called chromatic scale, which consists of all the 12 note octave on a piano keyboard (counting both white and black keys).

How are the scales derived? There are two main aspects: the harmonic series and temperament. The harmonic series (closely related to the concept of an overtone) are derived from the physical behaviour of the musical instruments, and more precisely – oscillation (e.g. of a string). The harmonic (or overtone) series produce ever-increasing pitches, which are then transposed into a single octave (the pitch space between the fundamental frequency and 2 times that frequency). This is roughly how the chromatic scale is obtained. Then there is temperament – although the entirely physical explanation sounds a perfect way to link nature and music, in practice the thus obtained frequencies are not practical to play on musical instruments, and also yield some dissonances. That’s why musicians are tuning their instruments by changing the frequencies obtained from the harmonic series. There are multiple ways to do that, one of which is that 12-tone equal temperament, where an octave is divided in 12 parts, which are equal on a logarithmic scale (because pitch changes are perceived as the logarithm of their frequencies).

But what does that have to do with programming? Computers can generate an almost infinite amount of musical scales that follow the rules of the scales already proven to be good. Why limit ourselves to 7-tone scales out of 12 tones, when we can divide the octave into 24 parts and make a scale of 15 tones? In fact, some composers and instrument makers, the most notable being Harry Partch, have experimented with such an approach, and music has been written in such “new” scales (although not everyone would call it “pleasant”). But with computers we can test new scales in seconds, and write music in them (or let the computer write it) in minutes. In fact, I see this as one way for advancing the musical landscape with the help of computers (algorithmic composition aside).

That’s why I wrote a scale generator. It takes a few input parameters – the fundamental frequency, on which you want to base the scale (by default C=262.626); the size of the scale (by default 7); the size of the ‘chromatic scale’ out of which the scale will be drawn (by default 12); and the final parameter specifies whether to use equal temperament or not.

The process, in a few sentences: it starts by calculating the overtones (harmonics), skipping the 7th (for reasons I don’t fully understand). Then transposes all of them into the same octave (it does so, by calculating the ratio from a given harmonic to its tonic (the closest power-of-two multiple of the fundamental frequency), and then using that ratio calculates the frequency from the fundamental frequency itself. It does that until the “chromatic scale size” parameter value is reached. Then it finds the perfect interval (perfect fifth in case of heptatonic (diatonic) scale), i.e. the one with ratio 3/2. If equal temperament is enabled, the previous chromatic scale is replaced with an equal-tempered one. Then the algorithm makes a “circle” from the tones in the chromatic scale (the circle of fifths is one example), based on the perfect interval, and starting from the tone before the fundamental frequency, enumerates N number of tones, where N is the size of the scale. This is the newly formed scale. Note that starting from each note in the scale we just obtained (and continuing in the next octave when we run out of tones) would yield a completely different scale (this is the difference between C-major and a A-minor – they use the same notes)

Finally, my tool plays the generated scale (using low-level sound wave generation, which I copied from somewhere and is beyond the scope of this discussion) and also, using a basic form of my music composition algorithm, composes a melody in the given scale. It sounds terribly at first, because it’s not using any instrument, but it gives a good “picture” of the result. And the default arguments result in the familiar major scale being played.

Why is this interesting? Because hopefully music will evolve, and we will be able to find richer scales pleasant to listen to, giving composers even more material to work with.