An exploration into musical scales. How do we choose the notes for a scale?
In all the music systems I looked at, you have some rules for generating "all the possible notes in music". These tend to be rules to generate a set of notes within an octave, which is a fundamental divison in music because of how similar two tones in an octave are.
Equal Temperament Scale. A scale where the interval between adjacent notes is a constant. This means that the interval will generally be an irrational number and be slightly dissonant. However, the scale has no singularly bad interval (a "wolf" interval) and transposes easily.
The perfect fifth is an interval where the frequency ratio is 3/2. You can extend this chord 12 times to get a near-octave - 3^12 is within 1.4% of 2^19. All notes on the scale are a product of some power of 3 and 2. This scale is called Pythagorean tuning in the West and is attributed to Pythagoras, but was also used as the basis for ancient Chinese musical scales, where it is known as the Shi Er Lu (The Twelve Notes).
The Shruti are a set of 22 notes used in Indian music; They are chosen by a type of 5-limit tuning. It is not used directly; rather subsets of the notes in the Shruti form musical modes called ragas that inform the theme of the composition. The wikipedia shruti scale uses at most one perfect third, i.e. a 5/4 interval in combination with up to 6 perfect fifths, i.e. a 3/2 interval.
A base scale tends to have too many notes, and not all of them sound good with each other. Can we choose a subset of the base scale so that it sounds more pleasing? I'm going to call this concept a mode, which in music theory is a set of notes with some melodic characteristic. For example, in Western music, you can pick 7-note Greek modes out of a 12-note chromatic scale, or you can pick 5-note pentatonic modes out of a variety of base scales.
Smallest Fractions. Maybe the "best" notes are the notes that have the "nicest" fractions, i.e. the numerator and denominator are small. By this logic, 9/8 is a better fraction than 256/243.
Uniform Division. Try to break the scale down into N equally spaced tones, picking the closest match.
Minimize Dissonance. Many scales will have one or more "wolf" intervals which sound really discordant. Can we take a selection of notes and find a subset that minimizes pairwise dissonance over all pairs of notes in the scale?