The following description of the development of modern musical scales from acoustics exemplifies what is meant by instantiation and generalisation, which is one of the ordering principles used within subject domains in the Glass Bead Game variant proposed in 2010. The idea of this type of game, and the difficult challenge it presents, is then to identify an analogous ordering principle in other subject domains in the same game.
Each successive stage represents a special case of the preceding more general phenomenon. In this case, the order proceeds from general acoustics to the particularities of Western art music.
1. Moving objects move any atmosphere around them, producing sound waves.
2. Natural selection in most species favours binaural hearing which can distinguish between the pitches and harmonics produced by the sound source, because the inner ear’s analysis of the different pitch components of the sound waves, with the slightly different phase information in the left and right ears, facilitates the perception of the spatial position of complex sound sources such as predators or prey. In humans, complex sound signals are simplified by the brain which reduces natural harmonics to their fundamental tone, and perceives tones an octave apart as similar. These two related phenomena of sound perception (“low pitch” and “octave equivalence”), together with the laws of acoustics, determine how music has subsequently developed.
3. The ancient Greeks (the legend of the discovery is told in Plato’s Timaeus) discovered the mathematical relationship between sounds and their sources:
a. successive harmonics are produced by frequencies of vibration which are in arithmetic progression (and string lengths which are in harmonic progression);
b. successive musical intervals of equal size (e.g. each of the following intervals: the octave, perfect fifth and major third) are produced by frequencies of vibration which are in geometric progression (with a multiplicand of 2, 3 and 5 respectively for the above intervals).
4. Ancient Greek aesthetics favoured pitch intervals in the form of epimoric ratios (i.e. of the form n+1/n – e.g. octave = 2/1, perfect fifth = 3/2, major third = 5/4). The first and second overtones – the octave and the perfect fifth (and its inverse, the perfect fourth), together representing the three simplest epimoric ratios: 2/1, 3/2, 4/3 – became the basic structural foundation for Greek music.
a. The Pythagorean scale was formed from a “cycle” of 12 perfect fifths (which does not fit perfectly into the octave and leaves a “Pythagorean comma” of value 531,441/524,288 = (3/2)12/27 ).
b. Other scales were formed by dividing the perfect fourth into three further epimoric ratios (e.g. 9/8, 10/9, 16/15 – called the “diatonic” scale by Didymus and Ptolemy, and later forming the basis for the major and minor scales in European music). There are only 26 ways of combining three epimoric intervals in a perfect fourth, many of which were given individual names by Ancient Greek theorists.
Outside these developments recorded by theorists, in practice the octave was also divided by equal division of a string or pipe to give various modes, but no two intervals within the octave would be the same on such an instrument.
5. It is necessary, when constructing an instrument, to choose the intervals to be included in its pitch vocabulary, and common harmonic and modulatory practice are taken into account to optimise flexibility.
6. Eventually, the harmonic and modulatory limitations of instrument design based on integer ratios led to the invention of equal temperament, which is based on the division of the octave into twelve identical pitch intervals – i.e. successive semitones produced by frequencies of vibration which are in geometric progression with a multiplicand of the twelfth root of 2. The resulting pitch classes adequately approximate the main integer ratio intervals, and the scales and harmonic structures based on these integer ratios continued to be used and developed, until increasing chromaticism became atonality, and in particular in the early 20th century the definition of 12 tone serial music set an independent course for musical composition.