Natural Evolution of our Modern Tuning System

The diatonic or natural scale, consisting of five whole tones and two opposed semitones, is most familiar today in the white notes of the piano [Apel. see Diatonic]. On the piano this would be called C-major, which imposes the sequence of tones (T) and semitones (S) as T-T-S-T-T-T-S in which the initial and final tetrachords are identically T-T-S, leaving a tone between F and G, the two fixed tones of the Greek tetrachordal system

The diatonic scale is … an abstractum; for all we have is five tones and two semitones a fifth apart [until] we fix the place of the semitones within the scale, thereby determining a definite succession …, [and] we create a mode. [Levarie. 213].

Musical Morphology,. Sigmund Levarie and Ernst Levy. Ohio:Kent State 1983. 213.

One can see that the tones are split by the major diatonic into one group of two (T-T) and one group of three (T-T-T), so the semitones are opposed (B-F) towards the tonic C as in figure 1.

Figure 1 Tone circle and tetrachords for C-Major also called the modal scale of Ionian

Letters such as C are called note classes so as to label the tones of a diatonic scale which, shown on the tone circle, can be rotated into any key signature of twelve keys including flattened or sharpened notes, shown in black in figure 1. We will first show how these black notes came about naturally, due to two aspects of common usage.

The note classes arose from the need of choral music to notate music so that it could be stored and distributed. When we “read music” today, the tablature consists of notes placed within a set of five lines with four gaps, and two extendable areas above and below in which only seven note classes can be placed, seven being the number of note classes in the modal diatonic and the number of white keys on the keyboard, which is the other aspect of usage.

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Equal Temperament through Geometry and Metrology

The form of musical scale we use today is the (apparently modern) equal tempered scale. Its capabilities express well the new mind’s freedom of movement in that it allows us to change key to play compositions that move between alternative frameworks. This possibility was known to ancient tuning theory, could be approximated within Just intonation’s chromatic notes and was discussed by Plato as forming the constitution of one of his harmonic city states called Magnesia.

Relationship of the Equal Temperament Keyboard to the (logarithmic base-2) tone circle of an octave. We choose D (the Dorian scale) because it is symmetrical on both keyboard and tone circle. Equal Temperament supplies tones which enable any scale to be played starting from any note, though it is the Ionian (C-major) which defines modern key signatures.
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