Abstract
The usual diatonic system is “dyadic,” for it privileges two intervals, the perfect octave and fifth; the usual harmonic system is “triadic,” for it privileges, in addition, the major and minor thirds (Sect. 9.1). The dyadic and triadic privileged intervals support, respectively, a dyadic/triadic notion of “consonance.” Every consonance other than the perfect prime has a unique “root,” such that, if the root is also the lower note, the consonance is “stable.” Section 9.2 studies the non-diatonic subset of the “cluster” (the set of all notes that may be received relative to the diatonic core, reduced to their register-zero representatives). It is shown that the subset consists of two length-five segments of the line of fifths, extending the seven-element core at either end to form a line-of-fifths segment totaling 17 elements exactly.
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- 1.
The mid ninth-century treatise Musica enchiriadis states as follows (Erickson 1995, p. 13): “Just as letters, when they are randomly combined with each other, often will not make acceptable words or syllables, so too in music there are certain intervals which produce the symphonies. A symphony is a sweet combination of different pitches joined to one another. There are three simple or prime symphonies, out of which the remaining are made. Of the former, they call one diatessaron, another diapente, the third diapason.”
- 2.
Writing on organum in the eleventh century, Guido d’Arezzo seems to have preferred a three-voice setting with the principle voice “sandwiched” between two organal voices, one situated a perfect fourth below, and the other, a perfect fifth above (Babb 1978, p. 77). This may well reflect the finding that the root of the perfect fifth is the lower note, whereas the root of the perfect fourth is the upper note.
- 3.
A chord, to be exact, is a sequence of notes. By ignoring order and repetition, a sequence is reduced to a set. Definitions 9.4 and 9.6 may be applied to chords as sequences reduced to sets.
- 4.
“Musica falsa est, quando de tono faciunt semitonium, et e converso. Omnis tonus divisibilis est in duo semitonia et per consequens signa semitonia designantia in omnibus tonis possunt applicari.” Latin text (after Coussemaker , 1864–1876, Vol. 1, p. 166, from Introductio secundum Johannem de Garlandia) and corresponding English translation from Dahlhaus (1990, p. 173).
- 5.
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Agmon, E. (2013). Tonal Preliminaries. In: The Languages of Western Tonality. Computational Music Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39587-1_9
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