Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

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strips of different width. The diatonic system consists of all pairs $(a.q,b.o) (- P$ where $a$ varies between $- 1$ und $5$ (c.f. Figure 1).

Figure 1:

The diatonic tone system uses only a vertical strip of the full pythagorean tone space. The fifth coordinates vary between -1 and 5, corresponding to tone names $F$ and $B$ . The tones are numbered according to their pitch height order yielding a zick-zack line.

The formal context for Carey’s and Clampitt’s contrafactual experiment consists of all vertical strips $P[n] = {(a .q,b.o)|- 1 \< a \< n - 2}$ for any width $n = 1,2,3,4,...$ . The prominent cases are $n = 5,7,12$ . Which additional property might be relevant for the understanding of these tone systems?

The restriction of the pitch height function $h : P --> R$ to the strip $P[n]$ and the linear order of the pitch height values induce the generic step numbering $s[n] : Z --> P[n]$ . To each integer $k (- Z$ there corresponds exactly one tone $s[n](k) (- P[n]$ and vice versa (We assume $s[n](0) = (0,0)$ ). Differences between these step numbers $k (- Z$ are called generic intervals.⁶

⁶: Up to translation between cardinal- and ordinal numbers we see that for $n= 7$ the generic intervals correspond to the diatonic interval names $0~$ »prime«, $1~$ »second«,....

Figure 1 displays the function $s[7] : Z --> P[7]$ . Each generic step number $k$ (between -7 and 21) is displayed at the position $s[7](k) (- P[7]$ .

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Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory