Abstract
A “semi-efficient” tone system consists of a composite tone system and a set of “primary” intervals generated by a privileged note interval (c, d), the “quintic element” (Sect. 5.1). In such a system, the note-interval transmission function, restricted to the primary intervals, is assumed to be one-to-one. A semi-efficient system is efficient if, in addition, the difference between the pitch images of two primary intervals (under the transmission function) is not arbitrarily small. It is proven that efficient tone systems are not type-3 systems (of which the complete “Pythagorean” system is a familiar example); on the other hand, equal-tempered systems, which are special types of naturally oriented type-1 or 2 tone systems, are efficient. An efficient tone system is “coherent” if a certain algorithm, by which a primary interval may easily be computed from its transmitted image, exists (Sect. 5.2). It is proven that coherent tone systems satisfy cb − ad = ±1, where (a, b) is the cognitive octave and (c, d) is the quintic element. Finally (Sect. 5.3), since efficient tone systems are equal tempered and not dense, one may relax the unrealistic assumption of absolutely accurate note transmission. By the theory of categorical equal temperament deviations from strict ET of up to one-half of one equal-tempered increment are allowed, in either direction.
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- 1.
“Indirect,” because it concerns an auxiliary communication system for (privileged) intervals. The present theory of “efficient primary intervals” is not unrelated to Agmon ’s (1986, 1989) theory of “efficient diatonic intervals.” A comparison of the two theories would be a valuable study in descriptive and, particularly, explanatory adequacy .
- 2.
Cf. Example 3.40B. Since the psychoacoustical octave ψ is set to twelve the logarithm base is the twelfth root of 2.
- 3.
Although a device capable of transmitting notes with absolute accuracy cannot be constructed in principle, neither can one restrict a priori the degree of precision such a device is hypothetically capable of attaining. On the other hand, one cannot assume as much concerning reception since in general, reception parameters are not under one’s control in the same strong sense as are transmission parameters.
- 4.
One may object to Theorem 5.8 on the grounds that it strongly depends on Definition 5.4 (“semi-efficiency”), where the Maximalist Principle is applied in a weak sense (i.e., systems of the same type t are compared; cf. Definition 5.6, “efficiency,” where systems of type t are compared to systems of type t′). However, one cannot simultaneously apply the Maximalist Principle in a strong sense (comparing two systems, not necessarily of the same type) and ask for non-density, since by Lemmas 5.9 and 5.10 these two conditions are mutually exclusive (i.e., strong maximality is equivalent to dense systems). Suppose that instead of requiring efficiency, one simply rules out type-3 systems a priori. One saves nothing in terms of the Maximalist Principle, since systems of type 1 and 2 still have to satisfy this principle. Since the non-density requirement comes at no extra cost, the choice is therefore between (a) ruling out type-3 systems a priori, and (b) applying the Maximalist Principle, albeit in a weak sense. Between these two choices, (a) is unquestionably ad hoc. I am indebted to Nori Jacoby for helping clarify these and other important considerations relating to Theorem 5.8.
- 5.
Since a and c are coprime the existence of m is assured. The so-called “Extended Euclidean Algorithm ” will find the desired m.
- 6.
- 7.
Since the coherence algorithm assumes, via Definition 5.11, a naturally oriented system, one may wonder why natural orientation was not assumed in the first place, say, as part of semi-efficiency (Definition 5.4). However, from Lemma 5.9 it is clear that such an assumption would have amounted to ruling out type-3 systems a priori. By Theorem 5.8, on the other hand, the ruling out of type-3 systems is not orientation-dependent.
- 8.
- 9.
An instrument capable of producing a continuum of pitches (violin, the human voice) is assumed. However, depending on one’s personal taste, even such a “discrete” instrument as the modern piano may be purposefully “mistuned” such that strict equal temperament is only approximated.
References
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Agmon, E. (2013). Communicating the Primary Intervals. In: The Languages of Western Tonality. Computational Music Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39587-1_5
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