The mathematical physics of music

Journal of Statistical Physics

Volumn 121, Issue 5-6, 2005, Pages 1097-1104

  Zweifel, P F ^a  

^a NONE

Author keywords

Chords; Microtonality; Music scales

Indexed keywords

EID: 29044447893     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-005-7581-1     Document Type: Review

References (12)

1. In the Middle Ages music was part of the Quadrivium, the higher division of the seven liberal arts. Music was thus grouped with arithmetic, astronomy and geometry and hence, evidently, considered a science. (The lower division, the trivium, consisted of grammar, rhetoric and logic.)
2. Thus "The music of the spheres." John Fauvel, Raymond Flood and Robin Wilson, eds. Music and Mathematics. Oxford University Presss, 2003. Chapter 2.
3. Donald E. Hall, Musical Acoustics, 2nd Ed. (Brook/Cole Publishing Company, Pacific Grove, CA 1980), p. 407.
4. An abuse of nomenclature has crept in, namely referring to the frequency (or "pitch") of a note. In first approximation, this means the frequency of the fundamental. However, a few instruments, e.g. tympani and some gongs, produce no fundamental; the ear constructs the pitch from the overtone series. (See Ref. 3, pp. 429-430.) It is the frequency of the fundamental, real or implied, that I mean when I write of the frequency of a note.
5. See Hermann Helmholtz, On the Sensations of Tone (2nd English Ed. Dover, New York, 1954). pp. 237 ff. It is possible, although unproved, that the Greeks had already used diatonic scales (at least whoever invented the Greek names like "Dorian," "Dorian," etc. for the diatonic modes must have thought so).
6. A pentatonic scale can be played on the five black keys of a piano (just as a diatonic scale can be played on the seven white keys). A familiar example of a pentatonic melody is Auld Lang Syne. Richard Wagner used pentatonic melodic lines in Der Ring des Nibelugen when he was dealing with voices of nature, for example the Forest Bird in Act II of Siegfried.
7. The fifth degree of every mode is called the "dominant" because in plainsong chant it was the usual reciting tone.
8. Much of the treatment in this section was taken from Gerald J. Balzano, Computer Music J. 4:66-84 (1980).
9. Inter alia, it gives the order in which sharps and flats appear in key signatures. (The "key" of a scale is simply its tonic note.)
10. Paul F. Zweifel, Perspectives of New Music 34:140-161 (1996).
11. pq are isomorphic. She or he pointed out further that the equivalent result in number theory is sometimes called the Chinese Remainder Theorem.
12. Harold C. Schonberg, The lives of the great composers. W. W. Norton and Co., New York (revised edition, 1981). p. 554.