Chapter 2 WELL-FORMED SCALES

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Music Theory Spectrum, 1989

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Journal of Mathematics and Music, 2007

2007

We divide the set of all diatonic scales into three classes P,G,R, the intersection of which contains the major diatonic scale. The class G contains Gypsy scales, the class P – Pythagorean heptatonic, and the class R – Redfield Scale. The paper could be of interest for the music theorists, mathematicians, as well as producers of modern key music instruments, interpreters, composers, and electro-acoustical studios.

It is to the author's credit that while writing this book he composed and recorded a set of twelve pieces in a series of unorthodox scales dividing the octave equally into thirteen, fourteen, fifteen etc. parts (the last piece being in equal temperament with quarter-tones). Yet the book is not presented as an adjunct to his probing compositional experiments. Instead, the publisher says on the dust-jacket that it 'analyzes all the important historical tunings', and Milton Babbitt says: 'In unprecedented detail and with unparalleled rigour, Blackwood determines and examines the various quantizations of the frequency continuum that have been employed and suggested for the "chromatic" systems of Western music'.

The topic of this thesis is the properties of scales and modes other than the familiar major and minor, and the possibilities these afford for extended tonality and alternative functional harmony. In chapter one, I develop a systematic account of how to describe a large set of musically useful scales by using a generated line of intervals. This account is based on Hook's (2011) use of the line of fifths, but generalized to any eligible generating interval. In chapter two, I review the theoretical literature on various relevant scale properties, such as transpositional asymmetry and intervallic content. I discuss the implications of these properties when modes of these scales are realized in actual music. In chapter three, I discuss the placement of "tendency tones"—minor seconds that resolve to a member of the tonic triad—in various scales and modes, and how this placement affects the tonal implications of a given mode. In connection with this argument I introduce the idea of modal pitch-class sums and show that a mapping exists between these sums and a scale's structure upon the line of fifths. Finally, chapter four discusses how different structures of harmonic functions (analogous to the familiar tonic, subdominant, and dominant functions) can be realized within a mode. Building on the account of tendency tones developed previously, I discuss the contribution of these functions to the tonal dynamics of a given mode.

We all know that diatonic modes can be obtained by stacking fifths. But this way you can find not only diatonic (and pentatonic) scales. Scales that cannot be arranged in fifths can also be obtained by stacking fifths. If we add a fifth to the sequence F C G D A E B and use the new note F# instead of F, we just change the key. The double step adds two fifths: B - F# (up) and F - Bb (down). Using the new notes F# and Bb instead of F and B, we can obtain a new scale (D melodic major). In this way we have obtained all scales that are not random sets of notes and are not atonal, like symmetrical scales. Once we have this limited set of scales, we can test various hypotheses about their properties. The work features a naming system that comprises 77 scales and modes.

In this work we provide an original approach to the classification of seven tone scales from the view point of composition and improvization. There are a total of 66 seven tone (heptatonic) scale formulas for equitempered systems out of which we eliminate 34 of them in view of the principles of tonal harmony and classify the remaining 32 scales into seven groups which are associated with the seven modes of the diatonic major scale. Such a classification is claimed to provide a perspective in understanding harmonic progressions and improvizational techniques for both the educator and the composer. We also provide a number of musical examples that demonstrate our methodology.

The Psychology of Music (D. Deutsch, Ed.), 2012

This chapter examines human sensitivity to pitch relations and the musical scales that help us to organize these relations. Tuning systems – the means by which scales and pitch relations are created and maintained within a given musical tradition – are also discussed. Questions addressed in this chapter include: How are pitch intervals processed by the auditory system? Do certain intervals have a special perceptual status? What is the relation between intervals formed by pitches sounded sequentially and those formed by pitches sounded simultaneously? Why is most music organized around scales? Are there similarities in the scales used in different musical systems across cultures? Is there an optimal tuning system?