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Russian Pitch-Class Set Analysis and the Music of Webern
Russian Pitch-Class Set Analysis and the Music of Webern

... help to analyze atonal music.7 In so doing, Forte essentially gave us many of pc set theory’s basic notions: “‘PC set,’ ‘normal’ or ‘prime form,’ and ‘interval vector’ have become household concepts for many music theorists, but in the mid-1960s they met a very specific need: the need for definition ...
Pentatonic Scales
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Generalized Interval System and Its Applications
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... A diagram of a keyboard. Each key is associated with repeating sequence of notes. ...
Nicola Vicentino and the Enharmonic Diesis
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... sources. I will then discuss Vicentino‟s adaptation of the Greek genera, his theory of interval affect, and the features of his 31-tone tuning system. The discussion will focus on Vicentino‟s advocacy for the use of microtonal step of the minor enharmonic diesis as a means of affective expression. C ...
Tripod Notation article
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... tone that you find between a pure, perfect fourth and fifth. Magic temperament has two different sizes of whole tone in the nine-limit that both end up as major seconds in mean1.5 ...
Lesson 10: The Power of Algebra—Finding Pythagorean Triples
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Mazzola`s Counterpoint Theory - Dmitri Tymoczko
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Chapter One Introduction to Quarter
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... So, the focus of this paper is on the easier (albeit still challenging!) problem of developing scored one-dimensional indices for processing multi-dimensional events. Given the above system architecture, the main technical challenge is devising efficient Scored Interval Indices. Existing interval in ...
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... according to various mathematical considerations. The result is an alternative harmonic system that opens new possibilities to contemporary and future music. Acoustically speaking, the octave's frequency ratio 1:2 is replaced by the ratio 1:3 in the Bohlen-Pierce scale, making the perfect twelfth an ...
Mathemusical Thought - LUC Sakai
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... length 5 half steps, as per Table 2.1. Thus this is an augmented fourth, also called a tritone. (b) IntervalName=“fifth”. IntervalLength=6= 7 -1. A perfect fifth is of length 7 half steps. Thus this is a diminished fifth, likewise called a tritone. (c) IntervalName=“third”. IntervalLenth=3. A third ...
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Quarter-comma meantone

Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning. The difference is that in this system the perfect fifth is flattened by one quarter of a syntonic comma, with respect to its just intonation used in Pythagorean tuning (frequency ratio 3:2). The purpose is to obtain justly intoned major thirds (with a frequency ratio equal to 5:4). It was described by Pietro Aron (also spelled Aaron), in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be ""sonorous and just, as united as possible."" Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.
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