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Overview and Introduction.

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This presentation uses 3:2 ratios to create a 12-tone set very much like the twelve tone chromatic system in common use today. Despite its out-of-tune triads, this scale seems very functional but suffers from a gap in its frequencies, another comma, meaning it is not circular. Since the tonal pitch system relies on circularity (the ability

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This presentation explains the octave, a 2:1 ratio of frequencies, and octave equivalence, the core concept in tonal systems. An oscilloscope is used to visualize various musical ratios including the perfect fifth and perfect fourth. The connection is made between musical intervals and simple whole number ratios.

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This presentation will expand the concept of octaves as a spectrum to incorporate the circle of fifths, a common diagram for study of music theory. Understanding the circle of fifths leads to a larger realization; tonal pitch systems are circular. The 5:4 major third will be added to the set of simple number ratio intervals we've studied, 2

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This presentation will demonstrate how the 4:5:6 ratio known as a major triad works within the circular pitch system of tonal music. The concept of I, IV, and V (1, 4, and 5) harmonic functions will be explained using the basic mathematical structures already demonstrated.

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This presentation explores how the major triads formed by the Pythagorean tuning system sound when compared with pure 4:5:6 major triads. The differences are subtle, but easy to hear and see using an oscilloscope, and begin to complicate the apparently simple system of using 3:2 and 4:5:6 ratios to form a tonal pitch system.

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This presentation shows how 3:2 frequency ratios can be stacked to form a major scale (also known as a diatonic scale, diatonic collection, or diatonic set). Now we've progressed to the point of generating functional sets of frequencies--scales--using simple math and an understanding of how tonal pitch systems work. This kind of approach to

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This presentation introduces the 6:5 minor third and 10:12:15 minor triad. The tonal pitch system uses major and minor triads, and the pitch system in use has to accommodate this. These sounds and ratios lead to the construction of a diatonic circle of thirds and a further goal; tune a diatonic set so that all I IV V major and i iv v minor

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This presentation uses 3:2 ratios to create a 12-tone set very much like the twelve tone chromatic system in common use today. Despite its out-of-tune triads, this scale seems very functional but suffers from a gap in its frequencies, another comma, meaning it is not circular. Since the tonal pitch system relies on circularity (the ability

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This series is one part of UC Irvine's Musicianship 15 ABC sequence for music majors. An understanding of music notation and basic musical terms is helpful but not required for these presentations. The math involved is basic. Pitch systems use mathematics to organize audible phenomenon for creative expression. The cognitive processes we d

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This presentation defines basic terms and compares the range of audible sound to the visual spectrum. The fundamental principles of sound waves are illustrated and the simplified waves (sine waves) used throughout the series are explained.