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The Role of Continued Fractions in Rediscovering a Xenharmonic Tuning

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The Role of Continued Fractions in Rediscovering a Xenharmonic Tuning The Role of Continued Fractions in Rediscovering a Xenharmonic Tuning Jordan Schettler University of California, Santa Barbara 10/11/2012 Outline 1 Motivation 2 Physics 3 “Circle” of Fifths 4 Continued Fractions 5 A Tuning of Wendy Carlos Motivation Helped to popularize and improve the Moog synthesizer Used creative and unconventional tunings in original compositions Wendy Carlos American composer and Grammy winner Used creative and unconventional tunings in original compositions Wendy Carlos American composer and Grammy winner Helped to popularize and improve the Moog synthesizer Wendy Carlos American composer and Grammy winner Helped to popularize and improve the Moog synthesizer Used creative and unconventional tunings in original compositions Carlos’ Best Known Works: Soundtracks 1971 1980 1982 Carlos’ Best Known Works: Albums 1968 1984 1986 pγ 35:097:::q The Title Track on Beauty in the Beast uses equally spaced notes with α and β cents between consecutive notes where α 77:995::: β 63:814::: An Interesting Tuning On a standard keyboard or guitar, notes are equally spaced with 100 cents between consecutive notes. pγ 35:097:::q An Interesting Tuning On a standard keyboard or guitar, notes are equally spaced with 100 cents between consecutive notes. The Title Track on Beauty in the Beast uses equally spaced notes with α and β cents between consecutive notes where α 77:995::: β 63:814::: An Interesting Tuning On a standard keyboard or guitar, notes are equally spaced with 100 cents between consecutive notes. The Title Track on Beauty in the Beast uses equally spaced notes with α and β cents between consecutive notes where α 77:995::: β 63:814::: pγ 35:097:::q Physics The period T is the number of seconds in one cycle. The (fundamental) frequency f 1{T (in Hz 1{sec) is the number of cycles per second. If L is the length of the string, then v f 2L where v depends only on the density and tension. An Ideal String with Fixed Endpoints (Or Open Pipe) Vibrations periodic compression waves in air (sound). The (fundamental) frequency f 1{T (in Hz 1{sec) is the number of cycles per second. If L is the length of the string, then v f 2L where v depends only on the density and tension. An Ideal String with Fixed Endpoints (Or Open Pipe) Vibrations periodic compression waves in air (sound). The period T is the number of seconds in one cycle. If L is the length of the string, then v f 2L where v depends only on the density and tension. An Ideal String with Fixed Endpoints (Or Open Pipe) Vibrations periodic compression waves in air (sound). The period T is the number of seconds in one cycle. The (fundamental) frequency f 1{T (in Hz 1{sec) is the number of cycles per second. An Ideal String with Fixed Endpoints (Or Open Pipe) Vibrations periodic compression waves in air (sound). The period T is the number of seconds in one cycle. The (fundamental) frequency f 1{T (in Hz 1{sec) is the number of cycles per second. If L is the length of the string, then v f 2L where v depends only on the density and tension. 2f f 3f 4f 2f f 5f Likewise, 6f 3f , 8f f , 10f 5f , etc. Harmonics (Octaves Give Same “Note”) Standing Wave Frequency f 3f 4f 2f f 5f Likewise, 6f 3f , 8f f , 10f 5f , etc. Harmonics (Octaves Give Same “Note”) Standing Wave Frequency f 2f f 4f 2f f 5f Likewise, 6f 3f , 8f f , 10f 5f , etc. Harmonics (Octaves Give Same “Note”) Standing Wave Frequency f 2f f 3f 5f Likewise, 6f 3f , 8f f , 10f 5f , etc. Harmonics (Octaves Give Same “Note”) Standing Wave Frequency f 2f f 3f 4f 2f f Likewise, 6f 3f , 8f f , 10f 5f , etc. Harmonics (Octaves Give Same “Note”) Standing Wave Frequency f 2f f 3f 4f 2f f 5f Harmonics (Octaves Give Same “Note”) Standing Wave Frequency f 2f f 3f 4f 2f f 5f Likewise, 6f 3f , 8f f , 10f 5f , etc. Which harmonics are emphasized and to what extent determines the sound quality. Waveform Sound Link Link Link Table: Different instruments playing the same frequency Timbre In reality, a string vibrates at multiple harmonics simultaneously. Waveform Sound Link Link Link Table: Different instruments playing the same frequency Timbre In reality, a string vibrates at multiple harmonics simultaneously. Which harmonics are emphasized and to what extent determines the sound quality. Timbre In reality, a string vibrates at multiple harmonics simultaneously. Which harmonics are emphasized and to what extent determines the sound quality. Waveform Sound Link Link Link Table: Different instruments playing the same frequency If hpxq is real and odd, then Euler’s formula implies ¸8 hpxq an sinpnxq n1 where an 2icn P R is the amplitude of the nth harmonic. A Little Functional Analysis In the Hilbert space L2pr0; 2πsq, Dorthonormal decompositions ¸8 inx hpxq cne n¡8 inx inx where |cn| |xhpxq; e y| length of the projection onto e . A Little Functional Analysis In the Hilbert space L2pr0; 2πsq, Dorthonormal decompositions ¸8 inx hpxq cne n¡8 inx inx where |cn| |xhpxq; e y| length of the projection onto e . If hpxq is real and odd, then Euler’s formula implies ¸8 hpxq an sinpnxq n1 where an 2icn P R is the amplitude of the nth harmonic. 8 ¸ 1 spxq sinpnxq n n1 Fourier series Graph Analog synthesizers use sawtooths via subtractive synthesis. Example: Sawtooth Wave Consider the 2π-periodic function spxq s.t. π ¡ x spxq on p0; 2πq 2 Analog synthesizers use sawtooths via subtractive synthesis. Example: Sawtooth Wave Consider the 2π-periodic function spxq s.t. π ¡ x spxq on p0; 2πq 2 8 ¸ 1 spxq sinpnxq n n1 Fourier series Graph Example: Sawtooth Wave Consider the 2π-periodic function spxq s.t. π ¡ x spxq on p0; 2πq 2 8 ¸ 1 spxq sinpnxq n n1 Fourier series Graph Analog synthesizers use sawtooths via subtractive synthesis. If we plug in the sawtooth spxq, ¸8 ¸8 1 1 | |2 || p q||2 2 cn s x 2 n n¡8 n 1 » ¢ 1 2π π ¡ x 2 π2 dx 2π 0 2 12 A Famous Identity The Pythagorean Theorem holds for Hilbert spaces: ¸8 2 2 |cn| ||hpxq|| n¡8 A Famous Identity The Pythagorean Theorem holds for Hilbert spaces: ¸8 2 2 |cn| ||hpxq|| n¡8 If we plug in the sawtooth spxq, ¸8 ¸8 1 1 | |2 || p q||2 2 cn s x 2 n n¡8 n 1 » ¢ 1 2π π ¡ x 2 π2 dx 2π 0 2 12 The octaves A1 55 Hz, A3 220 Hz, etc., are also A notes. There is an equivalence relation on frequencies f ; g P p0; 8q: n f g ô f 2 g for some n P Z Back to Frequencies and Notes We define A2 110 Hz. There is an equivalence relation on frequencies f ; g P p0; 8q: n f g ô f 2 g for some n P Z Back to Frequencies and Notes We define A2 110 Hz. The octaves A1 55 Hz, A3 220 Hz, etc., are also A notes. Back to Frequencies and Notes We define A2 110 Hz. The octaves A1 55 Hz, A3 220 Hz, etc., are also A notes. There is an equivalence relation on frequencies f ; g P p0; 8q: n f g ô f 2 g for some n P Z We have isomorphisms of topological groups: p q p q p0; 8q log2 R exp 2πi / / S1 C Z frequencies notes circle group Take unit of frequency 110 Hz. Then for n P Z n 2πin r2 s (any A note) ÞÑ n Z ÞÑ e 1 In general, p q r s ÞÑ p q ÞÑ 2πilog2 f f log2 f Z e A Little Algebra Also, A1 and A2 are the same “distance” apart as A2 and A3. Take unit of frequency 110 Hz. Then for n P Z n 2πin r2 s (any A note) ÞÑ n Z ÞÑ e 1 In general, p q r s ÞÑ p q ÞÑ 2πilog2 f f log2 f Z e A Little Algebra Also, A1 and A2 are the same “distance” apart as A2 and A3. We have isomorphisms of topological groups: p q p q p0; 8q log2 R exp 2πi / / S1 C Z frequencies notes circle group In general, p q r s ÞÑ p q ÞÑ 2πilog2 f f log2 f Z e A Little Algebra Also, A1 and A2 are the same “distance” apart as A2 and A3. We have isomorphisms of topological groups: p q p q p0; 8q log2 R exp 2πi / / S1 C Z frequencies notes circle group Take unit of frequency 110 Hz. Then for n P Z n 2πin r2 s (any A note) ÞÑ n Z ÞÑ e 1 A Little Algebra Also, A1 and A2 are the same “distance” apart as A2 and A3. We have isomorphisms of topological groups: p q p q p0; 8q log2 R exp 2πi / / S1 C Z frequencies notes circle group Take unit of frequency 110 Hz. Then for n P Z n 2πin r2 s (any A note) ÞÑ n Z ÞÑ e 1 In general, p q r s ÞÑ p q ÞÑ 2πilog2 f f log2 f Z e “Circle” of Fifths (Every class in R{Z has a unique representative in r0; 1q.) r1; 2q spans one octave. Donly one A note in r1; 2q, namely f 1 (in units 110 Hz).
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