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

4f 2f f

Likewise, 6f 3f , 8f f , 10f 5f , etc.

Harmonics (Octaves Give Same “Note”) Standing Wave Frequency

f 3f

4f 2f f

Likewise, 6f 3f , 8f f , 10f 5f , etc.

Harmonics (Octaves Give Same “Note”) Standing Wave Frequency

2f f 4f 2f f

Likewise, 6f 3f , 8f f , 10f 5f , etc.

Harmonics (Octaves Give Same “Note”) Standing Wave Frequency

2f f

3f 5f

Likewise, 6f 3f , 8f f , 10f 5f , etc.

Harmonics (Octaves Give Same “Note”) Standing Wave Frequency

2f f

4f 2f f Likewise, 6f 3f , 8f f , 10f 5f , etc.

Harmonics (Octaves Give Same “Note”) Standing Wave Frequency

2f f

4f 2f f

5f Harmonics (Octaves Give Same “Note”) Standing Wave Frequency

2f f

4f 2f f

Likewise, 6f 3f , 8f f , 10f 5f , etc. Which harmonics are emphasized and to what extent determines the sound quality.

Waveform Sound

Link

Link Table: Different instruments playing the same frequency

Timbre

In reality, a string vibrates at multiple harmonics simultaneously. Waveform Sound

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 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).

The third harmonic of f 1 gives us a new note r3s r3{2s { P r q p { q P r q with 3 2 1, 2 . (log2 3 2 0, 1 )

The distance (ratio) between f 1 and 3{2 is a perfect ﬁfth. It’s the most signiﬁcant interval behind the octave.

The Perfect Fifth

Every class in p0, 8q{ has a unique representative in r1, 2q. r1, 2q spans one octave. Donly one A note in r1, 2q, namely f 1 (in units 110 Hz).

The third harmonic of f 1 gives us a new note r3s r3{2s { P r q p { q P r q with 3 2 1, 2 . (log2 3 2 0, 1 )

The distance (ratio) between f 1 and 3{2 is a perfect ﬁfth. It’s the most signiﬁcant interval behind the octave.

The Perfect Fifth

Every class in p0, 8q{ has a unique representative in r1, 2q. (Every class in R{Z has a unique representative in r0, 1q.) Donly one A note in r1, 2q, namely f 1 (in units 110 Hz).

The third harmonic of f 1 gives us a new note r3s r3{2s { P r q p { q P r q with 3 2 1, 2 . (log2 3 2 0, 1 )

The distance (ratio) between f 1 and 3{2 is a perfect ﬁfth. It’s the most signiﬁcant interval behind the octave.

The Perfect Fifth