Flexibly Defined Tuning Systems Using Continuous Fourier Transforms

Jason Yust, .

Mathematical and Computational Models in Music Workshop University of Pavia, June 2021 Remembering Jack Douthett (1942–2021)

Two important ideas: • Evenness • Continuous functions underlying discrete phenomena

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Outline (1) Flexible tuning systems (2) Periodic functions: heptatonicity, triadicity, etc. (3) Fourier coefficients, spectra, phases (4) Application: Balinese Pelog (5) Application: Persian Dastgah tuning

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Flexibly Defined Tuning Systems

A critique of defining tunings as idealized pitch sets Tuning Theory

The usual framework: • A scale or tuning can be represented by a set of points in frequency space. • Pitches occurring in practice are approximations of these — i.e., these are the intended frequencies, realized to within some tolerance (degree of precision).

What’s wrong here?

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Tuning Theory

Tuning variability is viewed as error. This is a distinctly classical-European attitude. Other traditions have a positive attitude towards tuning flexibility.

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Tuning Theory

Example, Violinist vs. fiddler:

In both cases the tuning system is considered to be 12-tET.

The violinist prizes accuracy of intonation and recognizes theoretical point in frequency space as ideal representations of notes. However, the violinist also recognizes the possibility of expressive intonation (frequency vibrato, sharpened leading tones), within narrow constraints. The fiddler also recognizes note identities within a 12-t system, tied to regions of frequency space. The fiddler requires a wider range of tuning flexibility for expressive intonation, such as portamento.

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Tuning Theory

An alternative framework: • A scale or tuning is defined by flexible interval categories. • Within some scale-identifying constraints, tuning flexibility is a resource of the scale system.

Example, violinist and fiddler: Both require scale-defining tuning constraints. Both also prize flexibility of intonation within that system (to differing degrees). Pythagorean and just intonation fall within that range of flexibility. Rather than distinct tuning systems, these are intonational variations of a system that can be applied ad hoc as the musical situation demands (tuning up a chord in a string quartet, making a melody stand out, etc.).

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Tuning Theory

Interval categories are transposable. A closed system is therefore periodic (in pitch / log-freq.) An open system can be understood as a subset of closed systems.

Multiple “grains” of interval categorization can exist simultaneously. (Ex.: Generic and specific intervals.)

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Towards Fourier Theory, Some Concepts

Heptatonicity Chromaticity Triadicity Dyadicity Diatonicity Hexatonicity Octatonicity Heptatonicity Heptatonic: Division of the 8ve into 7 equally spaced bins. Heptatonicity: Pitch collection viewed through such a division.

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Heptatonicity Example: C Diatonic scale Phase: 2!/6

Magnitude: 0.5 + 1.0 + 0.5 + 0 + 0.87 + 0.87 + 0 = 2.28

Formula: f7 = ∑cos(2!xi · 7/8ve) + j ∑sin(2!xi · 7/8ve)

Magnitude = |f7|, Phase = arg(f7)

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Heptatonicity Heptatonicity is an evenness measure for scales C diatonic scale

0.5 + 1.0 + 0.5 +0 + 0.87 + 0.87 + 0 = 3.73

C harmonic minor

0.97+0.26+0.26 +0.97 +0.71–0.26 – 0.97 = 1.93

C Hungarian minor (CDEbF#GAbB)

1.0 + 0.5 + 0 – 1.0+0.87–0.5 – 0.87 = 0

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Heptatonicity But: Notes are not forced to cover scale degree “bins”

Example: C-D#-E-F-G-A-Bb

1.5

0.5

0 C C# D Eb E F F# G Ab A Bb B C -0.5

-1 1.0 + 0 – 0.5 + 0.87 + 0.87 + 0 + 0.5 = 2.73 -1.5

This bin is empty This bin contains two notes The heptatonic spans of this scale are 2-0-1-1-1-1-1 Heptatonicity only measures evenness when interval spans are 1-1-1-1-1-1-1

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Heptatonicity Heptatonicity does not assume any temperament

1 0.5 Pythagorean 0 C C# D Eb E F F# G Ab A Bb B C diatonic -0.5

-1 0.37 + 1.0 + 0.37 – 0.21 + 0.82 + 0.82 – 0.21 = 2.97

0.5 Just

0 C C# D Eb E F F# G Ab A Bb B C diatonic -0.5

-1 0.64 + 0.95 + 0.76 + 0.10 + 0.96 + 0.99 + 0.26 = 4.65

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Heptatonicity Heptatonicity can also be measured for subsets of scales

C major triad 2-2-3

0.5 + 0.5 + 0.87 = 1.87

C dominant seventh 2-2-2-1

0.97 – 0.26 + 0.97 + 0.26 = 2.46

C add 6 2-2-1-2 0.5 + 0.5 + 0.87 + 0.87 = 2.73

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Chromaticity We can similarly define chromaticity as approximation to a subset of a 12-tone equal tuning

All ordinary pcsets have perfect chromaticity

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Triadicity Triad: Relatively even spacing of three pitch-classes Triadicity: A cosine function over the pitch-classes with frequency 8ve/3:

• Positive values are good representatives of the triadicity; negative values are poor representatives. • The curve can vary in phase (different triadicities).

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Triadicity Example: Triadicity of some triads

12-t C major triad

0.89 + 0.89 + 0.45 = 2.23

JI C major triad

0.86 + 0.97 + 0.52 = 2.35

No assumption is made of a 12-tone grid

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Triadicity Example: Triadicity of scales C diatonic scale Spans: C D E F G A B ½ ½ 0 1 0 1 0 1.0 – 1.0 + 1.0 +0 + 0 + 0 + 0 = 1.0 C harmonic minor Spans: C D Eb F G Ab B 1 0 0 1 0 1 0 0.45 –0.45+0.89 –0.89 +0.89+0.45 + 0.89 = 2.23 • Cardinality-flexible: applies to chords of any size • Not all triad positions need to be represented; multiple notes can represent a single category

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Dyadicity

Dyadicity: A cosine function over the pitch-classes with frequency 8ve/2:

0.5

0 C C# D Eb E F F# G Ab A Bb B C -0.5

-1

• Positive values are good representatives of the dyadicity; negative values are poor representatives. • The curve can vary in phase (different dyadicities).

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Dyadicity

Example: Dyadicity of a diatonic scale

0.5

0 C C# D Eb E F F# G Ab A Bb B C -0.5

-1 0.5 – 1.0 + 0.5 + 1.0 – 0.5 – 0.5 + 1.0 = 1.0

Spans: Dyadicity divides the scale into C D E F G A B tetrachords ABCD and DEFG ½ ½ 0 0 1 0 0

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Heptatonicity vs. Diatonicity

The diatonic scale is a prototype of heptatonicity: it maximizes heptatonicity for a 7-note subset of 12-tET.

Therefore heptatonicity in a 12-tET context equates to similarity to characteristic diatonic subsets. i.e. Diatonicity = Heptatonicity + chromaticity

The distinction is only relevant in the context of alternate or flexible tuning, but it is also conceptually important.

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Hexatonicity and Octatonicity The 12-tET prototype of triadicity is a hexatonic scale:

0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 = 3.0 Hexatonicity = Triadicity + chromaticity The 12-tET prototype of tetradicity is an octatonic scale:

0.5+0.5 + 0.5+0.5 + 0.5+0.5 + 0.5+0.5 = 4.0 Octatonicity = Tetradicity + chromaticity

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Fourier Transform

• Coefficient spaces (complex plane)

• Spectra

• Phase spaces

• Coefficient multiplication Fourier Transform as Vector Sums

Fourier component fk can be derived as a vector sum with each pitch class as a unit vector, where the unit circle is the 8ve/k.

The length of the resulting vector is the magnitude of the component, and the angle f3 is its phase.

Example: 12-tET C maj. triad, k = 3

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Fourier Transform as Vector Sums

G Fourier component fk can be derived as a vector sum with each pitch class as a unit E vector, where the unit circle is C F G the 8ve/k. {CEG} B∫ D The length of the resulting vector is the magnitude of D#/E∫ A f the component, and the angle 5F3 is its phase. A∫ E

C# B Example: 12-tET C maj. triad, F# k = 5

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Single component spaces (complex plane)

Bb diatonic Example: Eb major scale 12t-ET sets in Eb diatonic F diatonic

f5 space C minor Btriadb maj. F maj. C major scale

Ab diatonic C diatonic C min. G min.C major triad Eb maj. C maj. Distance from the FSingle min. pitch-classD min. C F C G G maj. Ab maj. Bb A min. B min. center is the b D D diatonic b Eb magnitude of f A G diatonic G 5 Eb min. # E min. Db maj. E D maj. C# B F#

G# min. B min. Angle is the F# maj. A maj. Gb diatonic C# min. F# min. D diatonic phase of f5 B maj. E maj.

B diatonic A diatonic

E diatonic

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Spectra The spectrum of a pitch-class vector shows the magnitudes of all its Fourier coefficients (ignoring phases) The spectrum is invariant with respect to transposition and inversion (i.e. it is a set class property) Examples: 12tET major triad Just major triad

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Spectra The spectrum of a pitch-class vector shows the magnitudes of all its Fourier coefficients (ignoring phases) More examples: Just triad (dotted) compared to . . . 7tET triad

12tET triad

19tET triad

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Spectra The spectrum of a pitch-class vector shows the magnitudes of all its Fourier coefficients (ignoring phases) Example: Diatonic scales in 12-tET, Pythagorean, just tuning

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Spectra The spectrum of an interval gives the ET approximations A generated set intensifies the spectrum of the generating interval

Spectrum of the just perfect fifth

Diatonic Pentatonic Sus triad

Spectra of fifth-generated collections

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Spectra The spectrum tends to have peaks at sums and differences of other peaks, especially for generated collections Pythagorean diatonic: 12 + 17 = 29 5 + 12 = 17 (12 + 12 5 + 7 = 12 = 24) t

Approximate symmetry (5th in 53tET is very close)

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Phase Spaces A phase space uses the phases of two (or more) coefficients as coordinates. Phases are cyclic, so phase spaces are toroidal. Transpositions correspond to translations (rotations) of the phase space. Inversions correspond to reflections.

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Coefficient Products Spectral ⇒ Transposition invariant but Transposition invariant ⇒/ Spectral

For any a + b = c, _ fa fb fc is a transposition-invariant complex number, with phase #a + #b – #c

Thanks to Emmanuel Amiot

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Coefficient_ Products Example: f2 f3 f5 of 12tET sets

5 12t Pelog 1-2

Complementation4

(012479) (013568) 3 Mystic B

2 Minor pentachord ← Imaginary Minor 7th #11 Minor triad

1 Octatonic 1-2 pentachord Inversion (012579) Pentatonic Diatonic (013578) Real axis: 0 -15 -10 -5 0 5 10 15

Inversional Octatonic 2-1 pentachord -1

symmetry Major triad Dominant #9 → -2 Major pentachord

Mystic A Imaginary axis:-3 (023479) (023578) Complementation -4 symmetry ←Real→ 12t Pelog 2-1 -5

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Coefficient_ Products Example: f2 f3 f5 of 12tET sets Pentatonic: Diatonic: C D E G A C D E F G A B

f2 spans: 0 0 1 0 1 + f2 spans: ½ ½ 0 0 1 0 0 +

f3 spans: ½ ½ 1 0 1 ≠ f3 spans: ½ ½ 0 1 0 1 0 =

f5 spans: 1 1 1 1 1 f5 spans: 1 1 0 1 1 1 0 _ _ Negative f2 f3 f5 Positive f2 f3 f5

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Coefficient_ Products Example: f2 f3 f5 of 12tET sets (013568): (023578): B C D E F G A B C D E F

f2 spans: 0 0 1 0 0 1 + f2 spans: 0 0 1 0 0 1 +

f3 spans: 0 1 0 0 1 1 = f3 spans: 1 0 0 1 0 1 =

f5 spans: 0 1 1 0 1 2 f5 spans: 1 0 1 1 0 2 _ _ Positive real f2 f3 f5 Positive real f2 f3 f5 and positive imaginary and negative imaginary

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Coefficient_ Products Example: f2 f3 f5 of 12tET sets (012479): (023479): C C# D E G A C D Eb E G A

f2 spans: 0 0 0 1 0 1 + f2 spans: 0 0 0 1 0 1 +

f3 spans: 0 0 1 1 0 1 ≠ f3 spans: 1 0 0 1 0 1 ≠

f5 spans: 1 0 1 1 1 1 f5 spans: 1 0 1 1 1 1 _ _ Negative real f2 f3 f5 Negative real f2 f3 f5 and positive imaginary and negative imaginary

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Coefficient Products Spectra of fifth-generated collections

Generated collections are inversionally symmetrical ⇒ real only Well-formedness rule for coefficient product of spectral peaks cardinality < large coefficient ⇒ positive cardinality = large coefficient ⇒ negative Maximal evenness: cardinality = smaller coefficient, Use ET of large coefficient

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Balinese Pelog

Andrew Toth’s measurements Pelog spectra Begbeg–Sedeng–Tirus models _ f2 f7 f9 space Andrew Toth’s measurements

Toth measured 50 gamelans across all regions of Bali Thanks to Wayne Vitale and Bill Sethares for data. (“Balinese Gamelan Tuning: The Toth Archives” forthcoming in Analytical Approaches to World Music) Processing: • Average across instruments. • Average step sizes between second and third octave. • Stretch/compress to a 1200¢ octave.

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Toth Plots ar+cle: Figures

Models: Begbeg–Sedang–Tirus

i o e u a i

Figure 20 (recreated from Figure 2 of Toth 1980): The two types of interval proﬁles, begbeg and $rus, plus an intermediate one (Toth’s idealizedsedang). The numbers indicate interval sizes in cents; the doHed line shows how models of pelog tuning varieties (from testimony“Dang begbeg becomes of masterdung +rus.” tuners)

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Pelog Spectra

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

Average Range Standard deviation

Peaks at f2, f7, and f9 and troughs in between are consistent.

Above f9, little discernable consistency.

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Pelog Spans i o a e u i o a e u i o a e u

f2: 0 0 1 0 1 (all the same)

f3: 0 1 1 0 1 or 1 0 1 0 1

f5: 0 1 2 0 2 or 0 1 1 1 2 or 1 0 2 0 2 or . . .

f7: 1 1 2 1 2 (all the same)

f9: 1 1 3 1 3 (all but one the same)

f12: 1 2 4 1 4 or 2 1 4 1 4 or . . .

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Tuning Variants: Spectral

|f9| is highly correlated with |f7|, |f11|, |f13|, anti-correlated with |f5| and |f12| but relatively uncorrelated with |f2|

↪ |f9| × |f2| is a good space to distinguish pelog spectra

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Tuning Variants: Spectral

Begbeg |f2| is a 3.5 good

Br Tengah Sesetan model of Batur Kusamba

Sidemen Gladag “Begbeg– 3 Kekeran Sudembunut Kedis Kaja Menyali

← | Losan Takmung Pujung Kelod KuaDuahUbung Kokar Kecamatan Selat Banyuning Peliatan Prasi Tirus axis,” Bubunan Sumampan Teges Kanginan Sawan Selat Puri Kaleran Tabanan f Kalapaksa Jumpai Busungbiyu Belaluan Sadmerta 2 Kemoning Jagaraga Br Kukuh KrambitanUbud Kaja | Sidembunut Tunjuk Br Babakan Sukawati Pengosekan, Old

→ Sedang Br Pekandelan Sanur Br Anyar Perean Lebah Sadmerta Pengosekan, New but Sedang 2.5 Sengguan Kawan Budaga Pekandelan Kecamatan Kubuh Beng Sangkanabuana Getas Manggis also differs SidakaryaTengah Sima Puri Anung Loji Blungbang Bangli in |f9|.

Br Kawan Bangli 2

Tirus

←|f9|→

1.5 1.5 2 2.5 3 3.5 4 4.5 Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Tuning Variants: Spectral

5 ET models 01010

in |f2|-|f9| 4.5 space 4 15151 14141 3.5 Begbeg

3 13131 ← | 24141 Sedang 2.5 f 2 | 2 24142 → Tirus 12121

1.5

1 23232 (Slendro) 0.5

11111 ←|f9|→ 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Tuning Variants: Spectral

4 Most

tunings are Begbeg close to 3.5

Sedang. Br Tengah Sesetan Batur Kusamba

Sidemen Gladag Begbeg and 3 Kekeran Sudembunut Kedis Kaja Menyali

← | Losan Takmung Pujung Kelod KuaDuahUbung Kokar Kecamatan Selat Banyuning Peliatan Prasi Bubunan Tirus are Teges Kanginan Sawan Sumampan Selat Puri Kaleran Tabanan f Kalapaksa Jumpai Busungbiyu Belaluan Sadmerta 2 Kemoning Jagaraga Br Kukuh KrambitanUbud Kaja outside the | Br Babakan Sukawati Sidembunut Tunjuk Pengosekan, Old

→ Sedang Br Pekandelan Sanur Br Anyar Perean Lebah Sadmerta Pengosekan, New 2.5 range of Sengguan Kawan Budaga Pekandelan Kecamatan Kubuh Beng Sangkanabuana Getas Manggis

SidakaryaTengah Sima observed Puri Anung Loji Blungbang Bangli tunings.

Br Kawan Bangli 2

Tirus

←|f9|→

1.5 1.5 2 2.5 3 3.5 4 4.5 Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Spectral Tuning Variation by Region Only Bangli 4 region Begbeg (central 3.5 highlands) is reliably ← |

3 f

distinct 2

| Karangasem Buleleng (include all → Badung Klungkung most Tirus Sedang 2.5 Tabanan Gianyar tunings). Buleleng, Bangli 2 Tirus Klungkung ←|f |→ more Sedang 9

1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Ellipses show standard error

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 _ Tuning Variants: f2 f7 f9 space

Puri Anung Loji All tunings have 8

Getas positive real 6 SidakaryaTengah values. Sengguan Kawan Teges Kanginan 4 Lebah Sadmerta

Tirus Kemoning Manggis Begbeg 2 Beng Consistency of Tunjuk Sima Ubud Kaja 0 Budaga Pekandelan 0 5 10 15 20 25 30 35 40 45 50 spans: Br Anyar Perean Pengosekan, Old Sangkanabuana Sedang Sidembunut Blungbang Bangli Busungbiyu Belaluan Sadmerta -2 Gladag Menyali Sawan Br Kukuh Krambitan i o e u a Jumpai Banyuning Br Babakan Sukawati Sumampan -4 Sidemen Bubunan Kecamatan Kubuh Jagaraga Br Pekandelan Sanur Pengosekan, New Kekeran Kalapaksa Kokar 0 0 1 0 1 KuaDuahUbung Kedis Kaja Puri Kaleran Tabanan -6 Pujung Kelod Losan Takmung Peliatan Kecamatan Selat Selat 1 1 2 1 2 Prasi Batur Kusamba -8 Br Tengah Sesetan Br Kawan Bangli 1 1 3 1 3 Sudembunut

-10

-12

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 _ Tuning Variants: f2 f7 f9 space

Regions vary consistently on imaginary axis: Only Gianyar is balanced around zero Other regions consistently negative.

Ellipses show standard error.

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Persian Dastgah Tuning

• Farhat’s tuning and the Dastgah system • Spectral analysis of scales and tetrachords • Coefficient-product spaces Farhat’s Tuning

–Loosely empirical (based on measurements but no data reported) –Generated by two basic intervals: • Perfect fifth (two Pythagorean scales of 11 and 6 notes each) and • Neutral step, which Farhat estimates at 135¢ (Pythagorean second – koron 205¢ – 70¢) (Large neutral step is semitone + koron, 90¢ + 70¢ = 160)

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Some scales and tetrachords

b Shur

Bayāt-e b Tork Chahārgāh b Segāh

Homāyun Mahur

b Bayāt-e Esfahān

Arab Rast b b (quartertone)

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Spectra for scales

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

Mahur Sur1 Sur2 Quartertone Rast

7 Mahur 6 Rast

5 Segah Sur Mahur Sur 4 Segah Sur Mahur Rast 3 Segah 2 Rast 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Spectra for scales

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

Sur2 Chahargah Homayun

5 Chahargah 4 Homayun 3

1 Segah

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Spectra for tetrachords

Tetrachords show similar pattern in |f5|, |f7|, and |f12|

4 Segah Mahur 3.5 Sur Tork 3

2.5 Tork Sur

1.5

1 Segah 0.5 Mahur

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Spectra for tetrachords

Tetrachords show similar pattern in |f5|, |f7|, and |f12|

Esfahan Segah Tork

Chahargah

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Coefficient Product Space

20 _ • Tetrachords Complex space for f5 f7 f12 • Scales Esfahan 10

Segah 0 Minor Mahur -40 -20 0Esfahan 20 40 60 80 100 Chahargah Sur Tork Mahur Chahargah Sur -10 Segah

Homayun -20

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Coefficient Product Space

50 • Tetrachords 40 Segah • Scales Sur

Homayun 10 Tork Esfahan Complex_ space Sur Esfahan Mahur Mahur for f5 f12 f17 0 Segah Minor -20 0 20 40 60 80 100 120 140 160 180 Chahargah

-10

Chahargah

-20

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Coefficient Product Space 4 Esfahan Sur • Tetrachords 3 • Scales 2 Esfahan 1

0 Minor Homayun -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 -1 Sur Mahur Segah Segah -2 Chahargah Tork -3

-4 Mahur -5

-6 Chahargah -7 _ Complex space for f2 f5 f7

Jason Yust Fourier Models of Tuning Systems MCMM Workshop 6/29/21 Conclusions

• Interval categories defined by even divisions (heptatonicity, triadicity, chromaticity) have many theoretical uses. • Tuning systems can be defined by multiple interval categorizations. —These have the advantage of setting limits of intonation flexibility, rather than idealized “correct” interval sizes. • Real tuning systems relate to the WF sequence of the acoustical fifth. • Two kinds of transposition-invariant features of collections: —Spectrum —Coefficient product phases

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21 Bibliography

Amiot, Emmanuel (2007). “David Lewin and Maximally Even Sets.” Journal of Mathematics and Music 1/3. ——— (2016). Music Through Fourier Space: Discrete Fourier Transform in Music Theory. (Springer) Amiot, Emmanuel, and William Sethares (2011). “An Algebra for Periodic Rhythms and Scales.” Journal of Mathematics and Music 5/3: 149– 169. Block, Steven, and Jack Douthett (1994). “Vector Products and Intervallic Weighting.” Journal of Music Theory 38/1: 21–41. Callender, Cliff (2007). “Continuous Harmonic Spaces,” Journal of Music Theory 51/2 Carey, Norman, and David Clampitt (1989). “Aspects of Well-Formed Scales.” Music Theory Spectrum 11/2: 187–206. Clough, John, and Jack Douthett (1991). “Maximally Even Sets.” Journal of Music Theory 35: 93–173. Clough, John, and Jack Douthett, N. Ramanathan, and Lewis Rowell (1993). “Early Indian Heptatonic Scales and Recent Diatonic Theory.” Music Theory Spectrum 15/1: 36–58. Clough, John, and Gerald Myerson (1985). “Variety and Multiplicity in Diatonic Systems.” Journal of Music Theory 29: 249–270. Lewin, David (1959). “Re: Intervallic Relations between Two Collections of Notes,” Journal of Music Theory 3/2. ——— (2001). “Special Cases of the Interval Function between Pitch Class Sets X and Y.” Journal of Music Theory 45/1. Rahn, Jay (1978). “Javanese Pelog Tunings Reconsidered.” Yearbook of the International Folk Music Council 10: 69–82. Tymoczko, Dmitri. 2009. “Generalizing Musical Intervals.” Journal of Music Theory 53/2: 227–254. Vitale, Wayne, and William Sethares (2021). “Ombak and Octave Stretching in Balinese Gamelan.” Journal of Mathematics and Music. Vitale, Wayne, and William Sethares (2021). “The Toth Archives,” Forthcoming. Yust, Jason (2015). “Schubert’s Harmonic Language and Fourier Phase Spaces.” Journal of Music Theory 59/1. ——— (2016). “Special Collections: Renewing Forte’s Set Theory.” Journal of Music Theory 60/2. ——— (2019). “Probing Questions about Keys: Tonal Distributions through the DFT” Mathematics and Computation in Music, Fifth International Conference, ed. O. Agustín-Aquino, E.Lluis-Puebla, M. Montiel (Springer) ——— (2020) “Generalized Tonnetze and Zeitnetze and the Topology of Music Concepts.” Journal of Mathematics and Music 14/2. ——— (2020) “Stylistic Information in Pitch-Class Distributions” Journal of New Music Research 48/3

Jason Yust Fourier Models of Tuning Systems MCMM workshop 6/29/21