Optimal well-temperaments

PolanskyMusic 150UC Santa CruzClass notes2/4/02

“A Mathematical Model for Optimal Tuning Systems”

Perspectives of New Music, 47/1:69-110. Winter, 2009

Co-authored with Dan Rockmore, Kimo Johnson, Douglas Repetto, and Wei Pan

tuning systems

• Rational, naturalist, evolutionary, cognitive, experimental, logical, or acoustical motivations (harmonic series, simple integer relationships)

• Multiplicity (extended rational tunings, large number equal temperaments)

• approximation (optimal small number equal, mean-tone, well-temperaments, large-space rational tunings approximated by fixed sets)

• Non-”pitch fundamental” based• Stylistic (keys, modulation, interval hegemonies, harmony,

melody, modes, formal uses of pitch)• Extra-musical: Spiritual, visionary, allegorical, metaphorical,

mystical, pharmacological, chimerical • Extra-musical: Historical, social, economic, political concerns• Just like the sound of them; newness; novelty• Practicality• Whimsy

Like….• Historical: well-temperaments, just, meantone, equal-temperaments (Harrison)• Extended rational systems: Partch, Tenney, Johnston, Wilson, Polansky, Sims• Multiple equal-divisions: Fokker, Sims, Vïshnegradsky, Carillo, Darreg, Ives• Harmonic series based: Cowell, Tenney, Polansky, Scholtz• New Scales and Logical systems: Wilson, Chalmers, Carlos• Detuned systems: Balinese, Sonic Youth• “world music”: Harrison, Carlos• Who knows: slendro• Adaptive, real-time, paratactical, free-style (Harrison), intelligent (Polansky)

Harrison:Simfony in Free Style (example)

Paratactical or

“adaptive” or free-

style intonation

Arion’s leap

“scales” in Arion’s Leap

Ya chengs3-part chord with the intervals 7/6 and 4/3, tuned as A-C-D (4/3, 14/9, 16/9), transposed up 25 /24, 16/15, 6/5, down 25/24.

Troubadour HarpAdds Eb (50/27), Bb (25/18), G (32/27) to the total fabric

E E F F F# Ab A Bb Bb B B C# C# D D# D#

0 20 22 112 182 337 498 520 610 680 729 858 884 996 1040 1108

1/1 2048/

2025

40/39

16/

15

10/9 243

/200

4/3 27/

20

64/

45

40/

27

32/

21

64

/339

5/3 16/9

640

/351

256/135

Metal Strung Harp

Catherine Lamb: Frame for Flute (March, 2009)

(excerpt)

tyvarb (B’rey’sheet) (in the beginning ... ) (Cantillation Study #1)

(1985; revised 1987, 1989)for voice and live interactive computer

Jody Diamond, voiceLarry Polansky and Phil Burk, live computer systems

from The Theory of Impossible MelodyNew World Records, 2009(reissue of Artifact CD, #4, 1991)

(Piece done in HMSL computer language)

Five Constraints(informally)

Tuning systems through history and across cultures have used a set of complex compromises to account for some or all of the following constraints:

1. Pitch set: use of a fixed number of pitches (and consequently, a fixed number of intervals);

2. Repeat factor: use of a modulus, or repeat factor for scales, and for the tuning system itself (i.e., something like an octave);

3. Intervals: an idea or set of ideas of correct or ideal intervals, in terms of frequency relationships;

4. Hierarchy: a hierarchy of importance for the accuracy of those intervals in the system;

5. Key: a higher-level hierarchy of the relative importance of the “in-tuneness” of specific scales or modes begun at various pitches in the system.

hypotheses

Most tuning systems attempt to resolve some or all of these five constraints.

Rationally based tuning systems

• collision of primes• “historical tuning problem”

– (pn ≠ qm for distinct primes p and q (and n, m > 0 ))

• “Canidae” interval (LP)

Simple Example: Pythagorean commaand the historical tuning problem

(only two primes: 3, 2)

review: just intonation

1/1 9/8 5/4 4/3 3/2 5/3 15/82/1

C D E F G A B C 0 204¢ 386¢ 498¢ 702¢ 88¢ 1088¢ 1200¢

A standard just diatonic scale (primes 2, 3, 5)

1/2 – matrix for just diatonic scale

Diagonals are same “intervals,” but are tuned variously.The top row might be considered as a kind of “ideal,” but the intervals in that row are not exactly propagated thru the matrix.

C (1/1)

D

(9/8)

E (5/4)

F (4/3)

G

(3/2)

A

(5/3)

B

(15/8)

C

(2/1)

C 9/8 5/4 4/3 3/2 5/3 15/8 2/1

D 10/9 32/27 4/3 40/27 35/18 16/9

E 16/15 6/5 4/3 3/2 8/5

F 9/8 5/4 45/32 3/2G 10/9 5/4 4/3

A 9/8 6/5

B 16/15

C C#

D Eb E F F# G G# A Bb B CC 90 192 294 390 498 588 696 792 888 996 1092 1200

C#

102 204 300 408 498 606 702 798 906 1002 1110

D 102 198 306 396 504 600 696 804 900 1008

Eb 96 204 294 402 498 594 702 798 906

E 108 198 306 402 498 606 702 810

F 90 198 294 390 498 594 702

F# 108 204 300 408 504 612

G 96 192 300 396 504

G#

96 204 300 408

A 108 204 312

Bb 96 204

B 108

W3 1/2 – Matrix

W3 is top row. ”keys” correspond to rows

Error Matrix for W3(M3, P4, P5 only)

C C# D D# E F F# G G# A Bb B C

C 4 0 6

C# 22 0 0

D 10 6 0

D# 16 0 0

E 16 0 0

F 4 0 0

F# 0 22 6

G 6 10 6

G# 0 0 22

A 16 6 0

Bb 10 0 0

B 16 0 6

W3 Error DistributionMajor Triads by “key”

Error distribution for W3 (in cents, from one specific set of ideal intervals) can be seen, showing an almost symmetrical increase (via the circle of 5ths) around the central or “best” key of F.

Key C C# D Eb E F F# G G# A Bb B

M3 4 22 10 16 16 4 22 10 22 16 10 16

P4 0 0 6 0 0 0 6 6 0 6 0 0

P5 6 0 6 0 0 0 0 0 0 0 0 6

Error

10 22 22 16 16 4 28 16 22 22 10 22

Ideal Tuning

An ideal tuning would be one in which the i,j entry only depends on |i-j| — each entry of the matrix is equal to an ideal interval.

In the ideal interval matrix, diagonal values are constant and equal to the ideal interval. The ideal interval matrix is equivalent to the interval matrix only in ET.

The entries of the error matrix of a tuning system are the differences between the entries of the interval matrix and the respective entries in the

ideal interval matrix.

Formal Definition of Constraints

1. Pitch set: let a1, … an be a set of n pitches, none equal to 0.

2. Repeat factor: let > an be the repeat factor of the tuning system.

3. Intervals: let I1, … In represent the ideal intervals.

4. Hierarchy: let i1, …, in be interval weights used to represent the desired accuracy of the n intervals in the tuning system.

5. Key: let k0 , …, kn be key weights used to represent the fixed pitches in the tuning system from which intervals are measured.

Least squares formulation of optimization

— where M is the interval matrix, and L is the ideal interval matrix

(unweighted version)

(weighted version)

— where W is the weight matrix (the product of key and interval weights), or:

(Note: Other norms are possible, as in the L1 norm, used in our GA solution).

Two historic: W3 and Young 2Two synthetic: OWT1 and OWT2

All four are minimally “mean-tempered” by Rasch’s measure. That is, they are equivalent to 12TET in terms of the mean consonance of major triads.

Four Optimal Temperaments(with minimal “mean-tempering”)

Comparison of four optimal scalesby Rasch “mean temperament” measure

Two “septimal” optimal temperaments

Septimal OWT1

Septimal OWT2

Intervals such as 267 and 969 are septimal (7/6, 7/4)

Sound examples of synthetic and historical well-temperaments

Bach WTC in different optimal temperaments

For more informationhttp://eamusic.dartmouth.edu/~larry/owt/index.html

For real-time software (written by Wei Pan)http://www.cs.dartmouth.edu/~pway/owt/index.html

(Thanks to Ron Nagorcka for making these examples)

Jogyakarta gamelan statistics(Gadjah Madah study)

Solo (Surakarta) gamelan statistics(Gadjah Madah study)

Mean and ranges of adjacent slendro intervalsGadjah Madah study

Note: Slendro is numbered 1, 2, 3, 5, 6, 1’There is no 4.

Slendro Pilot Experiment 1

Procedure: Generate a set of new scales, some stretched, some non-stretched, with only two ideal intervals specified (3/2, 8/7) (all others weighted to zero). Fit those scales to the 27 GM scales, record average error. Next generate 27 random slendros with overall mean and variance matching the GM scales

The two best fits for the randomly generated scales were significantly worse than the two best GM scales (1/1 = 4.10; 1/3 = 6.22, both unstretched). That is, there is some structure in the optimally generated scales that in some way reflects the structure of this dataset.

Statistics of GM gamelan by city

The maximum and minimum ranges of intervals in Solo and Jogya are {52, 36} and {38, 21}. The variation in GM Jogya tunings is “flatter” than Solo (especially around the “middle” of the scale).

Slendro Pilot Experiment 2

Average fitting error for GM study gamelans, using fixed ideal intervals and varying key weights. “3x” means that the “key” on the specified pitch was set 3x higher than all the others (which were equal). The last line of the table sets all weights equal, except for the key based on pitch 3, which is set to 0.

What does this mean?

Sindusawarno uniquely excludes pitch 6 as an important note (either first, second or third) from any of the three pathets.

The results of this pilot experiment (2) may suggest that since pitch 6 is in some respects the least important in terms of pathet identification, giving it an unusually large value magnifies some tendency in city-specific tuning systems.

However, it might also indicate that a high key weight on pitch 6 “makes no sens

e” in any slendro, and generates in general, much larger fitting errors.

Future Directions

1) Further exploration of the parameter space. Given a specified tuning, set of ideal ratios and repeat factor, there is not necessarily a unique set of corresponding weightings. What is the geometry of that space?

2) Constraint-based system. Adding or modifying constraints may affect the mathematical solution(s) considerably, as well as the geometry of the weighting space discussed above. For example, a particular form of design caprice might be incorporated, that of desiring one particular interval in one particular key to be “just so.”

3) Multiple Interval Representations: the possibility of having more than one ideal interval for a given position, such as the familiar situation of using either 81/64 or 5/4 for the M3rd.The framework might also be extended to facilitate the choice of several weighted alternatives for certain ideal intervals, such as the variant 2nd in the Just Diatonic scale.

acknowledgements

Thanks to Tim Polashek, Chris Langmead, Jody Diamond, Peter Kostelec and Dennis Healy for valuable advice in this project