Optimal%20well-temperaments

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Optimal well-temperaments
polansky 8/11/09
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• Talk based on article
• A Mathematical Model for Optimal Tuning Systems
• Forthcoming in Perspectives of New Music
• Co-authored with Dan Rockmore, Kimo Johnson,
  Douglas Repetto, and Wei Pan

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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)
• 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
• Practicality
• Whimsy
• Whatever

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Like.

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

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Paratactical or adaptive or free-style
intonation
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Arions leap
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scales in Arions Leap
Metal Strung Harp
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
Ya chengs 3-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 Harp Adds Eb (50/27), Bb (25/18), G
(32/27) to the total fabric
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Catherine Lamb Frame for Flute (March,
2009)(excerpt)
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tyvarb (Breysheet) (in the beginning ... )
(Cantillation Study 1)

• (1985 revised 1987, 1989)for voice and live
  computerJody Diamond, voiceLarry Polansky and
  Phil Burk, live computer systemsfrom The
  Theory of Impossible MelodyNew World Records,
  2009(reissue of Artifact CD, 4, 1991)

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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
• Pitch set use of a fixed number of pitches (and
  consequently, a fixed number of intervals)
• Repeat factor use of a modulus, or repeat factor
  for scales, and for the tuning system itself
  (i.e., something like an octave)
• Intervals an idea or set of ideas of correct or
  ideal intervals, in terms of frequency
  relationships
• Hierarchy a hierarchy of importance for the
  accuracy of those intervals in the system
• 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.

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hypotheses

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

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Rationally based tuning systems

• collision of primes
• historical tuning problem
• (pn ? qm for distinct primes p and q (and n, m gt
  0 ))
• Canidae interval (LP)

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Simple Example Pythagorean commaand the
historical tuning problem(only two primes 3, 2)
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review just intonation

• 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
• 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)
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1/2 matrix for just diatonic scale
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 3/2
G 10/9 5/4 4/3
A 9/8 6/5
B 16/15

• 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.

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W3 1/2 Matrix
W3 is top row. keys correspond to rows
C C D Eb E F F G G A Bb B C
C 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
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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
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W3 Error DistributionMajor Triads by key
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

• 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.

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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.

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Formal Definition of Constraints

• 1. Pitch set let a1, an be a set of n pitches,
  none equal to 0.
• 2. Repeat factor let w gt 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.

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Least squares formulation of optimization
(unweighted version)
where M is the interval matrix, and L is the
ideal interval matrix
(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).
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Four Optimal Temperaments(with minimal
mean-tempering)

Two historic W3 and Young 2 Two synthetic OWT1
and OWT2
All four are minimally mean-tempered by Raschs
measure. That is, they are equivalent to 12TET in
terms of the mean consonance of major triads.
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Comparison of four optimal scalesby Rasch mean
temperament measure
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Two septimal optimal temperaments

• Septimal OWT1
• Septimal OWT2

Intervals such as 267 and 969 are septimal (7/6,
7/4)
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Sound examples of synthetic and historical
well-temperaments

• Bach WTC in different optimal temperaments
• For more information
• http//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)
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Jogyakarta gamelan statistics(Gadjah Madah study)
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Solo (Surakarta) gamelan statistics(Gadjah Madah
study)
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Mean and ranges of adjacent slendro
intervalsGadjah Madah study
Note Slendro is numbered 1, 2, 3, 5, 6, 1 There
is no 4.
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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.
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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).
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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.

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

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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.

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acknowledgements

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