The Lambdoma Keyboard: Applying and Experiencing Mathematics Barbara Hero Fourth International Confe

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The Lambdoma Keyboard Applying and Experiencing
MathematicsBarbara Hero Fourth International
Conference of Applied Mathematics and
ComputingAugust 12-18, 2007 Plovdiv, Bulgaria

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Pythagorean ChiX
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Lambdoma Matrix of Ratios
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Lissajous figures on Matrix
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Lambdoma Matrix Color-Coded
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Color Coding of Notes
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Lambdoma Reference Octave
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Computer Generated Lambdoma Mandala
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Lambdoma Mandala Drawing
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Triangles in a Square Drawing
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Drawings of Tone Systems

• Tatra Mountains Mathematical Publications,
  Harmonic Analysis and Tone Systems, Ed Jan
  Haluska, Vol 23, Mathematical Institute Slovak
  Academy of Sciences Bratislava, Vol. 23, 2001.
• Mathematics to Music
• Music to Mathematics

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3-D Interference PatternsMusical Color Triad of
Red-C, Yellow-E, Blue-G
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Circle of 5ths around an OctaveThe Pythagorean
Comma and the Petzval System 53 intervals
within the octaveThe ratio 32 iterated 12 times
259.4926758 1.013643265
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Expanding-ContractingA circular grid of inverse
waveforms, convex-concave, inside-outside and
solid-space.
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Hidden Pythagorean Formulain a 345 triangle
where 32G-blue, 43F-green and 54A-purple
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Lambdoma Overtone-UndertoneA vibrating string
with tangents to the ascending and descending
harmonics and sub-harmonics.Red-C, Orange-D,
Yellow-E, Green-F, Blue-G, Indigo-A Violet-B
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Triangular LambdomaArithmetic, Geometric
Harmonic MeansThe hidden origins of the Lambdoma
rays begin at both corners of the base of the
triangle indicating that the apex is the seed.
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Musical Triangles within CirclesLying on
different Planes
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Pyramids as Multiple Lensesin Audio Holography
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The 4 Quad Lambdoma Color-Coded
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Lambdoma Keyboard 4th quadrant with Lissajous
figures, ratios, colors frequencies
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Multidimensional Scaling Analysis

• The MDS analysis tool visualizes relationships in
  a 256-entry Lambdoma matrix.
• The 60 subjects interviewed chose
• 1. A frequency in Hertz representing a musical
  keynote.
• 2. A chord of harmonic ratios referenced to the
  keynote.
• 3. An emotive word or phrase.
• Sub-harmonic and super-harmonic choice correlates
  into active or passive individual responses.

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2-D ISOMAP embedding of 39 dimensions
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2-D ISOMAP Embeddingof 39 Dimensions

• A metric MDS method was applied to the set of
  39-dimensional Euclidean part of the response
  space to find the best 2-dimensional subspace for
  visualization.
• Each word was attached as a label to each
  response vector.
• The ISOMAP algorithm by Tanenbaum etal was
  applied to best preserve inter-subject geodesic
  distances in the process of embedding.

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Histogramlog(n) of (m,n)
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Population Histogram of log of second element
(log(n)) of harmonic pair (m,n)

• The responses were obtained by pooling all
  subjects together.
• Each subject is encoded with a response vector of
  dimension 1 16 22 39 and a single word or
  phrase.
• A real valued (39 Euclidean dimensions) response
  space and discrete valued (1 categorical
  dimension) response space corresponds to a 40
  dimensional encoding.

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Histogram oflog ratio (log(m/n)) of (m,n)
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Population Histogram of log ratio (log(m/n) of
harmonic pair (m,n)

• This histogram shows all log ratio selected
  harmonics.
• A separate normalization of the two histograms
  was performed prior to embedding.
• Each of the 60 subjects is described by a
  response array consisting of an integer valued
  keynote from 0 - 255, one or more relative
  harmonic pairs mn(m,n) taking one or 162 256
  possible values, and a word or phrase.
• Alfred Hero, hero_at_eecs.umich.edu was the source
  of the histogram-encoding concept. The code is
  available on request.

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Helmholtz Table of Wavelengths Modified to
Frequencies V WF

• Intervals Unison, 2nds thru 7ths, Octave
• Notations A, B, C, D, E, F and G (s, bs)
• Ratios p/q q/p where p 1-16, q 1-16
• Frequencies in Hertz multiplied by p/q ratios
  and their inverses q/p ( Hero)
• Resonances 100 (Unison) to 1.4 (7th)

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A Sample from 60 Experiments

• Keynote F
• Ratio 8/12
• Frequency 171Hz
• Chord 8/12Bb, 12/6F, 8/4F, 8/3Bb, 12/3F and
  5/15Db
• Words Elevated, Flying
• Interval Inverse of Fourth is Fifth
• Helmholtz Resonance 8.3 or 16.7

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Hypothesis of Players Choices

• The hypothesis is that Helmholtzs resonance
  intensities from 100 to 1.4 are linked to the
  emotions of players who chose their ratio
  intervals as their fundamental keynote of 1/1.
• The generating frequency of 256Hz was chosen as a
  control frequency for all the experiments based
  upon finding a players keynote.

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Conclusion

• The experiment is repeatable based upon the
  selection of the frequency of the keynote
  multiplied by the harmonic ratios of the chord.
• The ratios may be compared with Helmholtz
  intervals to determine which ratios have the most
  resonant effects depending upon the word chosen.
• A configuration of a mathematical harmonic matrix
  of ratios may become a psychologically
  therapeutic musical tool.