Geek File – Warning heavy doses of maths and science contained within -Awareness of sound and Multi Dimensional Scaling

multidimensional scaling

As I sit in St Georges concert hall, I hear the anticipatory opening of Brahms first symphony in C minor, a piece that took Brahms 14 years to write. It commences lento, deceiving the listener before an aggressive allegro begins. Despite comparisons being made, and in fact the piece being dubbed “Beethoven’s 10th”, the orchestration lends itself more so to Bach, with the polyphonic counterpoint, descending and lamenting lines in the woodwinds and violas……Brahms exonerates himself from the oppressing burden of the contrasts in likeness between this composition and Beethoven’s 9th, with its thematic chambers, rich chromaticism and lyrical progressions. So here it is, that I start to contemplate musical timbre and its mechanisms…….

One could describe a myriad of characteristics when describing sound: the attack of the brass section, the sharpness of the wind instruments, brightness of the clarinet, nasality of the trombone and richness of the cello.

And yet we can be wooed by some more discrete offerings. As an example…..two different instruments playing the same note in differing ways: accented note on piano verses a soft vibrato sound of a flute.

What makes music and in particular the study of timbre interesting, is the complex set of auditory attributes which covers many parameters of perception that are not referred to by pitch, loudness, spatial position, duration, or room reverberation.

Timbre is a tool used to identify and monitor a derivation of sound. However, it is not an absolute classification of sound itself: Handel 1995 McAdams 1993 Risset 2004

Timbre enables one to define elements of vibrating/resonating instruments and their sound waves. Create systems, formula and matrix that describe the neurological process undergone to decipher sound waves. A way for listeners to create a form of descriptive language as a means of decoding on an abstract level the perceived patterns and ways in which music can be expressed and therefore replicated.

In this particular article I will be covering timbre in terms of timbre space and multi- dimensional scaling as a theoretical approach to understanding and applying on a practical level.

McAdams (1989) was interested in the form-bearing dimensions in music as inference to timbre. Form-bearing dimensions are to do with the receivership of sound due to its structural form on a psychological level. To do this, experimentation is conducted to conceive the relationship between the perception process and memory structures of the listener. Three aspects of psychology and it processes are discussed:

  1. Perceptual grouping processes – which simply means putting parts together into a whole
  2. Abstract musical knowledge structures – as an example dynamic processes in music perception
  3. Event structure processing – so, events can be understood by their temporal structure (a patterned organization of time, used to help us understand and manage time). Research is conducted into an analysis of how people use event structuring in understanding and then planning. There are 2 levels of organization events …. One being taxonomies (a scheme of classification) and one being partonomies (how people catagorize events when they occur) and their cause and goal relationship which deals with understanding memory and planning. Models such as these enlighten us to how these organizations may be improved.

Timbre, Psychophysics of Timbre and Multidimensional Scaling

How we hear sound and how that differs is classified as timbre. Before the application of multidimensional scaling, which is unbiased and scientifically sound, timbre was assessed more so by priori hypothesis (more presumptive or by reasoning) such as Slawson (1985) describing sound by weights of frequencies (sound colour). An example of this previous application was Mcadams, Deplane & Clarke 2004 when they assessed singing the same note with different vowels or for brass the same note with different embouchure – (the affects of sound produced by way of varying the way in which the player applies the mouthpiece on a brass instrument).

To explore timbre, McAdams and co played around with the spectral shape of various resonating apparatus that the ingeniously came up with. So here is my rabbit hole, hold on for the ride, as one thing led to another here and in order to truly understand this, I had to branch out. Here goes WARNING: SOME HEAVY DUTY MATHS AND SCIENCE INVOLVED******

Spectral shape analysis is a comparison of geometric shapes through the eigenvalues of the Laplace-Beltrami.

Eigenvalues are non zero vector collection of objects called vectors. Vectors in this case are sound waves and are predetermined by their positioning in relation to one another. Vectors are either added or multiplied (scaled), by real numbers whose direction does not change when the linear transformation is applied. Linear mapping or transformation is mapping between two modules (including Vector space) that preserves the operations of scalar multiplication. Scalar multiplication is simply taking a regular number (scalar) and multiply it on every entry in the matrix.

So going back to Eigenvalues more formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, ZeroVector_1000

then v is an eigenvector of T if T(v) is a scalar multiple of v. This condition can be written as the equation T ( v ) = λ v”

The Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space. and, more generally, on Riemannian and pseudo-Riemannian manifolds. (smooth function vector fields) on M, then p ↦ ↦ g p ( X ( p) , Y ( p ). Manifolds simply being charts or maps (see atlas image below).

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria The term “Euclidean” distinguishes these spaces from other types of spaces considered in modern geometry.

eucliden spaceEuclidian also generalise to higher dimensions

This more general operator goes by the name Laplace–Beltrami operator, after Laplace and Eugenio Beltrami. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient : the gradient replaces the the derivative when there are several variables concerned and vector valued instead of derivative which is scalar valued.

 

gradiant imageand is a linear operator taking functions into functions

 

The operator can be extended to operate on tensors as the separation of the covariant derivative which is a fundamental tool in maths. Covariant derivatives in simple terms is the measurement of the speed with which something changes with time and looks like this (see below).

 

 

derivative

 

Alternatively, the operator can be generalised to operate on differential forms where there are multiple variable using the divergence and exterior derivative. So if f is a smooth function then the exterior derivative of f is the differential of f. That is, df if is the unique 1-form (which is the same as a linear functional space). Manifolds are simply a collection of charts like an atlas as an example and are used in exterior derivatives: atlas charts

So in summary the The Laplace–Beltrami operator, like the Laplacian, is the divergence of the gradient:

f = ∇ ∇ ⋅ ⋅ ∇ ∇ f . {\displaystyle \nabla ^{2}f=\nabla \cdot \nabla f.}

Soooooooo controlling spectral shapes in sound is really controlling wave formations in certain ways to create differing sounds which is timbre….. very simply. I like the science bit.

Multidimensional scaling (MDS)

 

multi dimensional scaling

To try and tie all this science stuff together and so you see the relevance of understanding each factor….. MDS is a way of manifolding or charting and therefore visualising a set of data. In relation to this topic we are talking about sound tones or vector waves. This is done through strict analysis as oppose to previous priori techniques conducted by Slawson, which were assessed by preconception. This is done is by ordering sound tones in terms of the similarities or dissimilarities on a distance matrix.

keanu reeves in the matrixExperimentation was carried out purely on timbre alone and all other musical attributes remained fixed – dynamical, pitch and sustentation of note.

1.The timbre space model is a representation of the experiment and analysis and flows as such

2.The sound event – as an example synthetic sound (Miller and Carterette 1975

3. Perceptual relations – distinguishing 2 separate tones

4.  Perceptual dissimilarity matrix – Two experiments were performed to evaluate the perceptual relationships between 16 music instrument tones. The stimuli were computer synthesised based upon an analysis of actual instrument tones, and they were perceptually equalised for loudness, pitch, and duration. Experiment 1 evaluated the tones with respect to perceptual similarities, and the results were treated with multidimensional scaling techniques and hierarchic clustering analysis. A three‐dimensional scaling solution, well matching the clustering analysis, was found to be interpretable in terms of (1) the spectral energy distribution; (2) the presence of synchronicity in the transients of the higher harmonics, along with the closely related amount of spectral fluctuation within the the tone through time; and (3) the presence of low‐amplitude, high‐frequency energy in the initial attack segment; an alternate interpretation of the latter two dimensions viewed the cylindrical distribution of clusters of stimulus points about the spectral energy distribution, grouping on the basis of musical instrument family (with two exceptions). Experiment 2 was a learning task of a set of labels for the 16 tones. Confusions were examined in light of the similarity structure for the tones from experiment 1, and one of the family‐grouping exceptions was found to be reflected in the difficulty of learning the labels. – John M Grey

4. Geometric configuration (model section) – finite arrangement of lines.

5. Psychophysical explanation – Experimenters interpretation of acoustic properties

So empirical observation and live experimentation rather than logic or theory to determine that listeners use the same perceptual dimensions when comparing timbres.

I am going to press pause on this topic here mainly so I can get a cup of tea and allow the information presented so far to cook a little before I delve deeper into the more complex models and specific timbres and models: EXSCAL, INDSCAL and CLASCAL as this will take another article I feel. Perhaps now would be a sensible time to stick a funny movie or youtube clip with a slice of cake……

I make reference to the following authors and sources in this article: John M Grey, Diana Deutsch, Wikipedia and Stephen McAdams.


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