The Living Thing / Notebooks : Musical metrics and manifolds

Douthett and Steinbach’s “Cube Dance” as reinterpreted by Dmitri Tymoczko

Douthett and Steinbach’s “Cube Dance” as reinterpreted by Dmitri Tymoczko

See also machine listening, Arpeggiate by Numbers, clustering, manifold learning

Features, descriptors, metrics, kernels and affinities and the spaces and topologies they induce and what they reveal about composition. This has considerable overlap with machine listening, but there I start from audio signals, and here I usually think about more “symbolic” data such as musical scores, and with rhythm, but there I care only about the time axis. There is overlap also with psychoacoustic units, which are the pure elements from which these amalgams are made.

This is a specialized area with many fine careers built on it, and many schools with long and proud histories of calling adherents of other schools wrong. A short literature search will find many worth distinctions drawn between differences, cultural versus the biological, general versus individual, the different models applicable at different time scales, of masking effects and such.

I will largely pass over these fine contrasts here, in my quest for some pragmatically useful, minimally complex features for using in machine-learning algorithms, which do not require truth but functionality.

General musically-motivated metric spaces

Chords etc.

Consider grouping chords played on a piano with some known (idealised) 16-harmonic spectrum. How can we understand the harmonic relations of chords played upon this keyboard?

We know a-priori that this projection can be naturally embedded in a Euclidean space with something less than \(16\times 12 = 192\) dimensions, since there are at most that many spectral bands. In fact, since there are only 12 notes, we can get this down to 12 dimensions. The question is, can we get to an even smaller number of dimensions? How small? By “small” here, I mean, can I group chords on a lower number of dimensions than this in some musically interesting way? For bonus points, can I group chords into several different such maps and switch between them?

Without explanation, here is one similarity-distance embedding of all the chords using an ad hoc metric based on thinking about this question. (Brightness: The more notes in the chord, the darker. Hue: I forget.)

Question: can we use word-bag models for note adjacency representation? Since I asked that question I discovered the Chord2vec model of Walder’s lab, whose answer is “yes”. See MaQW16.

3 dimensional MDS 1
3 dimensional MDS 2

To understand

Spectral roughness” Dissonance

A particular empirically-motivated metric, with good predictive performance despite its simplicity, and willful lack of concern for the actual mechanisms of the ear and the brain, or modern nuances such as masking effects and the influence of duration on sound perception etc. Invented by Plomp and Levelt (PlLe65), and developed by variously, Sethares, Terhardt and Parncutt and others.

Some sources seem to distinguish roughness in the sense of Sethares from the Plomp and Levelt sense, although they use qualitatively equations. I suspect therefore that the distinction is philosophical, or possibly pointed failure to cite one another because someone said something rude at the after-conference drinkies.

An overview by Vassilakis might help, or the app based on his work by Dr Kelly Fitz.

The plot for Plomp and Levelt dissonance

Plomp and Levelt’s dissonance curves (PlLe65, Seth97).

Juan Sebastian Lach Lau’s produced some actual open-source software (DissonanceLib) that attempts to action this stuff in a musical setting. There is a slightly different version below attributed to Parncutt.

A convenient summary of both is in the DissonanceLib code.

It’s most useful for things where you are given the harmonics a priori; I’m not especially convinced about the tenability of directly inferring this metric from an audio signal (“how dissonant is this signal?”). We should be cautious about the identifiability of this statistic from signals nonparametrically e.g. windowed DTFT power-spectrogram peaks, just because beat frequency stuff is complicated and runs into the uncertainty principle. Lach11, Seth98b and TeSS82 give it a go, though. Inferring dissonance between two signals known to be not dissonant might work though, or parametric approaches, as in linear system identification

Dissonance an interesting measure, despite these problems, though because it is very much like a Mercer kernel, in that it constructs a distance defined on an (explicit) high-dimensional space; Also, the “nearly circular” geometry it induces is interesting; For harmonic spectra, you recover the equal-tempered 12-tone scale and the 2:1 octave by minimising dissonance between twelve notes with harmonic spectra (i.e. plucked string spectra), which is suggestive that it might do other useful things.

Also, it’s almost-everywhere differentiable with respect to your signal parameters, which makes fitting it or optimising its value easy.

Anyway, details.

Plomp and Levelt’s dissonance curves

Attributed to Plomp and Levelt’s (PlLe65), here is Sethares’ version (Seth98a), also summarised on Sethares’ web page.

Dissonance between two pure sinusoidal frequencies, \(f_1 \leq f_2\), with amplitudes respectively \(v_1, v_2\), is given by:

\begin{equation*} d_\text{PL}(f_1,f_2, v_1,v_2) := v_1v_2\left[ \exp\left( -as(f_2-f_1) \right) - \exp\left( -bs(f_2-f_1) \right) \right] \end{equation*}


\begin{equation*} s=\frac{d^*}{s_1 f_1+s_2} \end{equation*}

and \(a=3.5, b=5.75, d^*=.24, s_1=0.21, s_2= 19\), the constants being fit by least-squares from experimental data.

If your note has more than one frequency, one sums the pairwise dissonances of all contributing frequencies to find the total dissonance, which is not biologically plausible but seems to work ok. Other ways of working out differences between two composite sounds could be possible (Hausdorff metric etc). [1]

This looks to me like the kind of situation where the actual details of the curve are not so important as getting the points of maximal and minimal dissonance right. Clearly we have a minimal value at \(f_1=f_2\). We solve for the maximally dissonant frequency \(f_2\) with respect to a fixed \(f_1, v_1, v_2\):

\begin{align*} -as\exp( -as(f_2-f_1) ) &= -bs\exp( -bs(f_2-f_1) )\\ a\exp( -as(f_2-f_1) ) &= b\exp( -bs(f_2-f_1) )\\ \ln a - as(f_2-f_1) &= \ln b -bs(f_2-f_1)\\ \ln a - \ln b &= as(f_2-f_1) -bs(f_2-f_1)\\ \ln a - \ln b &= s(a-b)(f_2-f_1) \\ f_2 &= f_1+\frac{\ln b - \ln a}{s(b-a)}\\ f_2 &= f_1(s_1+C)+s_2C \end{align*}


\begin{equation*} C:=\frac{\ln b - \ln a}{d^*(b-a)} \end{equation*}

That affine difference is reminiscent of resolvability criteria in functional bases.

Parncutt and Barlow dissonance

Differences of exponentials are computationally tedious because of numerical concerns with large frequency values; this is suggestive of approximation by something more convenient, maybe of this form:

\begin{equation*} d_\text{simple}(f_1,f_2,v_1, v_2):=C_1(f_2-f_1)\exp -C_2(f_2-f_1) \end{equation*}

The Parncutt approximation takes this approach and additionally transforms the units into heuristically preferable psychoacoustic ones.

Cribbed from Lach Lau’s source code and thesis (Lach12), where he attributes it to Parncutt and Barlow, although I can’t find any actual articles by Parncutt and/or Barlow which use this. Mash06 implies it might be unpublished. Hans14 gives a squared version of the same formula.

For this we take frequencies \(b_1\leq b_2\) and volumes \(s_1, s_2\) in, respectively, barks and sones. Then

\begin{align*} d_\text{PB}(b_1, b_2, s_1, s_2) &:=\sqrt{(s_1 s_2)}(4 ( b_2- b_1) \exp(1 - 4 ( b_2- b_1)))\\ &= \sqrt{(s_1 s_2)}(4 ( b_2- b_1) e \exp( - 4 ( b_2- b_1))) \end{align*}

Since this scale is relative, I’m not quite sure why we have constants everywhere. Why not

\begin{equation*} d_\text{PB}'(b_1, b_2, s_1, s_2) := \sqrt{(s_1 s_2)}\frac{ b_2- b_1}{ \exp(b_2-b_1)}? \end{equation*}

Possibly in order to more closely approximate Sethares?

[1]I thought I was being clever to consider such alternatives, but Budney and Sethares (BuSe14) scooped me.

Induced topologies

TBD. Nestke (2004) and Mazzola (2012). Tymozcko.

To Read

Barlow, C., & Lohner, H. (1987) Two Essays on Theory. Computer Music Journal, 11(1), 44. DOI.
Bartlett, M. S., & Medhi, J. (1955) On the Efficiency of Procedures for Smoothing Periodograms from Time Series with Continuous Spectra. Biometrika, 42(1/2), 143. DOI.
Bigand, E., & Parncutt, R. (1999) Perceiving musical tension in long chord sequences. Psychological Research, 62(4), 237–254.
Bigand, E., Parncutt, R., & Lerdahl, F. (1996) Perception of musical tension in short chord sequences: The influence of harmonic function, sensory dissonance, horizontal motion, and musical training. Perception & Psychophysics, 58(1), 125–141. DOI.
Bigo, L., Giavitto, J.-L., & Spicher, A. (2011) Building Topological Spaces for Musical Objects. In Proceedings of the Third International Conference on Mathematics and Computation in Music (pp. 13–28). Berlin, Heidelberg: Springer-Verlag DOI.
Bingham, C., Godfrey, M., & Tukey, J. W.(1967) Modern techniques of power spectrum estimation. Audio and Electroacoustics, IEEE Transactions on, 15(2), 56–66.
Bod, R. (2002) Memory-based models of melodic analysis: Challenging the Gestalt principles. Journal of New Music Research, 31(1), 27–36. DOI.
Boggs, P. T., & Rogers, J. E.(1990) Orthogonal distance regression. Contemporary Mathematics, 112, 183–194.
Budney, R., & Sethares, W. (2014) Topology of Musical Data. Journal of Mathematics and Music, 8(1), 73–92. DOI.
Callender, C., Quinn, I., & Tymoczko, D. (2008) Generalized voice-leading spaces. Science (New York, N.Y.), 320(5874), 346–348. DOI.
Cancho, R. F. i, & Solé, R. V.(2003) Least effort and the origins of scaling in human language. Proceedings of the National Academy of Sciences, 100(3), 788–791. DOI.
Carlos, W. (1987) Tuning: At the Crossroads. Computer Music Journal, 11(1), 29–43. DOI.
Casey, M. A., Veltkamp, R., Goto, M., Leman, M., Rhodes, C., & Slaney, M. (2008) Content-Based Music Information Retrieval: Current Directions and Future Challenges. Proceedings of the IEEE, 96(4), 668–696. DOI.
Casey, M., Rhodes, C., & Slaney, M. (2008) Analysis of Minimum Distances in High-Dimensional Musical Spaces. IEEE Transactions on Audio, Speech, and Language Processing, 16(5), 1015–1028. DOI.
Chon, S. H.(2008) Quantifying the consonance of complex tones with missing fundamentals.
Cohen, D., & Dubnov, S. (1997) Gestalt phenomena in musical texture. In M. Leman (Ed.), Music, Gestalt, and Computing (pp. 386–405). Springer Berlin Heidelberg
Cook, N. D., Fujisawa, T. X., & Konaka, H. (2007) Why Not Study Polytonal Psychophysics?. Empirical Musicology Review, 2(1).
Cooper, J., & Fazio, R. H.(1984) A new look at dissonance. Advances in Experimental Social Psychology, 17, 229–268.
Corral, A. del, León, T., & Liern, V. (2009) Compatibility of the Different Tuning Systems in an Orchestra. In E. Chew, A. Childs, & C.-H. Chuan (Eds.), Mathematics and Computation in Music (pp. 93–103). Springer Berlin Heidelberg
Cousineau, M., McDermott, J. H., & Peretz, I. (2012) The basis of musical consonance as revealed by congenital amusia. Proceedings of the National Academy of Sciences, 109(48), 19858–19863. DOI.
de Cheveigné, A. (2005) Pitch Perception Models. In C. J. Plack, R. R. Fay, A. J. Oxenham, & A. N. Popper (Eds.), Pitch (pp. 169–233). Springer New York DOI.
Demaine, E. D., Gomez-Martin, F., Meijer, H., Rappaport, D., Taslakian, P., Toussaint, G. T., … Wood, D. R.(2005) The Distance Geometry of Deep Rhythms and Scales. In CCCG (pp. 163–166).
Demaine, E. D., Gomez-Martin, F., Meijer, H., Rappaport, D., Taslakian, P., Toussaint, G. T., … Wood, D. R.(2009) The distance geometry of music. In Computational Geometry (Vol. 42, pp. 429–454). DOI.
Demopoulos, R. J., & Katchabaw, M. J.(2007) Music Information Retrieval: A Survey of Issues and Approaches. . Technical Report
Du, P., Kibbe, W. A., & Lin, S. M.(2006) Improved peak detection in mass spectrum by incorporating continuous wavelet transform-based pattern matching. Bioinformatics, 22(17), 2059–2065. DOI.
Duffin, R. J.(1948) Function classes invariant under the Fourier transform. Duke Mathematical Journal, 15(3), 781–785. DOI.
Ferguson, S., & Parncutt, R. (2004) Composing In the Flesh: Perceptually-Informed Harmonic Syntax. In Proceedings of Sound and Music Computing.
Févotte, C., Bertin, N., & Durrieu, J.-L. (2008) Nonnegative Matrix Factorization with the Itakura-Saito Divergence: With Application to Music Analysis. Neural Computation, 21(3), 793–830. DOI.
Fitzpatrick, J. M., Hill, D. L. G., & Maurer, Jr., C. R.(2000) Image Registration. In M. Sonka & J. M. Fitzpatrick (Eds.), Handbook of Medical Imaging, Volume 2. Medical Image Processing and Analysis (p. Chapter 8). 1000 20th Street, Bellingham, WA 98227-0010 USA: SPIE
Flamary, R., Févotte, C., Courty, N., & Emiya, V. (2016) Optimal spectral transportation with application to music transcription. In arXiv:1609.09799 [cs, stat] (pp. 703–711). Curran Associates, Inc.
Fokker, A. D.(1969) Unison vectors and periodicity blocks in 3-dimensional (3-5-7-) harmonic lattice of notes. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen Series B-Physical Sciences, 72(3), 153.
Gashler, M., & Martinez, T. (2011) Tangent space guided intelligent neighbor finding. (pp. 2617–2624). IEEE DOI.
Gashler, M., & Martinez, T. (2012) Robust manifold learning with CycleCut. Connection Science, 24(1), 57–69. DOI.
Hall, R. W.(2008) Geometrical Music Theory. Science, 320(5874), 328–329. DOI.
Hansen, B. (2014) Spatial Utilization of Sensory Dissonance and the Creation of Sonic Sculpture. . Ann Arbor, MI: Michigan Publishing, University of Michigan Library
Haussler, D. (1999) Convolution kernels on discrete structures. . Technical report, UC Santa Cruz
Hawe, S., Kleinsteuber, M., & Diepold, K. (2013) Analysis operator learning and its application to image reconstruction. IEEE Transactions on Image Processing, 22(6), 2138–2150. DOI.
Hemmen, J. L. van, & Vollmayr, A. N.(2013) Resonating vector strength: what happens when we vary the “probing” frequency while keeping the spike times fixed. Biological Cybernetics, 107(4), 491–494. DOI.
Hermes, D. J.(1988) Measurement of pitch by subharmonic summation. The Journal of the Acoustical Society of America, 83(1), 257–264. DOI.
Honingh, A., & Bod, R. (2011) In Search of Universal Properties of Musical Scales. Journal of New Music Research, 40(1), 81–89. DOI.
Huron, D. (1994) Interval-Class Content in Equally Tempered Pitch-Class Sets: Common Scales Exhibit Optimum Tonal Consonance. Music Perception: An Interdisciplinary Journal, 11(3), 289–305. DOI.
Huron, D., & Parncutt, R. (1993) An improved model of tonality perception incorporating pitch salience and echoic memory. Psychomusicology: A Journal of Research in Music Cognition, 12(2), 154–171. DOI.
Jewell, M. O., Rhodes, C., & d’Inverno, M. (2010) Querying improvised music: Do you sound like yourself?. In ISMIR (pp. 483–488). International Society for Music Information Retrieval
Kameoka, A., & Kuriyagawa, M. (1969a) Consonance theory part I: Consonance of dyads. The Journal of the Acoustical Society of America, 45(6), 1451–1459. DOI.
Kameoka, A., & Kuriyagawa, M. (1969b) Consonance theory part II: Consonance of complex tones and its calculation method. The Journal of the Acoustical Society of America, 45(6), 1460–1469. DOI.
Katz, J., & Pesetsky, D. (2009) The recursive syntax and prosody of tonal music. Ms., Massachusetts Institute of Technology.
Krumhansl, C. L.(2000) Rhythm and pitch in music cognition. Psychological Bulletin, 126(1), 159. DOI.
Kuswanto, Heru. (2011) Comparison Study of Saron Ricik Instruments’ Sound Color (Timbre) on the Gamelans of Nagawilaga and Gunturmadu from. International Journal of Basic & Applied Sciences.
Kuswanto, H. (2012) Saron demung’s timbre and sonogram of gamelans gunturmadu from Keraton Ngayogyakarta. Jurnal Pendidikan Fisika Indonesia, 8(1).
Kuswanto, Heru, Sumarna, Purwanto, A., & Cipto Budy Handoyo. (2012) Saron demung instruments timbre spectrum comparison study on the gamelans sekati from Karaton Ngayogyakarta. International Journal of Basic & Applied Sciences.
Lach Lau, J. S.(2011) Dissonance Curves as Generating Devices for Dealing With Harmony. ICMC2011.
Lach Lau, J. S.(2012, December 13) Harmonic duality : from interval ratios and pitch distance to spectra and sensory dissonance (Dissertatie).
Leman, M. (1997) Music, Gestalt, and computing: studies in cognitive and systematic musicology. (Vol. 1317). Springer
Lerdahl, F. (1988) Tonal Pitch Space. Music Perception: An Interdisciplinary Journal, 5(3), 315–349. DOI.
Lerdahl, F. (1996) Calculating Tonal Tension. Music Perception: An Interdisciplinary Journal, 13(3), 319–363. DOI.
Levitin, D. J., Chordia, P., & Menon, V. (2012) Musical rhythm spectra from Bach to Joplin obey a 1/f power law. Proceedings of the National Academy of Sciences of the United States of America, 109(10), 3716–3720. DOI.
Li, G. (2006) The effect of inharmonic and harmonic spectra in Javanese gamelan tuning (1): a theory of the sléndro. In Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications (pp. 65–71). World Scientific and Engineering Academy and Society (WSEAS)
Macke, J. H., Berens, P., Ecker, A. S., Tolias, A. S., & Bethge, M. (2009) Generating spike trains with specified correlation coefficients. Neural Computation, 21(2), 397–423. DOI.
Madjiheurem, S., Qu, L., & Walder, C. (2016) Chord2Vec: Learning Musical Chord Embeddings.
Manaris, B., Machado, P., McCauley, C., Romero, J., & Krehbiel, D. (2005) Developing Fitness Functions for Pleasant Music: Zipf’s Law and Interactive Evolution Systems. In F. Rothlauf, J. Branke, S. Cagnoni, D. W. Corne, R. Drechsler, Y. Jin, … G. Squillero (Eds.), Applications of Evolutionary Computing (pp. 498–507). Springer Berlin Heidelberg
Mashinter, K. (2006) Calculating sensory dissonance: Some discrepancies arising from the models of Kameoka & Kuriyagawa, and Hutchinson & Knopoff.
Mazzola, G. (2012) Singular homology on hypergestures. Journal of Mathematics and Music, 6(1), 49–60. DOI.
McFee, B., & Ellis, D. P.(2011) Analyzing song structure with spectral clustering. In IEEE conference on Computer Vision and Pattern Recognition (CVPR).
McKinney, M. F.(2001) Neural correlates of pitch and roughness: toward the neural code for melody and harmony. . Massachusetts Institute of Technology
Moustafa, K. A.-, Schuurmans, D., & Ferrie, F. (2013) Learning a Metric Space for Neighbourhood Topology Estimation: Application to Manifold Learning. In Journal of Machine Learning Research (pp. 341–356).
Nicolls, F., & de Jager, G. (2001) Uniformly most powerful cyclic permutation invariant detection for discrete-time signals. (Vol. 5, pp. 3165–3168). IEEE DOI.
Nordmark, J., & Fahlen, L. E.(1988) Beat theories of musical consonance. Speech Transmission Laboratory, Quarterly Progress and Status Report.
Park, C. G. C. G.(2004) Construction of random vectors of heterogeneous component variables under specified correlation structures. Computational Statistics & Data Analysis, 46(4), 621–630. DOI.
Parncutt, R. (1989) Harmony: a psychoacoustical approach. . Berlin ; New York: Springer-Verlag
Parncutt, R. (1994) A Perceptual Model of Pulse Salience and Metrical Accent in Musical Rhythms. Music Perception: An Interdisciplinary Journal, 11(4), 409–464. DOI.
Parncutt, R. (1997) A model of the perceptual root(s) of a chord accounting for voicing and prevailing tonality. In M. Leman (Ed.), Music, Gestalt, and Computing (pp. 181–199). Springer Berlin Heidelberg
Parncutt, R. (2005) Psychoacoustics and music perception. Musikpsychologie–das Neue Handbuch.
Parncutt, R. (2006) Commentary on Keith Mashinter’s “Calculating Sensory Dissonance: Some Discrepancies Arising from the Models of Kameoka & Kuriyagawa, and Hutchinson & Knopoff”.
Parncutt, R. (2013) Psychoacoustics and cognition for musicians. Sound Musicianship: Understanding the Crafts of Music, 2.
Parncutt, R., & Hair, G. (2011) Consonance and dissonance in music theory and psychology: disentangling dissonant dichotomies. J. Interdiscipl. Music Stud, 5, 119–166.
Parncutt, R., & Strasburger, H. (1994) Applying Psychoacoustics in Composition: “Harmonic” Progressions of “Nonharmonic” Sonorities. Perspectives of New Music, 32(2), 88–129. DOI.
Perchy, S., & Sarria, G. (2009) Dissonances: Brief description and its computational representation in the RTCC calculus. In Proc. of SMC2009, Porto, Portugal.
Plomp, R., & Levelt, W. J.(1965) Tonal consonance and critical bandwidth. The Journal of the Acoustical Society of America, 38(4), 548–560. DOI.
Rasch, R., & Plomp, R. (1999) The perception of musical tones. The Psychology of Music, 2, 89–112.
Reese, K., Yampolskiy, R., & Elmaghraby, A. (2012) A framework for interactive generation of music for games. In 2012 17th International Conference on Computer Games (CGAMES) (pp. 131–137). Washington, DC, USA: IEEE Computer Society DOI.
Reitboeck, H., & Brody, T. P.(1969) A transformation with invariance under cyclic permutation for applications in pattern recognition. Information and Control, 15(2), 130–154. DOI.
Reuter, C. (1997) Karl Erich Schumann’s principles of timbre as a helpful tool in stream segregation research. In M. Leman (Ed.), Music, Gestalt, and Computing (pp. 362–374). Springer Berlin Heidelberg
Rohrmeier, M., Zuidema, W., Wiggins, G. A., & Scharff, C. (2015) Principles of structure building in music, language and animal song. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 370(1664), 20140097. DOI.
Serrà, J., Corral, Á., Boguñá, M., Haro, M., & Arcos, J. L.(2012) Measuring the Evolution of Contemporary Western Popular Music. Scientific Reports, 2. DOI.
Sethares, W. A.(1997) Specifying spectra for musical scales. The Journal of the Acoustical Society of America, 102(4), 2422–2431. DOI.
Sethares, W. A.(1998a) Consonance-Based Spectral Mappings. Computer Music Journal, 22(1), 56. DOI.
Sethares, W. A.(1998b) Tuning, Timbre, Spectrum, Scale. . Springer London
Sethares, W. A., Milne, A. J., Tiedje, S., Prechtl, A., & Plamondon, J. (2009) Spectral Tools for Dynamic Tonality and Audio Morphing. Computer Music Journal, 33(2), 71–84. DOI.
Shiv, V. L.(2011) Improved frequency estimation in sinusoidal models through iterative linear programming schemes. In Sound and Music Computing Conference, Proceedings.
Smola, A. J., Williamson, R. C., Mika, S., & Schölkopf, B. (1999) Regularized Principal Manifolds. In P. Fischer & H. U. Simon (Eds.), Computational Learning Theory (pp. 214–229). Springer Berlin Heidelberg
Sriperumbudur, B. K., Fukumizu, K., Gretton, A., Schölkopf, B., & Lanckriet, G. R. G.(2012) On the empirical estimation of integral probability metrics. Electronic Journal of Statistics, 6, 1550–1599. DOI.
Stevens, S. S., & Volkmann, J. (1940) The Relation of Pitch to Frequency: A Revised Scale. The American Journal of Psychology, 53(3), 329–353. DOI.
Stolzenburg, F. (2013) Harmony Perception by Periodicity Detection. arXiv:1306.6458 [Cs].
Terhardt, E. (1974) Pitch, consonance, and harmony. The Journal of the Acoustical Society of America, 55(5), 1061–1069. DOI.
Terhardt, E., Stoll, G., & Seewann, M. (1982) Algorithm for extraction of pitch and pitch salience from complex tonal signals. The Journal of the Acoustical Society of America, 71(3), 679–688. DOI.
Thompson, W. F., & Parncutt, R. (1997) Perceptual judgments of triads and dyads: Assessment of a psychoacoustic model. Music Perception, 263–280.
Tillmann, B., Bharucha, J. J., & Bigand, E. (2000) Implicit learning of tonality: a self-organizing approach. Psychological Review, 107(4), 885.
Toiviainen, P. (1997) Optimizing self-organizing timbre maps: Two approaches. In M. Leman (Ed.), Music, Gestalt, and Computing (pp. 335–350). Springer Berlin Heidelberg
Toussaint, G. (2005) Mathematical Features for Recognizing Preference in Sub-saharan African Traditional Rhythm Timelines. In S. Singh, M. Singh, C. Apte, & P. Perner (Eds.), Pattern Recognition and Data Mining (Vol. 3686, pp. 18–27). Berlin, Heidelberg: Springer Berlin Heidelberg
Toussaint, G. T.(2004) A Comparison of Rhythmic Similarity Measures. In ISMIR.
Tymoczko, D. (2006) The Geometry of Musical Chords. Science, 313(5783), 72–74. DOI.
Tymoczko, D. (2009a) Generalizing Musical Intervals. Journal of Music Theory, 53(2), 227–254. DOI.
Tymoczko, D. (2009b) Three Conceptions of Musical Distance. In E. Chew, A. Childs, & C.-H. Chuan (Eds.), Mathematics and Computation in Music (pp. 258–272). Springer Berlin Heidelberg DOI.
Tymoczko, D. (2011a) A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. (1 edition.). New York: Oxford University Press
Tymoczko, D. (2011b) Mazzola’s Counterpoint Theory. Mathematics and Computation in Music, Springer, 297–310.
Tymoczko, D. (2012) The Generalized Tonnetz. Journal of Music Theory, 56(1), 1–52. DOI.
Tymoczko, D. (2013) Geometry and the quest for theoretical generality. Journal of Mathematics and Music, 7(2), 127–144. DOI.
Vassilakis, P. N., & Kendall, R. A.(2010) Psychoacoustic and cognitive aspects of auditory roughness: definitions, models, and applications. In IS&T/SPIE Electronic Imaging (p. 75270O–75270O). International Society for Optics and Photonics
Wagh, M. D.(1976) Cyclic autocorrelation as a translation invariant transform. India, IEE-IERE Proceedings, 14(5), 185–191. DOI.
Xin, J., & Qi, Y. (2006) A Many to One Discrete Auditory Transform. arXiv:math/0603174.
Zanette, D. H.(2006) Zipf’s law and the creation of musical context. Musicae Scientiae, 10(1), 3–18. DOI.
Zhao, Z., & Singer, A. (2013) Fourier–Bessel rotational invariant eigenimages. Journal of the Optical Society of America A, 30(5), 871. DOI.