The inharmonicity of vibrating strings can easily be estimated from recordings of isolated tones. Likewise, the tuning system (temperament) of a keyboard instrument can be ascertained from isolated tones by estimating the fundamental frequencies corresponding to each key of the instrument. This paper addresses a more difficult problem: the automatic estimation of the inharmonicity and temperament of a harpsichord given only a recording of an unknown musical work. An initial conservative transcription is used to generate a list of note candidates, and high-precision frequency estimation techniques and robust statistics are employed to estimate the inharmonicity and fundamental frequency of each note. These estimates are then matched to a set of known keyboard temperaments, allowing for variation in the tuning reference frequency, in order to obtain the temperament used in the recording. Results indicate that it is possible to obtain inharmonicity estimates and to classify keyboard temperament automatically from audio recordings of standard musical works, to the extent of accurately (96%) distinguishing between six different temperaments commonly used in harpsichord recordings. Although there is an interaction between inharmonicity and temperament, this is shown to be minor relative to the tuning accuracy.
REFERENCES
1.
Abe
, M.
, and Smith
, J.
(2004
). “CQIFFT: Correcting bias in a sinusoidal parameter estimator based on quadratic interpolation of FFT magnitude peaks,”
Technical Report No. STAN-M-117, Center for Computer Research in Music and Acoustics, Department of Music, Stanford University.2.
Anderson
, B.
, and Strong
, W.
(2005
). “The effect of inharmonic partials on pitch of piano tones,”
J. Acoust. Soc. Am.
117
, 3268
–3272
. 3.
4.
Benetos
, E.
, and Dixon
, S.
(2010
). “Multiple-F0 estimation of piano sounds exploiting spectral structure and temporal evolution,”
in International Speech Communication Association Tutorial and Research Workshop on Statistical and Perceptual Audition (SAPA2010)
, pp. 13
–18
.5.
de Cheveigné
, A.
(2006
). “Multiple f0 estimation,”
in Computational Auditory Scene Analysis: Principles, Algorithms and Applications
, edited by D.
Wang
and G.
Brown
(IEEE Press/Wiley
, Piscataway, NJ
), pp. 45
–79
.6.
de Cheveigné
, A.
, and Kawahara
, H.
(2002
). “YIN, a fundamental frequency estimator for speech and music,”
J. Acoust. Soc. Am.
111
, 1917
–1930
. 7.
Di Veroli
, C.
(2009
). Unequal Temperaments: Theory, History, and Practice
(Bray Baroque
, Bray, Ireland
).8.
Dixon
, S.
(2001
). “Automatic extraction of tempo and beat from expressive performances,”
J. New Mus. Res.
30
, 39
–58
. 9.
Earis
, A.
, Daly
, M.
, Fei
, S.
, and Thompson
, R.
(2007
). “Acoustical studies of historical keyboard instruments in the Royal College of Music museum of instruments,”
in Proceedings of the 19th International Congress on Acoustics
, MUS-02-006.10.
Emiya
, V.
, Badeau
, R.
, and David
, B.
(2010
). “Multipitch estimation of piano sounds using a new probabilistic spectral smoothness principle,”
IEEE Trans. Audio Speech Lang. Process.
18
, 1643
–1654
. 11.
Fletcher
, H.
(1964
). “Normal vibration frequencies of a stiff piano string,”
J. Acoust. Soc. Am.
36
, 203
–209
. 12.
Fletcher
, H.
, Blackham
, E.
, and Stratton
, R.
(1962
). “Quality of piano tones,”
J. Acoust. Soc. Am.
34
, 749
–761
. 13.
Fletcher
, N.
(1977
). “Analysis of the design and performance of harpsichords,”
Acustica
37
, 139
–147
.14.
Fletcher
, N.
, and Rossing
, T.
(1998
). The Physics of Musical Instruments
(Springer
, New York, NY
).15.
Gerhard
, D.
(2003
). “Pitch extraction and fundamental frequency: History and current techniques,”
Technical Report No. TR-CS 2003-06, Department of Computer Science, University of Regina, Regina, Canada.16.
Järveläinen
, H.
, Välimäki
, V.
, and Karjalainen
, M.
(2001
). “Audibility of the timbral effects of inharmonicity in stringed instrument tones,”
Acoust. Res. Lett. Online
2
, 79
–84
. 17.
Klapuri
, A.
(2003
). “Multiple fundamental frequency estimation based on harmonicity and spectral smoothness,”
IEEE Trans. Speech Audio Process.
11
, 804
–816
. 18.
Klapuri
, A.
(2009
). “A method for visualizing the pitch content of polyphonic music signals,”
in 10th International Society for Music Information Retrieval Conference
, pp. 615
–620
.19.
Klapuri
, A.
, and Davy
, M.
, eds. (2006
). Signal Processing Methods for Music Transcription
(Springer
, New York, NY
).20.
Lundin
, R.
(1947
). “Toward a cultural theory of consonance,”
J. Psych.
23
, 45
–49
. 21.
Mauch
, M.
, and Dixon
, S.
(2010
). “Simultaneous estimation of chords and musical context from audio,”
IEEE Trans. Audio Speech Lang. Process.
18
, 1280
–1289
. 22.
McDermott
, J.
, Lehr
, A.
, and Oxenham
, A.
(2010
). “Individual differences reveal the basis of consonance,”
Current Biol.
20
, 1035
–1041
. 23.
Moore
, B.
, Glasberg
, B.
, and Peters
, R.
(1985
). “Relative dominance of individual partials in determining the pitch of complex tones,”
J. Acoust. Soc. Am.
77
, 1853
–1860
. 24.
Noland
, K.
, and Sandler
, M.
(2006
). “Key estimation using a Hidden Markov Model,”
in 7th International Conference on Music Information Retrieval
, pp. 121
–126
.25.
Palisca
, C.
, and Moore
, B.
(2010
). “Grove music online,” http://www.grovemusic.com/ (Last accessed 15 January 2010).26.
Pianoteq
(2010
). “Pianoteq 3 true modeling
,” http://www.pianoteq.com (Last accessed 23 November 2010).27.
Rasch
, R.
(2002
). “Tuning and temperament,”
in The Cambridge History of Western Music
, edited by T.
Christensen
(Cambridge University Press
, Cambridge, UK
), pp. 193
–222
.28.
Rauhala
, J.
, Lehtonen
, H.-M.
, and Välimäki
, V.
(2007
). “Fast automatic inharmonicity estimation algorithm,”
J. Acoust. Soc. Am.
121
, EL184
–EL189
. 29.
30.
Shankland
, R.
, and Coltman
, J.
(1939
). “The departure of the overtones of a vibrating wire from a true harmonic series,”
J. Acoust. Soc. Am.
10
, 161
–166
. 31.
Smith
, J.
(2010
). “Spectral audio signal processing: March 2010 draft,” http://ccrma.stanford.edu/~jos/sasp/ (Last accessed 23 November 2010).32.
Smith
, J.
, and Serra
, X.
(1987
). “PARSHL: An analysis/synthesis program for non-harmonic sounds based on a sinusoidal representation,”
in Proceedings of the International Computer Music Conference
, 290
–297
.33.
Terhardt
, E.
(1977
). “The two-component theory of musical consonance,”
in Psychophysics and Physiology of Hearing
, edited by E.
Evans
and J.
Wilson
(Academic Press
, London
), pp. 381
–390
.34.
Tidhar
, D.
, Fazekas
, G.
, Mauch
, M.
, and Dixon
, S.
(2010a
). “TempEst: Harpsichord temperament estimation in a semantic-web environment,”
J. New Mus. Res.
39
, 327
–336
. 35.
Tidhar
, D.
, Mauch
, M.
, and Dixon
, S.
(2010b
). “High precision frequency estimation for harpsichord tuning classification”
, in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing
, pp. 61
–64
.36.
Tzanetakis
, G.
, and Cook
, P.
(2002
). “Musical genre classification of audio signals,”
IEEE Trans. Speech Audio Process.
10
, 293
–302
. 37.
Välimäki
, V.
, Penttinen
, H.
, Knif
, J.
, Laurson
, M.
, and Erkut
, C.
(2004
). “Sound synthesis of the harpsichord using a computationally efficient physical model,”
EURASIP J. Appl. Sign. Process.
2004
, 934
–948
.38.
von Helmholtz
, H.
(1863
). Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik (On the Sensations of Tone as a Physiological Basis for the Theory of Music)
(Friedrich Vieweg und Sohn, Braunschweig
, Germany
).39.
Wen
, X.
, and Sandler
, M.
(2009
). “Notes on model-based non-stationary sinusoid estimation methods using derivatives,”
in 12th International Conference on Digital Audio Effects
, pp. 113
–120
.40.
Young
, R.
(1952
). “Inharmonicity of plain wire piano strings,”
J. Acoust. Soc. Am.
24
, 267
–273
. © 2012 Acoustical Society of America.
2012
Acoustical Society of America
You do not currently have access to this content.