The Temperament Police: The Truth, the Ground Truth, and Nothing but the Truth

The tuning system of a keyboard instrument is chosen so that frequently used musical intervals sound as consonant as possible. Temperament refers to the compromise arising from the fact that not all intervals can be maximally consonant simultaneously. Recent work showed that it is possible to estimate temperament from audio recordings with no prior knowledge of the musical score, using a conservative (high precision, low recall) automatic transcription algorithm followed by frequency estimation using quadratic interpolation and bias correction from the log magnitude spectrum. In this paper we develop a harpsichord-specific transcription system to analyse over 500 recordings of solo harpsichord music for which the temperament is specified on the CD sleeve notes. We compare the measured temperaments with the annotations and discuss the differences between temperament as a theoretical construct and as a practical issue for professional performers and tuners. The implications are that ground truth is not always scientific truth, and that content-based analysis has an important role in the study of historical performance practice.

[1]  H. Fletcher.  Normal Vibration Frequencies of a Stiff Piano String , 1964 .

[2]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[3]  H. Sebastian Seung,et al.  Learning the parts of objects by non-negative matrix factorization , 1999, Nature.

[4]  Masataka Goto,et al.  RWC Music Database: Music genre database and musical instrument sound database , 2003, ISMIR.

[5]  J. Barbour.  Tuning and Temperament: A Historical Survey , 2004 .

[6]  B. Lehman.  Bach's Extraordinary Temperament: Our Rosetta Stone--1 , 2005 .

[7]  Anssi Klapuri,et al.  Signal Processing Methods for Music Transcription , 2006 .

[8]  Guy J. Brown,et al.  Multiple F0 Estimation , 2006 .

[9]  D. Hall.  How Equal Temperament Ruined Harmony (and Why You Should Care) , 2007 .

[10]  Raul Kompass,et al.  A Generalized Divergence Measure for Nonnegative Matrix Factorization , 2007, Neural Computation.

[11]  Dan Tidhar,et al.  High precision frequency estimation for harpsichord tuning classification , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[12]  Guillaume Lemaitre,et al.  Real-time Polyphonic Music Transcription with Non-negative Matrix Factorization and Beta-divergence , 2010, ISMIR.

[13]  Simon Dixon,et al.  Polyphonic music transcription using note onset and offset detection , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[14]  Dan Tidhar,et al.  Estimation of harpsichord inharmonicity and temperament from musical recordings. , 2012, The Journal of the Acoustical Society of America.

[15]  J. Smith,et al.  CQIFFT : Correcting Bias in a Sinusoidal Parameter Estimator based on Quadratic Interpolation of FFT Magnitude Peaks October 13 , 2004 , .

[16]  J. Smith,et al.  Design Criteria for the Quadratically Interpolated FFT Method ( II ) : Bias due to Interfering Components October 13 , 2004 , .