mel-cepstral-distance 0.0.4

A Python library for computing the Mel-Cepstral Distance (also known as Mel-Cepstral Distortion, MCD) between two inputs. This implementation is based on the paper 'Mel-Cepstral Distance Measure for Objective Speech Quality Assessment' by Kubichek (1993).

mel-cepstral-distance

A Python library for computing the Mel-Cepstral Distance (also known as Mel-Cepstral Distortion, MCD) between two inputs. This implementation is based on the method proposed by Robert F. Kubichek in Mel-Cepstral Distance Measure for Objective Speech Quality Assessment.

• Compute MCD between two inputs: audio files, amplitude spectrograms, mel spectrograms, or MFCCs.
• Remove pauses from audio files or feature representations (amplitude spectrograms, mel spectrograms, or MFCCs) using a threshold.
• Align feature representations using either Dynamic Time Warping (DTW) or zero-padding.
• Calculate an alignment penalty as an additional metric to indicate the extent of alignment applied.

Getting Started

Installation

pip install mel-cepstral-distance

Example usage

Compare two audio files with default parameters:

from mel_cepstral_distance import compare_audio_files

mcd, penalty = compare_audio_files(
  'examples/GT.wav',
  'examples/WaveGlow.wav',
)

print(f'MCD: {mcd:.2f}, Penalty: {penalty:.4f}')
# MCD: 4.03, Penalty: 0.0197

Calculation

Spectrogram

$$ X(k, m) = \text{FFT of } x_k(n), \text{ for real input.} $$

Where:

• $X(k, m)$: The result (amplitude spectrogram) of the real-valued FFT for the $k$-th frame at frequency index $m$.
• $x_k(n)$: The time-domain signal of the $k$-th frame.
• $\text{FFT}$: The real-valued discrete Fourier transform, computed using np.fft.rfft.

Mel spectrogram

$$ X_{k,n} = \log_{10}\left\lbrace\sum_m^M |X(k, m)|^2 \cdot w_n(m)\right\rbrace $$

Where:

• $X_{k,n}$: The logarithmic Mel-scaled power spectrogram for the $k$-th frame at Mel frequency $n$.
• $X(k, m)$: The amplitude spectrum of the $k$-th frame at frequency $m$.
• $M$: The total number of Mel frequency bins.
• $w_n(m)$: The Mel filter bank weights for Mel frequency $n$ and frequency bin $m$.

Mel-frequency cepstral coefficients

$$ MC_X(i, k) = \sum_{n=1}^{M} X_{k,n} \cos\left[i\left(n - \frac{1}{2}\right)\frac{\pi}{M}\right] $$

Where:

• $MC_X(i, k)$: The $i$-th Mel-frequency cepstral coefficient (MFCC) for the $k$-th frame.
• $X_{k,n}$: The logarithmic Mel-scaled power spectrogram for the $k$-th frame at Mel frequency $n$.
• $M$: The total number of Mel frequency bins.
• $i$: The index of the MFCC being computed.

Mel-cepstral distance

Per frame

$$ MCD(k) = \alpha\sqrt{\sum_{i=s}^{D} \left(MC_X(i, k) - MC_Y(i, k)\right)^2} $$

Where:

• $MCD(k)$: The Mel-cepstral distance for the $k$-th frame.
• $MC_X(i, k)$: The $i$-th MFCC of the reference signal for the $k$-th frame.
• $MC_Y(i, k)$: The $i$-th MFCC of the target signal for the $k$-th frame.
• $D$: The number of MFCCs used in the computation.
• $\alpha$: Optional scaling factor used in some literature, e.g. $\frac{10\sqrt{2}}{\ln 10}$.
  • Note: Kubichek didn't use it, so it has value 1
• $s$: Parameter to exclude the 0th coefficient (corresponding to energy):
  • $s = 0$: Includes the 0th coefficient
  • $s = 1$: Excludes the 0th coefficient

Mean over all frames

$$ MCD = \frac{1}{N} \sum_{k=1}^{N} MCD(k) $$

Where:

• $MCD$: The mean Mel-cepstral distance over all frames.
• $N$: The total number of frames.
• $MCD(k)$: The Mel-cepstral distance for the $k$-th frame.

Alignment penalty during dynamic time warping (DTW)

$$ PEN = 2 - \frac{N_X + N_Y}{N_{XY}} $$

Where:

• $N_X$: The number of frames in the reference sequence.
• $N_Y$: The number of frames in the target sequence.
• $N_{XY}$: The number of frames after alignment (same for X and Y).
• $PEN$: A value in interval [$0$, $1$), where a smaller value indicates less alignment.

Used parameters in literature

Literature	Sampling Rate	Window Size	Hop Length	FFT Size	Window Function	$M$	Min Frequency	Max Frequency	$s$	$D$	Pause	DTW	$\alpha$	Smallest MCD	Largest MCD	Citation MCD	Domain
[1]	8kHz	32ms/256	<16ms/128*	32ms/256*	?	20	0Hz*	4kHz*	1	16	no	no	1	~0.8	~1.05	original	generic
[2]	?	?	?	?	?	80*	80Hz*	12kHz*	1	13	yes*	no	1	0.294	0.518	[3]	TTS
[3]	24kHz*	?	?	?	?	80	80Hz	12kHz	1	13	yes*	no	1	6.99	12.37	[1]	TTS
[4]	16kHz*	25ms	5ms	?	?	?	0Hz*	8kHz*	1	24	yes*	no	$\frac{10}{\ln(10)}$	~2.5dB	~12.5dB	[5]	TTS
[5]	?	30ms	10ms	?	Hamming	?	?	?	1	10	yes*	yes	1	3.415	4.066	[1]	TTS
[6]	?	>10ms*	5ms	>10ms*	Gaussian*	?	?	8kHz*	1	24	no	no	$\frac{10 \sqrt{2}}{\ln(10)}$	~4.75	~6	[7]	VC
[7]	16kHz	40ms*	5ms	64ms/1024	Gaussian	?	?	12kHz	1	40	yes	no	$\frac{10 \sqrt{2}}{\ln(10)}$	2.32dB	3.53dB	none	TTS
[8]	24kHz	50ms/1200	12.5ms/300	2048/~85.3ms	Hann	80	80Hz	12kHz	1	13	yes*	yes	1	4.83	5.68	[1]	TTS
[9]	16kHz	64ms/1024	16ms/256	128ms/2048	Hann	80	125Hz	7.6kHz	1*	16*	yes*	yes	1*	10.62	14.38	[1]	TTS
[10]	16kHz	?	?	?	?	?	?	?	1	16*	yes*	yes	1*	8.67	19.41	none	TTS
[11]	16kHz*	64ms* (at 16kHz)/1024	16ms* (at 16kHz)/256	64ms*/1024*	Hann*	80	0Hz	8kHz	1	60	yes*	no	$\frac{10 \sqrt{2}}{\ln(10)}$	5.32dB	6.78dB	[12]	TTS

*Parameters are not explicitly stated, but were estimated from the information in the literature

Literature:

• [1] Kubichek, R. (1993). Mel-cepstral distance measure for objective speech quality assessment. Proceedings of IEEE Pacific Rim Conference on Communications Computers and Signal Processing, 1, 125–128. https://doi.org/10.1109/PACRIM.1993.407206
• [2] Lee, Y., & Kim, T. (2019). Robust and Fine-grained Prosody Control of End-to-end Speech Synthesis. ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 5911–5915. https://doi.org/10.1109/ICASSP.2019.8683501
• [3] Ref-Tacotron -> Skerry-Ryan, R. J., Battenberg, E., Xiao, Y., Wang, Y., Stanton, D., Shor, J., Weiss, R., Clark, R., & Saurous, R. A. (2018). Towards End-to-End Prosody Transfer for Expressive Speech Synthesis with Tacotron. Proceedings of the 35th International Conference on Machine Learning, 4693–4702. https://proceedings.mlr.press/v80/skerry-ryan18a.html
• [4] Nature/ansp19-503 Anumanchipalli, G. K., Chartier, J., & Chang, E. F. (2019). Speech synthesis from neural decoding of spoken sentences. Nature, 568(7753), Article 7753. https://doi.org/10.1038/s41586-019-1119-1
• [5] Shah, N. J., Vachhani, B. B., Sailor, H. B., & Patil, H. A. (2014). Effectiveness of PLP-based phonetic segmentation for speech synthesis. 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 270–274. https://doi.org/10.1109/ICASSP.2014.6853600
• [6] Kominek, J., Schultz, T., & Black, A. W. (2008). Synthesizer voice quality of new languages calibrated with mean mel cepstral distortion. SLTU, 63–68. http://www.cs.cmu.edu/~./awb/papers/sltu2008/kominek_black.sltu_2008.pdf
• [7] Mashimo, M., Toda, T., Shikano, K., & Campbell, N. (2001). Evaluation of cross-language voice conversion based on GMM and straight. 7th European Conference on Speech Communication and Technology (Eurospeech 2001), 361–364. https://doi.org/10.21437/Eurospeech.2001-111
• [8] Capacitron -> Battenberg, E., Mariooryad, S., Stanton, D., Skerry-Ryan, R. J., Shannon, M., Kao, D., & Bagby, T. (2019). Effective Use of Variational Embedding Capacity in Expressive End-to-End Speech Synthesis (No. arXiv:1906.03402). arXiv. http://arxiv.org/abs/1906.03402
• [9] Attentron -> Choi, S., Han, S., Kim, D., & Ha, S. (2020). Attentron: Few-Shot Text-to-Speech Utilizing Attention-Based Variable-Length Embedding. Interspeech 2020, 2007–2011. https://doi.org/10.21437/Interspeech.2020-2096
• [10] VoiceLoop -> Taigman, Y., Wolf, L., Polyak, A., & Nachmani, E. (2018). VoiceLoop: Voice Fitting and Synthesis via a Phonological Loop. 6th International Conference on Learning Representations (ICLR 2018), 2, 1374–1387. https://openreview.net/forum?id=SkFAWax0-
• [11] MIST-Tacotron -> Moon, S., Kim, S., & Choi, Y.-H. (2022). MIST-Tacotron: End-to-End Emotional Speech Synthesis Using Mel-Spectrogram Image Style Transfer. IEEE Access, 10, 25455–25463. IEEE Access. https://doi.org/10.1109/ACCESS.2022.3156093
• [12] Kim, J., Choi, H., Park, J., Hahn, M., Kim, S., & Kim, J.-J. (2018). Korean Singing Voice Synthesis Based on an LSTM Recurrent Neural Network. Interspeech 2018, 1551–1555. https://doi.org/10.21437/Interspeech.2018-1575

Default parameters

Based on the values in the literature the default parameters were set:

• Hop Length (hop_len): 8ms
  • Note: should be 1/2 or 1/4 of the window size
• Window Size (win_len): 32ms
• FFT Size (n_fft): 32ms
  • Should match the window size.
  • For faster computation, the sample equivalent should be a power of 2.
• Window Function (window): Hanning
• Sampling Rate (sample_rate): is taken from the audio file
• Min Frequency (fmin): 0Hz
• Max Frequency (fmax): sampling rate / 2
  • Cannot exceed half the sampling rate.
• Num. Mel-Bands ($N$): 20
  • Increasing the number will increase the resulting MCD values.
• $s$: 1
• $D$: 16
• $\alpha$: 1 (alternate values can be applied by multiplying the MCD with a custom factor)
• Aligning: DTW
• Align Target (align_target): MFCC
• Remove Silence: No
  • Silence should be removed from Mel spectrograms before computing the MCD, with dataset-specific thresholds.

License

MIT License

Test coverage

Name                                       Stmts   Miss  Cover   Missing
------------------------------------------------------------------------
src/mel_cepstral_distance/__init__.py          2      0   100%
src/mel_cepstral_distance/alignment.py        84      0   100%
src/mel_cepstral_distance/api.py             371      0   100%
src/mel_cepstral_distance/computation.py      69      0   100%
src/mel_cepstral_distance/helper.py           38      0   100%
src/mel_cepstral_distance/silence.py          55      0   100%
------------------------------------------------------------------------
TOTAL                                        619      0   100%

Citation

If you want to cite this repo, you can use the BibTeX-entry generated by GitHub (see About => Cite this repository).

Taubert, S., & Sternkopf, J. (2025). mel-cepstral-distance (Version 0.0.4) [Computer software]. https://doi.org/10.5281/zenodo.15213012

Acknowledgments

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 416228727 – CRC 1410