https://github.com/stefantaubert/mel-cepstral-distance
A Python library for computing the Mel-Cepstral Distance (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".
https://github.com/stefantaubert/mel-cepstral-distance
cepstral distance distortion divergence dtw dynamic-time-warping language linguistics mcd mel mfcc objective-evaluation spectrogram spectrum speech-quality speech-synthesis text-to-speech tts voice-cloning
Last synced: 11 days ago
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A Python library for computing the Mel-Cepstral Distance (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".
- Host: GitHub
- URL: https://github.com/stefantaubert/mel-cepstral-distance
- Owner: stefantaubert
- License: mit
- Created: 2021-03-03T12:15:33.000Z (about 4 years ago)
- Default Branch: main
- Last Pushed: 2025-04-30T16:53:58.000Z (18 days ago)
- Last Synced: 2025-05-07T20:18:27.900Z (11 days ago)
- Topics: cepstral, distance, distortion, divergence, dtw, dynamic-time-warping, language, linguistics, mcd, mel, mfcc, objective-evaluation, spectrogram, spectrum, speech-quality, speech-synthesis, text-to-speech, tts, voice-cloning
- Language: Python
- Homepage:
- Size: 59.8 MB
- Stars: 53
- Watchers: 3
- Forks: 10
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- Changelog: CHANGELOG.md
- Contributing: CONTRIBUTING.md
- License: LICENSE
- Code of conduct: CODE_OF_CONDUCT.md
- Citation: CITATION.cff
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[](https://doi.org/10.5281/zenodo.15213012)
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*](https://ieeexplore.ieee.org/document/407206).
- 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
```sh
pip install mel-cepstral-distance
```
### Example usage
Compare two audio files with default parameters:
```py
from mel_cepstral_distance import compare_audio_files
mcd, penalty = compare_audio_files(
'examples/GT.wav',
'examples/Tacotron-2.wav',
)
print(f'MCD: {mcd:.2f}, Penalty: {penalty:.4f}')
# MCD: 7.45, Penalty: 0.1087
```
## 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
## Citation
If you want to cite this repo, you can use the BibTeX-entry generated by GitHub (see *About => Cite this repository*).
```txt
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