temperament4fft

A tool to find rational approximations r = 2^{-a} × 3^b × 5^c of each semitone interval 2^{n/12} in 12-tone equal temperament.

When the FFT size is chosen as 2^N × r, having r composed only of small prime factors enables efficient FFT computation (e.g., FFTW). This makes it possible to perform pitch shifting by an arbitrary number of semitones at high speed.

The output of this tool is used in the Euterpe system.

Requirements

• GHC (Glasgow Haskell Compiler) for the Haskell version
• Python 3 for the Python version

Build & Run

Haskell

ghc -o temperament4fft temperament4fft.hs
./temperament4fft

Python

python temperament4fft.py

Output

$ ./temperament4fft
-11 semitone	≒ 135/256 (-8 cents)
-10 semitone	≒ 9/16 (+4 cents)
-9 semitone	≒ 75/128 (-25 cents)	≒ 625/1024 (+45 cents)
-8 semitone	≒ 5/8 (-14 cents)	≒ 81/128 (+8 cents)
-7 semitone	≒ 675/1024 (-22 cents)
-6 semitone	≒ 45/64 (-10 cents)	≒ 729/1024 (+12 cents)
-5 semitone	≒ 3/4 (+2 cents)	≒ 375/512 (-39 cents)
-4 semitone	≒ 25/32 (-27 cents)	≒ 405/512 (-6 cents)
-3 semitone	≒ 27/32 (+6 cents)
-2 semitone	≒ 225/256 (-23 cents)
-1 semitone	≒ 15/16 (-12 cents)	≒ 243/256 (+10 cents)
+0 semitone	≒ 1/1 (+0 cents)	≒ 125/128 (-41 cents)
+1 semitone	≒ 135/128 (-8 cents)
+2 semitone	≒ 9/8 (+4 cents)	≒ 1125/1024 (-37 cents)
+3 semitone	≒ 75/64 (-25 cents)	≒ 625/512 (+45 cents)	≒ 1215/1024 (-4 cents)
+4 semitone	≒ 5/4 (-14 cents)	≒ 81/64 (+8 cents)
+5 semitone	≒ 675/512 (-22 cents)
+6 semitone	≒ 45/32 (-10 cents)	≒ 729/512 (+12 cents)
+7 semitone	≒ 3/2 (+2 cents)	≒ 375/256 (-39 cents)
+8 semitone	≒ 25/16 (-27 cents)	≒ 405/256 (-6 cents)
+9 semitone	≒ 27/16 (+6 cents)
+10 semitone	≒ 225/128 (-23 cents)	≒ 1875/1024 (+47 cents)
+11 semitone	≒ 15/8 (-12 cents)	≒ 243/128 (+10 cents)

For example, to transpose up by +6 semitones, you can set the FFT window size ratio to 45/32 (with -10 cents deviation) or 729/512 (+12 cents). If the base FFT size is 2^N, the actual size becomes 2^N × 45/32 = 2^{N-5} × 45 = 2^{N-5} × $3^{2}$ × 5, which is still a product of small primes and thus efficient for FFT.

Options

Both versions accept the following command-line arguments:

Option	Description	Default
--primes	Comma-separated list of primes to use	2,3,5
--threshold	Maximum deviation in cents	50

Example:

./temperament4fft --primes 2,3,5,7 --threshold 30
python temperament4fft.py --primes 2,3,5,7 --threshold 30

Author

Hideyuki Tachibana (2013)

License

MIT