## Friday, July 17, 2015

### [ipjbducl] Octave and perfect fifth

Let "octave-tuning" be the very common 12 tone equal temperament tuning with a frequency multiplier of o=2^(1/12) between semitones.  An octave will exactly double the frequency.  Let "fifth-tuning" be an equal temperament tuning with a frequency multiplier of f=1.5^(1/7) between semitones.  The fifth will be "perfect", with frequency exactly 1.5 times the base note.

(Octave-tuning is often called 12-TET.  Fifth-tuning is often called 7-TET.)

Fifth-tuning causes intervals to be sharp by 0.27928583791248815 cents per semitone, 3.3514300549498590 cents per octave, in octave-tuned cents, compared to octave-tuning.  12 fifths equals 7 octaves, or 84 semitones, an interval which will be 23.460010384649013 cents sharp.  (A piano covers a range of 87 semitones, though the highest and lowest notes are tuned differently because of psychoacoustic effects.)

Inversely, octave-tuning has its fifth 1.9495560314892994 cents flatter than a perfect fifth, in fifth-tuned cents.

If we split the difference per semitone, then the frequency multiplier between successive semitones is exp(2/(1/log(o)+1/log(f))).  (Appropriately, harmonic mean in logarithmic space.)  This yields a semitone that is d cents sharp in octave-tuning and d cents flat in fifth-tuning, where d=0.13944818943408673.

Alternatively, we can split the difference unequally in the following way: a frequency multiplier of exp(19/(12/log(o) + 7/log(f))) yields an octave that is d cents sharp in octave-tuning and a fifth that is d cents flat in fifth-tuning, where d=1.2325632573261429.

Let "twelfth-tuning" be an equal temperament tuning dividing the just interval of an octave plus a fifth (a twelfth) into 19, so a frequency multiplier of 3^(1/19) per semitone.  Relative to octave-tuning, a semitone is 0.10289478238881145 cents sharp and the octave is 1.2347373886657375 cents sharp.  Relative to fifth-tuning, the fifth is 1.2312985462037681 cents flat.

Incidentally, a twelfth in the "split the difference unequally" tuning is extremely close to, just 0.0034388362379609146 cents flat of, a perfect twelfth in twelfth tuning.