Question

The classical guitar has 19 frets or 18 fret intervals. If each fret interval corresponds to a semitone interval (frequency ratio = 1.0595), then each string covers _____ semitones or _____ octaves.

Answer 1

Answer : Guitar will have a 3 octave range with 18 fret range.

The frequency ratio of a half step is

21/12 = 1.059

Human are capable of detecting a change in frequency of 1/10 of a half step, which corresponds to a frequency ratio of

21/120 = 1.0056

To appreciate a 12-tone scale one must have precision that is tangibly smaller than a half step. Humans are well within this bound.

For example, for the notes on an A-string with a frequency of 440 Hertz,

I  =  Index of a note.  I=0 for the tonic and I=12 for the octave
F  =  Frequency of a note on the A-string
   =  440 * 2I/12
f  =  Smallest frequency greater than 440 Hertz for which "f" sounds
      indistinguishable
      from "F"
   ~  443 Hertz
R  =  Characteristic frequency ratio for human sensitivity
   =  f / F
   ~  1.0058

1-R  =  .0058  =  1/173


Note  I      F

 A    0    440     Open A-string
       .1  442.5   Largest frequency that sounds indistinguishable from 440 Hertz
 Bb   1    466     Half step
 B    2    494     Whole step
 C    3    523
 C#   4    554
 D    5    587     Perfect fourth
 Eb   6    622     Tritone
 E    7    659     Perfect fifth
 F    8    698
 F#   9    740
 G   10    784
 G#  11    831
 A   12    880     Octave

If we start from a frequency of 880 Hertz then the frequencies are

Note  I      F

 A    0    880
       .1  885    Largest frequency that sounds indistinguishable from 880 Hertz
 Bb   1    932    Half step
 B    2    988    Whole step
 ...