Now I would like to go on to a problem that I have with how most musical text handle musical scales; they print something like this:
“the pentatonic scale is: FDCAG.”
Someone goes over to the piano and plays a few notes and asks what is the big deal. My point is: what is lacking is information on the intervals between the notes that the text has represented by letters. The text made an error; there was not an explanation that the intervals between the pitches are important.
To help solve this problem there is a musical measure called a “cent”. A cent is a logrithmetic scale based on the ratio of the frequency of two pitches or the length of two “equal” strings. If one has a calculator that can do “log” functions then cent are easy calculated. In this article I want to stress that it is my opinion that if two scales have the same cent values for each note, then they are the same scale no matter neither how they were formed nor what the names are. If two scales have even one note with a different cent vale it is a different scale (there are rounding problems, so I draw the line at equality at requiring the values to be different by at least an integer). The definition of a cent is simply the ratio of the larger frequency (or equal string vibrating length) divided by the smaller, then take the logarithm to the base 2, finally take the results and multiply by 1200.
The cent value of the scale found on the piano, the 12 Tone Equal Temperament Scale:
Tonic or
Keynote Octave
C C# D D# E F F# G G# A A# B C
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 cent value from tonic.
The cent scale is independent of frequency and it repeats for every scale octave. Even though there are a 100 cent to the step size for an equal-temperament semitone or half step, do not confuse it with percent.
Remember these cent values. They are important. Note that none of the following scales will have any of these cent value except for the tonic and octave. Now look at our pentatonic scale example again, it is now:
pentatonic scale:FDCAGF
0 294 498 792 996 1200 cent value from tonic.
294 204 294 204 204 step size, between notes in cent.
The 12 Tone Equal Temperament is a modern scale, it could not have existed before Neiper invented logarithms . To make an Equal Temperament, one needs to know what number, when multiplied by itself 8 or 12 times equals 2.
How here is a comparison of several scales, a musician can hear a difference of a cent. A difference between two pitches of between 20 to 50 cent sounds bad. A musical instrument tuned to one of the Ancient Music Scales being accompanied by a modern piano will sound bad.
Ascending tone: tonic234567octave
Leath Gleas (Just) 0 204 408 498 702 906 1110 1200
Paraguayan Just 0 205 396 500 706 884 1088 1200
Mean0 213.7 386.3 498.1 702.0 884.4 1088.3 1200.0
Well0 192 390 498 696 888 1092 1200
Equal0 200 400 500 700 900 1100 1200
My opinion is that:
1)different musical scales sound different.
2)music sounds best when it is played in the scale that it was composed in.
3)the more tonal the music the bigger the difference.
4)to get from one musical scale to another the musical instrument has to be re-tuned, moving the semitones will not change the scale.
5)use cent rather than letters to symbolize the tones/pitches one is talking about.