Frank Nordberg wrote:
>I posted this table at abcusers a year or so ago:
>
> Pythagorean Equal
>c 260.74 261.6
>cb 278.43 277.2
>d# 274.69 277.2
>d 293.33 293.7
>db 313.24 311.1
>e# 309.03 311.1
>e 330 329.6
>f 347.65 349.2
>fb 371.25 370
>g# 366.25 370
>g 391.11 392
>gb 417.66 415.3
>a# 412.03 415.3
>a 440 440
>ab 469.86 466.2
>b# 463.54 466.2
>b 495 493.9
>
>Just multiply the numbers with 2 to get the next octave.
>Lots of people have already posted descriptions of the formula for equal
>temperement, so I suppose I don't have to do that.
>For the pythagorean temperement you multiply with 2187/2048 to go up an
>augmented prime and 256/243 for a minor second.
That's not what I understand as a Pythagorean scale. Pythagoras considered
only the octave (x 2) and the fifth (x 3/2) to be perfect intervals,
so a pythagorean scale is constructed using only these ratios. So, if
you start with A = 440, the fifth E = 440 * 3/2 * 1/2 = 330.0. You get
all the other notes by continuing around the circle of fifths, multiplying
by 3/2 or 3/4 as necessary to stay within the octave. The problem is
that twelve fifths doesn't quite add up to seven octaves, so when you
get back to A it's seriously out of tune. The difference is called
the comma of Pythagoras :
C 264.298096
C# 278.4375
D 297.335358
Eb 313.242188
E 330
F 352.397461
F# 371.25
G 396.447144
Ab 417.65625
A' 446.003036
A 440
Bb 469.863281
B 495
>These two temperements have two things in common, they are simple to
>define mathematically and they are pretty useless musically.
It is indeed a pretty useless scale for any music which wanders very
far away round the circle of fifths.
We wouldn't get very far without the equal-temperament scale though
would we? The equally-tempered scale distributes the comma of Pythagoras
around all twelve intervals so all intervals are very slightly wrong.
It's the only way you can tune an instrument with fixed tunings and
have it sound reasonably OK in all keys.
Phil Taylor
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