Phil Taylor wrote:
>
> 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
>
> To subscribe/unsubscribe, point your browser to:
>http://www.tullochgorm.com/lists.html
There have been various interpretations on what the Pythagorian scale is
(search web with Google, even look at H. Partch, which I don't take very
seriously. There's also a website with lots of other scales). Can anyone
tell me where to find out what Pythagoras said in a reliable
translation?
Bruce Olson
Old English, Irish and, Scots: popular songs, tunes, broadside
ballads at my website (no advs-spam, etc)- www.erols.com/olsonw
or click below <A href="http://www.erols.com/olsonw"> Click </a>
To subscribe/unsubscribe, point your browser to: http://www.tullochgorm.com/lists.html
