Pythagoras and scales

[Billbrpt@AOL.COM]

---------------------- multipart/alternative attachment
In a message dated 9/3/02 6:23:55 PM Central Daylight Time, 
Richard.Brekne@grieg.uib.no (Richard Brekne) writes:


> For the straight skinny go to
> 
> http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html
> 
> Pythagoras was in the end a mystic.
> 
> 

Thanks for this, Richard.  I picked up on this quote right away: 

 "Of Pythagoras's actual work nothing is known. His school practiced secrecy 
and communalism making it hard to distinguish between the work of Pythagoras 
and that of his followers."

It seems that some people will grasp at *any* straw to try to show that ET 
always was, is now and forever shall be:

"He truly
> despised
> > carnivores and formulated the scale which we all
> use
> > in pianos ( and most Western music ) today."

Skip Becker wrote a series for the PTG Journal on the History of Tuning.  The 
first chapter begins with Pythagoras and what is known about his tuning 
discoveries and practices.  The entire chapter can be accessed through my 
website.

Here are a few pertinent quotes:

"Pythagoras was the first to quantify that dividing a string length in half 
produced an octave, a ratio of 2:1. A 5th was produced by a ratio of 3:2, and 
a 4th by a ratio of 4:3. He thus proved that musical tones were based in 
numbers. Leibnitz may have said it best: 'Music is mathematics for souls who 
don't know they are calculating.'  As tuners, we work everyday with the 
Pythagorean comma, which is commonly expressed mathematically by the ratio 
81:80.  For Pythagoreans this comma was not the troublesome obstacle it is 
for us. Instead, it was good news. The twelfth 5th in the cycle does not 
complete, but rather exceeds the octave. This implies not just completion, 
but a new beginning: renewal and rebirth."

Pythagorean Tuning

"Pythagoras was concerned with only three intervals. The octave (his 
favorite), the 5th, and the 4th. He called the octave the diapason , which 
means through all the chords (strings on the lyre). The intervals 4th and 5th 
get their names from the number of strings they encompass. These intervals 
had to be perfect, or pure. The interval of the 2nd was derived from dropping 
a pure 4th from the perfect 5th. The 6th was achieved by a perfect 5th above 
the 2nd. The 3rd came a 4th down from the 6th, and the final 7th was a 
perfect 5th above the 3rd. It should be noted that this is a tuning system, 
and not a temperament, as not a single note is tempered.

Pythagorean intervals are noted for being very large. In terms of beats per 
second, the major 3rd from C3 (C below middle C) to E3, beats about 8 BPS in 
Pythagorean intonation. That's about 3 BPS faster than the same third in 
equal temperament. Equal temperament cents (100 to a semi-tone) provide an 
excellent gauge for comparison to today's standard. The cents deviation from 
equal temperament for Pythagorean tuning is: C=0, D =+4 , E =+8, F = -2 , G = 
+2, A = +6, B =+10. The 3rd, for example, is over 8 cents larger than Equal 
Temperament's already very large major 3rds (13.7 cents above the just 5:4 
ratio). Compared to the just intonation of his time, Pythagorean 3rds sounded 
like a discordant jangle (22 cents wider than just)."

I'm sorry, but this doesn't sound like "the scale which we all use in pianos
( and most Western music ) today".

Although Pythagoras and his school are known for some of these early 
scientific discoveries, they were also believers in things we know today had 
no foundation in truth at all.  Not only did they not invent nor use ET, they 
thought any kind of tempered interval was destructive to society.  They also 
thought the earth was the center of the universe and that the planets each 
had their own tone (the Music of the Spheres).

Look what happened to the first guy who tried to tune a tempered scale:

"For example, when Timotheus, one of the most celebrated poets of antiquity, 
entered Sparta with his 12 string lyre he was arrested and brought to trial 
before the high council. The arrest warrant is extant:

      "Whereas Timotheus, the Milesean, coming to our city, has deformed the 
ancient Music; and laying aside
                  the use of the 8 string lyre, and introducing a 
multiplicity of notes, endeavors to corrupt our youth by
                  means of these novel and complicated conceits, which he 
calls chromatic; by him employed in the room
                  of our established, orderly, and simple music....It 
therefore seemed good to us, the King and Council,
                  after having cut off the superfluous strings of his lyre, 
to banish said Timotheus out of our dominions,
          that everyone beholding the wholesome severity of this city, may be 
deterred from bringing among us
                  any unbecoming customs."

It is ironic however, to note that the same kind of irrational and emotional 
thinking exists today as did 2500 years ago.

Bill Bremmer RPT
Madison, Wisconsin
 <A HREF="http://www.billbremmer.com/">Click here: -=w w w . b i l l b r e m m e r . c o m =-</A> 
Bill Bremmer RPT
Madison, Wisconsin
 <A HREF="http://www.billbremmer.com/">Click here: -=w w w . b i l l b r e m m e r . c o m =-</A> 

---------------------- multipart/alternative attachment
An HTML attachment was scrubbed...
URL: https://www.moypiano.com/ptg/pianotech.php/attachments/a2/64/0f/f3/attachment.htm

---------------------- multipart/alternative attachment--