<HTML>In a message dated 9/3/02 6:23:55 PM Central Daylight Time, Richard.Brekne@grieg.uib.no (Richard Brekne) writes: <BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">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. </BLOCKQUOTE> 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> </HTML>