commas and cents
My guitar teacher, Aldo Minella, has an absolute ear , and I remember he was suffering when he heard something not in tune, and by this I mean less than 5 commas difference ( books say human ear can't tell the difference below 5 commas, but I also met other musicians who could tell it , and I think I have the same problem). I don't have an absolute ear, but I would say I don't know what the usage in Italian is but so as not to confuse the English speakers - you probably mean cents and not commas. A cent is 1/1200 of an octave (100 cents = 1 equal tempered semione). Most books on musical acoustics quote a threshold of 4 cents for human pitch discrimination. I don't remember what the basis is or whether anyone has tested to see if those who claim they can do better really can. One of the the perceptual clues that things are not in tune is beats between the overtones. If the beats become so slow (because the overtones are so close) that the period of the beat is longer than the note, that clue disappears. Comma is usually used to refer to discrepencies in pitches. The main one you are concerned with is the Pythagorean comma - 12 perfect 5ths and 7 perfect octaves should get you to the same note - but they don't: 12 perfect 5ths get you about 23 cents higher (~1/4 semitone) than the seven octaves. Temperament is the business of where to sweep the 23 cents under the rug. ...Bob PS: If anyone asks you why temperament ?, the shortest answer is 2 to the N th power = 3 to M th power has no non-trivial solutions for integer N and M If nothing else that should leave the questioner in stunned silence while you make your escape. :-) Replies: (remove the ) Ekko Jennings: [EMAIL PROTECTED] Bob Clair: [EMAIL PROTECTED]
commas and cents
PS: If anyone asks you why temperament ?, the shortest answer is 2 to the N th power = 3 to M th power has no non-trivial solutions for integer N and M If nothing else that should leave the questioner in stunned silence while you make your escape. :-) Dear Bob, I know you're joking, but I'd like to make a serious point. Hearing phrases like square roots and numbers to the power something will drive many people away. Their brains will switch off. I think the important thing is to convey the idea that tuning is harder than some people think, because the sums don't add up. It took me a long time to realise that, but I felt a lot happier when I did. The irony is that the better you get at tuning, the worse you think you are getting. This is because beginners are more likely to accept dubious intonation than more experienced players do. Eventually there comes a point where one gets pretty good at recognising whether or not notes are in tune, but one hasn't yet acquired the sophistication needed to cope with tempering an instrument like the lute or guitar. I used to be able to tune the lute, but I just can't do it any more. I use simple sums to show that the figures don't add up: Let's say that middle C = 256 vibrations per second. Multiply by 2, and you get C an octave higher: C = 512. Divide by 2, and you get an octave lower: C ( 1 octave below middle C ) = 128 C ( 2 octaves below middle C ) = 64 C ( 3 octaves below middle C ) = 32 C ( 4 octaves below middle C ) = 16 C ( 5 octaves below middle C ) = 8 C ( 6 octaves below middle C ) = 4 C ( 7 octaves below middle C ) = 2 (You won't actually hear such a low note, but let's stay with it for the sake of easy sums.) To get a pure major 3rd, perfectly in tune, you multiply by 5: If C = 2, E = (2 x 5) = 10 If E = 10, G# = (10 x 5) = 50 If G# = 50, B# = (50 x 5) = 250 But B# needs to be the same note as C, which was 256, not 250. From this we learn that the numbers don't add up. We can also learn that three major 3rds (C-E + E-G# + G#-B#) are going to fall short of an octave, so it's probably a good idea to tune a lute with major 3rds slightly wider than pure to compensate for the discrepancy. Tuning-boxes can be a great help, as long as we keep using our ears, since they are a constant check on the accuracy of our tuning. They will be less help in the long run, if we don't listen, and rely on them to do all the work. Best wishes, Stewart.
Re: commas and cents
BobClair or EkkoJennings wrote: PS: If anyone asks you why temperament ?, the shortest answer is 2 to the N th power = 3 to M th power has no non-trivial solutions for integer N and M If nothing else that should leave the questioner in stunned silence while you make your escape. :-) Even a mathematical idiot will know that 2 is even and 3 is odd... Rainer adS
Re: commas and cents
Bob, and all, A cent is 1/1200 of an octave (100 cents = 1 equal tempered semione). Most books on musical acoustics quote a threshold of 4 cents for human pitch discrimination. I don't remember what the basis is or whether anyone has tested to see if those who claim they can do better really can. I don't know if the books on musical acoustics have used tests for pitch discrimination, but I can say they have been made. My A.B. in psych (of many, many years ago) came from a rat running department. We made mechanical tests of all aspects of the human (and animal) senses. I don't remember the results for pitch discrimination (it is a variable skill among people), but the test was a pair of pure tones in sequence - and the choice of lower, higher or same. That eliminates the question of overtones or beats. It is not a musical discrimination, but a pure physical capability. One could probably find an adept at pitch discrimination who had no sense of sonance or dissonance. That is, in part, a matter of training and/or accustomization. And one could probably find a musician with less physical pitch discrimination, yet a better ear because of his understanding and experience. [BYW, I found out why I wasn't a great athlete. We tested raw reaction times by asking the subject to release a key on the flash of a light. My time to remove my finger from a button was .25 seconds, the same time it took our 225 pound football captain to jump off a keyed platform. He moved his entire body in the time it took me to move one finger, his finger time was .1 sec. ]. Comma is usually used to refer to discrepencies in pitches. The main one you are concerned with is the Pythagorean comma - 12 perfect 5ths and 7 perfect octaves should get you to the same note - but they don't: 12 perfect 5ths get you about 23 cents higher (~1/4 semitone) than the seven octaves. Temperament is the business of where to sweep the 23 cents under the rug. I love that last sentence Bob, it sums it up. Stewart has made a fine and brief description of the actual numbers, and one of value to those who aren't familiar with the concepts of cents and commas. But to take your analogy to the fullest I might say how to spread those last 23 cents worth of dirt that the vacuum cleaner didn't pick up around the rug so your wife won't notice. But your sentence is briefer, and more to the point. Best, Jon
Re: commas and cents
Rainer, PS: If anyone asks you why temperament ?, the shortest answer is 2 to the N th power = 3 to M th power has no non-trivial solutions for integer N and M If nothing else that should leave the questioner in stunned silence while you make your escape. :-) Even a mathematical idiot will know that 2 is even and 3 is odd... I'm sure that Bob won't take the time to answer this, but you appear to have no knowledge of mathematical terminology. 2 to the Nth = 3 to the Mth when both M and N are zero. Any number to the power of zero is, by definition, one. So when Bob says there are no non-trivial solutions he means exactly that. The use of zero as the power factor would be a solution, but a trivial one. In mathematics there are elegant solutions, or proofs - and trivial ones - and a lot in between. There was a lot of press a few years ago about the final solving of Fermat's last theorum. But so far as I'm concerned his proof has not been found, although the theorum has been proven. His marginal notes mentioned an elegant solution, too long for the margin. The proof of today uses math Fermat never conceived of. So either Fermat was wrong, or there is yet an elegant proof awaiting discovery. Best, Jon
