If you want to play with some different historical solutions to the tempering problem, take a look at the spreadsheet at the bottom of the "Downloads" page on the LSA site.
http://www.cs.dartmouth.edu/~lsa/download/index.html Regards, Daniel Heiman On Thu, 22 Jul 2004 15:42:58 +0200 bill <[EMAIL PROTECTED]> writes: > dear stewart - > > staggering. i didn't mean for you (or jon) to compose anything so > comprehensive and thorough. while i'm sure you meant this for > general > edification i'd like to say thank you all very much from me. > > i wouldn't be even remotely interested in going to one of these but > i > understand there are oud camps in the us for people who want to > immerse > themselves in arabic music. one of the introductory exercises is > spent > inching your fingers up and down the neck while listening out for > the > infinite subtleties which can be produced on a stringed instrument > without frets. i tune my instruments with a digital auto tuner, > designed for a guitar and while the little light may indicate that > i'm > in perfect tune, my ear often tells me otherwise. i've often > wondered > if "being in perfect tune" was a purely subjective consideration > dependent on such things as what song was playing when the radio > alarm > went off; what the weather is like outside and barometric pressure > and > no q-tips left in the bathroom. > > i understand what you say and it all sounds to be a bit of a > mystery. > maybe the hillbilly prelude to some song or other in the key of > "...round 'bout 'c'..." may be the best that any of us can hope > for. > > thanks again stewart. i continue to be amazed at all the things i > don't know and very, very grateful to those people who do. > > sincerely - bill > > On Gioved�, lug 22, 2004, at 15:03 Europe/Rome, Stewart McCoy > wrote: > > > Dear Jon and Bill, > > > > I am no mathematician, so I cannot give a detailed account of > sixth > > comma meantone. You will find the relevant information in books > like > > Mark Lindley, _Lutes, Viols & Temperaments_ (Cambridge: Cambridge > > University Press, 1984). I'll try to explain how I understand it, > > without going into all the maths. A few simple sums are > inevitable > > though. > > > > The first thing to realise is that lutes, like all other > > instruments, cannot be tuned perfectly. The sums don't add up. We > > have to compromise, so, when we tune an instrument, we put up > with > > slightly squiffy intervals between notes, hopefully so slight > that > > no-one notices that anything is amiss. > > > > When you start to learn a string instrument, tuning may seem > easy, > > because you tolerate all sorts of bad tuning. As you get better > at > > it, your ear becomes more perceptive, and you find tuning more > > difficult, because you get to be more fussy. Eventually, in > despair, > > you get so good, that you simply cannot get the ****** instrument > in > > tune. Mercifully modern tuning boxes can help one cope with a > task > > which ultimately is theoretically impossible. > > > > Equal temperament > > > > Equal temperament is used nowadays on the piano. The piano tuner > > will tune all the semitones the same distance apart - hence > "equal" > > temperament, but it means that the major thirds (e.g. C+E, E+G#, > > etc.) are much wider than pure. (Pure = perfectly in tune.) The > > human ear can cope with this, but any wider, and you'd say the > notes > > were out of tune. At the same time the piano tuner will tune all > the > > fifths (e.g. C+G, E+B, etc.) very slightly narrower than they > would > > be if they were perfectly in tune. Again, the human ear can cope > > with slightly narrow fifths, because the difference between those > > fifths and pure fifths is hardly noticeable. > > > > The advantage of equal temperament is that you can play in any > key > > you like, and it will all sound OK. Vincenzo Galilei argued in > > favour of equal temperament for the lute. I think equal > temperament > > is essential for the baroque guitar. Mark Lindley presents much > of > > the evidence we have from the past, but he seems to conclude that > > equal temperament was more commonly used with some fretted > > instruments than I think he should. > > > > Meantone tunings > > > > The idea of a meantone temperament is to get the major thirds > > narrower than they are with equal temperament, so that they are > > better in tune, and will sound sweeter. You can do this for some > > major thirds, but it is impossible to do so with all of them. The > > trick is to favour the common thirds (e.g. C+E, F+A, etc.), and > have > > horrible out-of-tune intervals for the thirds you are unlikely to > > need (e.g. F#+A#, B+D#, etc.). > > > > The most extreme meantone tuning is quarter comma meantone, which > > was commonly used for early keyboard instruments. In this > > temperament all the common major thirds are pure. This works well > > for keyboard instruments, which have separate strings for each > note, > > but it is a bit risky on fretted instruments like lutes, which > use > > the same string (fretted) for many different notes. Discrepancies > > may creep in with lutes for all kinds of reasons, and you don't > want > > to risk having a major third less than pure. > > > > Fifth comma and sixth comma meantone are compromises, with the > good > > major thirds tuned somewhere between pure and equal. Sixth comma > > meantone is closer to equal temperament, and seems to work well > for > > most renaissance lute music. Occasionally you might have to move > a > > fret or two, if you want to play in an extreme key, which is what > we > > know some vihuela players did. > > > > The evidence supplied by surviving citterns is very important in > > knowing what temperament may have been used for fretted > instruments > > in the past. According to Peter Forrester, who has examined > > virtually all surviving citterns, the cittern fretting system is > > closest to sixth comma meantone. He has told me that later > > instruments tend to move towards equal, but they never actually > make > > it. Perhaps it is because it is so easy to distort the pitch of > the > > short wire strings, but equal temperament just doesn't work for > > citterns. There has to be some move in a meantone direction for > them > > to sound any good. > > > > People in the past argued about the suitability of different > > temperaments, just as they do now. We all have our preferences. > Mine > > is sixth comma meantone for renaissance lutes and citterns. > > > > The syntonic comma > > > > A comma is a discrepancy arising from the fact that the figures > > don't add up. The syntonic comma is the amount the 1st and 6th > > courses of the lute would be out, if you tuned all the intervals > > between the courses pure. This is how it works: > > > > Simple sums for simple intervals > > > > To get different intervals you multiply with simple numbers. For > > example, to get a note an octave higher, you multiply by 2. If > > middle C = 256, C an octave higher = 512. > > > > Let's think of 6-course renaissance lutes and the five intervals > > between the six courses. The 6th course is tuned to G. Let's say > > (for the sake of argument) that it vibrates at 81 cycles per > second. > > To get a note a fourth higher (the 5th course C), you multiply by > > 4/3: > > > > If G = 81, C = (81x4/3) = 108 > > > > You do the same to get the 4th course: > > > > If C = 108, F = (108x4/3) = 144 > > > > To get a major third higher for the 3rd course you multiply by > 5/4: > > > > If F = 144, A = (144x5/4) = 180 > > > > The 2nd course is a fourth higher, so you multiply by 4/3: > > > > If A = 180, D = 240 > > > > Finally, for the 1st course you multiply again by 4/3: > > > > If D = 240, G = 320 > > > > We should now have ended up two octaves higher than the 6th > course. > > We started with G = 81. To get an octave higher we multiply by 2, > so > > to get two octaves higher we have to multiply by 4: > > > > If G (6th course) = 81, G (1st course) = (81x4) = 324. > > > > Unfortunately that's not the same as the 320 we got earlier, by > > calculating from one course to the next across the neck of the > lute. > > The difference between 320 (four intervals of a fourth + one > major > > third) and 324 (two octaves) is called the syntonic comma. > Somehow > > or other we have to stretch our intervals, so that 320 can become > > 324. We can do this many ways: > > > > 1) Pythagorean temperament > > > > Keep the four intervals of a fourth (G+C, C+F, A+D, D+G) the same > > (pure), and widen the major third between F and A. This means > that > > all fourths (and consequently fifths) are pure, but the major > third > > is pretty foul. This sort of temperament was used for mediaeval > > music, when there were lots of fourths and fifths in the harmony, > > and when they understandably avoided bad-sounding thirds and > sixths. > > > > 2) Quarter comma meantone > > > > Keep the major third between F and A the same (pure), and widen > the > > other intervals (the four fourths). This is quarter comma > meantone, > > because you have divided the syntonic comma into four, and shared > it > > out amongst the four fourths (between G+C, C+F, A+D, D+G). > > > > 3) Fifth comma meantone > > > > Divide the syntonic comma into five, and spread the discrepancy > > amongst all the intervals between the strings. The four fourths > will > > be slightly wider than pure, and the major third between F and A > > will also be a little wider than pure. > > > > 4) Sixth comma meantone > > > > Again, share the discrepancy out between all the strings, but > this > > time divide the comma into six portions. Each of the four fourths > > will be very slightly closer to pure than with fifth comma > meantone, > > and the third between F and A will be very slightly wider. In > sixth > > comma meantone the major third is quite wide, but not as wide as > it > > would be with equal temperament. > > > > I think that's how it all works. Any comments or corrections will > be > > welcome. > > > > All the best, > > > > Stewart. > > > > > > ----- Original Message ----- > > From: "Jon Murphy" <[EMAIL PROTECTED]> > > To: "Stewart McCoy" <[EMAIL PROTECTED]>; "bill" > > <[EMAIL PROTECTED]> > > Cc: "Lute Net" <[EMAIL PROTECTED]> > > Sent: Thursday, July 22, 2004 8:37 AM > > Subject: Re: Sorry, help me....what to buy???? > > > > > >> bill, > >> > >> The cittern was originally designed for amateurs (according to > my > > books), > >> the pros preferred the lute. And I heard one played with a harp > > this weekend > >> (a commercial cittern, it had to be, as the pegs were guitar > style > >> machines). But the cittern wasn't intended to be a lute. > >> > >> As to the 6th comma meantone, Stewart will explain that. But > there > > are many > >> tunings for our western scale that are all compromises. If you > > want it I'll > >> scan a pictorial of the various compromises, and their > > relationship to the > >> pure tones, and send it. I haven't the vaguest idea what the > "6th > > comma" is, > >> but I do know the Pythagorean comma. Pythagoras made a board > with > > a string > >> (perhaps several, I don't know how good his pitch memory was - > > never met the > >> man). > >> > >> The natural overtone scale has a few fractions in it. They > confuse > > the > >> issue. In a tempered scale the octaves which are primary should > > come to the > >> same result as the fifths (the half lengths). (And if I'm a bit > > off in > >> saying the details, let the overall principal apply). The half > > should add to > >> the total, but it doesn't. Five octaves and eight fifths don't > > come to the > >> same pitch - and the difference is called the Pythgorean comma. > So > > whatever > >> the 6th comma is, it is a compromise in the scale. There are a > > number of > >> ways to do it, the orchestral piano has fixed pitches, as does > the > > harp. The > >> lute family may not, although once you have set your frets you > > have chosen a > >> temperament, but on the violin or any unfretted instrument that > > can vary. > >> Meantone is one choice (and there are several meantones, > depending > > on > >> whether you want to make the fifths closer to the natural, or > the > > thirds - > >> you can never do both). The standard solution is to divide the > > octave into > >> twelve equal parts by frequency, and this is a compromise. > Twelve > > hundred > >> "cents" to the octave, each half tone worth one hundred "cents". > > Works well > >> for digital tuners, but it is not the only solution. > >> > >> Best, Jon > >> > >> ----- Original Message ----- > >> From: "bill" <[EMAIL PROTECTED]> > >> To: "Stewart McCoy" <[EMAIL PROTECTED]> > >> Cc: "Lute Net" <[EMAIL PROTECTED]> > >> Sent: Thursday, July 22, 2004 2:37 AM > >> Subject: Re: Sorry, help me....what to buy???? > >> > >> > >> > >> On Gioved�, lug 22, 2004, at 00:46 Europe/Rome, Stewart McCoy > > wrote: > >> > >>> the grooves for the frets were already > >>> made, and they aren't at 6th-comma meantone > >> > >> dear stewart - > >> > >> i've tried to imagine what this might mean but haven't a clue. > > sounds > >> intriguing. would you please explain? > >> > >> sincerely - bill > > > > > > > > > > > ________________________________________________________________ The best thing to hit the Internet in years - Juno SpeedBand! 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