Frank Weston wrote Of course you can't have pure 5ths and pure 4ths in the same temperament! There are two immutable laws of the Universe which should concern persons attempting to tune a keyboard instrument. 1. You never get something for nothing. 2. n*log2(3/2) will never be equal to n for any value of n except zero. According to math, pure fourths and fifths cannot exist in the same temperement of a 2/1 octave. However it should be noted that one set of fourths and fifths in an octave beat at the same rate, while the other set of fifths and fourths, the fourths beat twice as fast as the fifths. >From the beat tables in Reblitz first ed. illus.4 - 13. For example F to Bb beats at ..790 and Bb to F fourth beats the same .790. However F to C beats at .590, while C to F beats 1.180, or exactly twice as fast. Looking at the minor third major sixth relation in the same octave, we find the F to Ab bewats at 6.390, while the Ab to F sixth gives the same 6.930 beats. OK so what is the significnace of these observations. Well right now in exploring Jim Coleman's widened temperament octave, (or WTO ? : )) it would serve us to know which fourth is at the same to which fifth and which one beats twice as fast. But that is way ahead of the scheme. First we need to see how it is based on the fundamentals, of both music and math. Frank gives the formula of n*log2(3/2), which I think means fifths afftecting n somehow. 3/2 is the ratio of the perfect fifth. But the explaination, "n*log2(3/2) will never be equal to n for any value of n except zero." I need some help with, as I was always slow in math. We know that according to the piano keyboard and the music staff, a series of 12 fifths equals a series of 7 octaves. The octave has a ratio of 2 to 1 and the fifth a ratio of 3 to 2. So the frequency of seven octaves is figured out as 2^7 or two the seventh power. If the beginning note has a frequency of 27.5 it's octave would be twice that 55. LIke wise the frequency of the fifth above 27.5 would be 27.5 times 3/2 or or 41.203. Now if you do the math of computing the frequency of seven octaves, multiply 27.5 times 2 to the 7th power (2^7 as the computer understands)(or 27.5*2^7 in a spread sheet) = 3520.000 We said 12 fifths equals 7 octaves, so mathematically in a spread sheet it would look like 27.5*(3/2)^12 = 3568.024. That is some what more that seven octaves, as 27.5*2^7 equals 3520.000. This difference is called a comma and it even has a name. (Pythagerous. )? So if 12 fifths are to fit into 7 octaves some compromises have to be made. The theorists then said shorten or narrow the fifth to somehting less than 3/2. Some more math people came along later on and said, the problem of fitting 12 notes in an octave in a maner musically acceptable, could be solved by the formula using the 12 th root of 2. That should look like 2^(1/12). The twelve means that 12 notes go into an interval that will be double at the 13th note. Here for now it is time to quit, check the math, and check the formulas in a spread sheet. What we are getting at is exploring the possibilities of octaves that are not the 2/1 ratio. As Jim has indicated in another post, it looks like the incredibly small widening of the octave from 2 to 2.004 will yield perfect 5ths for at least 7 octaves. Having tried a widened temperament octave, and hearing the resulting pure fifths, we might be on a break through as great as the discovery of ET itself. I think it all hinges on the fourth which Jim and the math predicts will be faster, and I for one hearing pure fifths would also like to hear pure fourths at the same time. It is impossible on paper, and even more unlikely for pure fourths if the octave is widened, as for pure fourths one needs a narrow octave. Like the old commercial said, "They said it couldn't be done, they said nobody could do it....." Richard Walkin the Dog. ---------- From: Frank Weston <waco@ari.net> To: pianotech@ptg.org Subject: Re: ramblin' (tunign by pure 5ths) Date: Monday, June 23, 1997 7:30 AM Jim wrote: > > As you spread the octave in order to fit in the pure 5ths, the 4ths > actually get faster. Now I'm learning that they will always be faster than > the stretch octaves. Sometimes more than twice the speed of the octave. > > I hope this clue helps those who are actually trying to set a scale with > pure 5ths. Doing it with the SAT was so easy, that almost all the pianos > I tune now are tuned this way. But doing it aurally is quite another > matter. I am working out a foolproof method now for doing it aurally. > It should be on the list in just the next few days. I would like to > actually do it a few more times before I publish. > > Jim Coleman, Sr. Of course you can't have pure 5ths and pure 4ths in the same temperament! There are two immutable laws of the Universe which should concern persons attempting to tune a keyboard instrument. 1. You never get something for nothing. 2. n*log2(3/2) will never be equal to n for any value of n except zero. The history of temperament is the history of attempts to reconcile these two laws. The tradeoff has always been more pure intervals vs. more freedom to modulate. When a new system of tuning is proposed, the question must be asked, "What does it give us?" Does the new system provide intervals which are more pure? Does the new system result in more freedom to modulate? Does the new system have some historical significance to the music that will be played? If the answer to these questions is no or mostly no, then what is the benefit of the new system? Well what "new" system would give mostly yes? How can we know until it is tried and either condemened or commended? We must bear in mind that the purpose of a tuning system or temperament is ultimately to make music, NOT to be convenient to the person doing the tuning or to the device used to do the tuning. Which brings to mind that keyboard tuning might also be regarded as an incumberance. How much time did the old masters spend on tuning or tuning schemes. One might think for a composition titled "Well Tempered Clavier" a tuning scheme or instructions would be with. Apparently Bach did not care to explain what he meant by "Well Tempered" The music was composed from the keyboard and most often conducted from the keyboard, and when the darned thing had to be tuned or retuned, it was an inconvience taken for granted. What would it be worth to see and hear Bach tuning his harpsichord for just one of the WTC? I wonder why he didn't think it was that important for us to know, or tune in a perscribed fashion? Its possible tuning execpt for the rudimentary aspects was not that important to him. So, the question: What does tuning by pure 5ths give us except for convenience? More musical intervals? No. Perfect fifths are traded for less than perfect octaves, thirds and sixths. More freedom to modulate? No. Modern equal temperament is the ultimate in this regard. Historical significance to music of a particular era? Possibly. According to Jorgensen, some English harpsicord tuners in the early 18th century claimed to tune 5ths perfect with good results. Now ask yourself, what English harpsicord music from the 1720’s would you like to reproduce? I heard that Bach did some transcriptions of Vivaldi to harpsichord, are these extant? For me, the ultimate temperament will always be a well temperament, and Vallotti is my favorite. Why? Because the musical intervals are relatively more pure (or at least more interesting) than equal or attempts at equal, and they are well suited to the characteristics of the piano. Because there is key-coloration. And finally because I am a big fan of Baroque and Classical Music. Actually a good well-temperament is usually suitable for most music except for modern art-music and competently played jazz. My two cents worth. Frank Weston ---------- It would be interesting to see these "Well Temepaments" in beat tables as Reblitz has done for ET. Richard Moodi
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