I would like to get some response on the following series of articles on tuning equal temperament use pure 5ths. AURAL EQUAL TEMPERAMENT BY PURE 5THS Jim Coleman, Sr. Impossible you say? That's what I always thought. The early organ and harpsichord tuners soon learned you can't tune a circle of pure fifths to complete an octave so they began various schemes to go as far as possible with the pure 5ths. This was the basis of Pythagorean tuning. Of course it would have been nice to have pure Maj 3rds 4ths 6ths and minor 3rds and 6ths as well. But alas, it was not to be. Actually, the 3rds and 6ths were quite busy. And watch out for those wolf 4ths or 5ths. Next came the Meantone tunings where it was possible to have as many as 8 pure 3rds and a number of pure 4ths and 5ths. But, alack and alas, there were those wolves. Then came the various Modified Meantone schemes. Now they were really cooking. It was possible to play in all keys with just a little careful attention to the writing and voicing of parts. Mini wolves were created to lessen the bite of larger wolves. This gradually developed a popularity for what we now call Well Temperament. Finally so many little wolves were created that it was decided to just go all the way. After all, so many other good intervals had already been compromised, so equal temperament seemed to be the next logical step. Equal temperament was found to be more of a struggle than at first expected. Not every musician had the patience to learn the techniques and skills to be able to tune equal temperament. This helped to more fully develop the professional piano technician. Here were people who were willing to devote their lives to practice and learn the art of equal temperament. Finally after about 75 years the art was well developed and standards of acceptable tempering were established. Yet, there remains even to this date some disagreement as to what constitutes an even balance of octaves. Since the advent of higher tensioned scaling in comparison to the lower tensioned scaling of the early Fortepianos and Harpsichords, it was discovered that octaves could not be tuned purely either. This seemed not to be a problem with the earlier instruments. But now, in order to make a piano have good balance in the octaves, some tempering is necessary. The main question today is: How wide can the octaves be and still not be an irritant? Also, is there an ideal trade off between the width of the 4ths or 5ths and the width of the octaves? Originally, the standard approach to setting equal temperament was to first establish the octave and then by using a circle of 4ths and 5ths, one could divide up the octave evenly by contracting the 5ths each by 1/12 of the 24 cent comma. The early attempts involved tuning two 5ths upward and then dropping down one pure octave and repeating the procedure until the circle of 5ths was completed. Later it was decided that steps could be saved if one tuned up one 5th, then down a 4th, up a 5th etc. until the circle of 4ths and 5ths was completed. There have been many schemes used to balance out a one octave scale. One special scheme was to divide the octave into 3 contiguous Maj 3rds then tuning down a 5th from the second one and then building two more Major 3rds, then dropping down a 5th etc. The earliest of this system was popularized by Faust and later made more popular by John Travis. Eventually a scientific system was developed where one could tune the circle by 4ths and 5ths and check with 3rds and 6ths. This was made very popular by Wm. Braid White. Later a system called "Both ways from the middle" was developed by Bill Stonaker where early on one had some checks and balances for the 3rds and 4ths. Several variations of this then have been used and made popular by such people as George Defebaugh, Don Morton, Bill Stegeman etc. Back in the early 70's, while working for the Baldwin Piano Company I tuned many Acrosonics. I discovered it was quite easy to set a temperament from A3 to A4 by using 3 contiguous 3rds as a basis from which to tune 4ths and 5ths to complete the scale and avoid the break area of the Tenor section. Later Dr Sanderson and Rick Baldassin extended that idea to covering two octaves and utilizing some bullet proof procedures to assure excellent results regardless of the difficult scaling challenges of some pianos. In all of the later developments, it was discovered that the 4ths would beat slightly faster than the theoretical rates. The 5ths would beat a little slower than the textbook values. As these later systems became more popular, the desire to stretch the octaves a little more to accommodate the effects of inharmonicity became more popular. At the present time it is common to find the octaves of the better tuners stretched by from 1/4 to 1/2 beats per second at the second coincident partials of an octave. ie when one listens to an octave at the partial one octave above the upper note of the octave there will be up to a 1/2 beat per second widening. Another way of checking would be to compare F3-A3 and F3-A4 where the beat of the latter would be faster than the former by up to a half beat per sec. The result of this causes the 5ths to beat even slower. But of course this is at the expense of the 4ths beating a little faster. The main purpose of this first article of the new series is to show that there are trade-offs when one attempts to favor one interval above others in stretching equal temperament. Many have held tenaciously to the idea that octaves should be pure. Can we talk about this? Is there a law of nature that says that octaves must be pure? What about double octaves, must they also be pure? What about triple octaves, or quadruple octaves? Must they also be pure? Who is to say? Contrary to what my good friend Virgil Smith says about always tuning pure octaves, we have been able to show that when comparing coincident partials, it is impossible to have all pure single, double and triple octaves. If one starts in the center of the piano and tunes a pure 2-1 type octave and then another contiguous pure 2-1 type octave, and then another contiguous 2-1 type octave, the resultant triple octave will be terribly flat on any typical modern piano. As a result of this the modern tendency is to stretch octaves judiciously in order to minimize this discrepancy. In spite of Virgil's explanation of his method of tuning, his results are very good and he actually is able to tune to greatly minimize the discrepancy created by inharmonicity. After studying Virgil's tuning style, I believe I have a clue as to what he is really doing. He is not the only one. Brent Fischer of Arizona State University is an excellent tuner who has also been able to realize this ideal type of stretch. Recent study of the aural tunings of Tom Kinney of La Crosse, WI have convinced me that for a long time we have ignored a very helpful interval in controlling our stretch tunings. This interval as you may have guessed is the pure 5th. As we have noted above, each effort to stretch octaves a little more has resulted in the 5ths being slower and the 4ths becoming faster than the theoretical values we all started with. Brent Fischer is the first one to bring to my attention the idea that the octaves should be stretched to the very edge of the limit. That is, as far as you can stretch without the octave beginning to sound bad. Perhaps others have tried to suggest this to me and at that time it just went over my head. The next logical question becomes: "What is the limit to the stretch of an octave?" Can single octaves be stretched to the point where triple octaves will sound good. I believe Virgil Smith does this. I also believe that this is why his tunings sound so good. Can octaves be stretched as much as 3 beats in 5 seconds? We've been listening to 5ths which are compressed that much for a long time and it has been tolerable. Worse yet, we have been listening to 4ths which theoretically beat 4 beats in 5 seconds. Now when you realize that we have been listening to 4ths that are more than 1 beat per second due to the recent trend to stretch octaves a little, just how far can we go in that direction? If we stretch our octaves just a little more, can we still tolerate even faster beating 4ths? I believe the answer to this is YES. After all, since the advent of equal temperament we have been condition- ed to listening to 3rds which beat more than 7 beats per second. What is the practical limit of beats which we can tolerate in the sound of the single octaves and the 4ths? Could they be balanced out equally? Should they be balanced out equally? I believe the answer to that question can be found in the 5ths. If one sets a temperament using pure 5ths, the 4ths will be faster and the octaves will be wider. In the early experiments thus far the results of this type of tuning has by the strangest coincidence provided just the right kind of octaves to produce tolerable triple octaves and the quadruple octaves are not bad. The overall sound of the piano is quite exciting. Next we will show some very simple practical ways of accomplishing this type of pure 5th tuning and at the same time keeping all the other intervals equally spaced (ie equal temperament). This next installment should appear tomorrow showing step by step tuning procedures. Jim Coleman, Sr.
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