In a message dated 12/6/99 5:24:32 PM Pacific Standard Time, dporritt@post.cis.smu.edu (David M. Porritt) writes: << When you strive for even progression of major 3rds, you are ignoring the pitch of the fundamental, ...(snip) Since partials get more irrational (non linear) the higher they are, you are tuning the fundamental based on information from flakier partials. If we really wanted true ET, we would measure the fundamental of each note in the temperament octave and space them evenly letting the partials fall where they will...(snip) I don't see how choosing one method (5ths) over the other (3rds) would cause an inside-out well temperament. >> These are very interesting ideas that you write because I really believe that it is mainstream thinking. But I have a different point of view for each of them. Just last week, I was tuning a Kimball console (for which quite incidentally, I drove the too- loose-to-hold-and-way-too-high tuning pins in to their proper depth about 7 years ago, got immediate positive results and they are still fine today-no CA was used, just a small sledge hammer and a punch) and forgot to set my SAT on the page number. I was tuning each note to the fundamental. I didn't realize my mistake until I had tuned almost the entire middle section center strings (with the strip mute in). Before starting over, I decided to see what kind of temperament it made. I'm afraid to say, somehow, that it was blatantly Reverse Well. I think I know why the buttered side of the toast always ends up on the floor but I can't explain why so many earnest attempts at ET end up in Reverse Well. But they surely do. This is why it is so important to use the higher partials when tuning in the Midrange. This is where the harmony that there is in music is found. This is where the most truly audible and discernible textures created by the Rapidly Beating Intervals (RBI) lie. I am not at all convinced that a true ET with both smooth 3rds and regular 4ths & 5ths is all that hard to do for a really good aural tuner on a piano like the Acrosonic. The scaling doesn't work quite right with programs that can only produce a smooth curve. Adjustments and manipulations of the calculation must be made for things to come out right. But a skilled aural tuner compares both Slowly Beating Intervals (SBI) and RBI's as the temperament construction progresses. The inharmonicity puzzle is solved piece by piece quite naturally. The idea of having "True ET" as some kind of ideal state of total and universal perfection just is not supported by common musical practice. There really is no music that requires the fundamentals of each note of the piano to match theoretical ET frequencies. Richard Brekne's response was right on the money this time. John Formsma writes: <<OK, this is all very interesting. Regardless of which temperament gets used, the octave stretching should be the same, right?>> No, the choice of octave width and temperament are separate decisions. Although it seems more logical to stretch the temperament and midrange octaves more on a high inharmonicity piano and less on a low inharmonicity one, it has become apparent lately that you really can do any kind of octave on any kind of scale. Fighting the natural, inharmonicity blending and absorbing stretch may be more difficult but the goal you have in mind may make it worth the trouble. A narrow octave will always give you slower, sweeter 3rds & 6ths (the RBI's) but will cause your 5ths to be more audibly tempered but your 4ths to be a little cleaner. A wide octave will give you very clean sounding 5ths, you can really get to the point where they are pure, but your 4ths will become very busy and your RBI's will get a steely fast sound to them in every key. It is very easy to make the Reverse Well error when attempting to make the 5ths sound pure or nearly so. <<So, by how little is the octave stretched when tuning the Acrosonic? I'm not using my SAT anymore, so how about some clues for aural tuners? Are the 3rd, 10ths, 17ths all the same...slower, or what? What about octave-fifths, and double octaves? Equal beating? Narrower octave-fifths than normal? Do you do 2:1 octaves up from the temperament octave? This concept is all new to me, so I need some specifics to have any idea of what you guys are doing.>> Remember, you can do the narrow octave on any piano and with any temperament you choose, if you have its effect as your goal. To tune in ET, start by tuning your A4 to the fork as usual. Then when you set A3, find the point at which it sounds pure to you then sharpen it just until you hear the slightest beat. About 1 beat in 2 seconds, for example. You may choose anything you like. If you want it to *sound* beatless but still be as narrow as possible, back off from the point where you begin to hear the beat until you hear none. The aural test for a 4:2 octave, the F3-A3 3rd compared to the F3-A4 10th should find the 10th a shade slower, certainly no faster. Now, estimate the speed of your F3-A3 3rd a bit slower than usual, about 6 beats per second. Then tune an octave similar to the A3-A4 octave from the F3 to F4 by finding the point where F4 sounds pure, then flattening it until there is the slightest beat, etc. Then, fill in your C#4 to see if you can create the proper 4:5 ratio of the contiguous 3rds: F3-A3-C#3-F4-A4. If it is not right, you may shift things around a bit until you do have two good sounding octaves and a true 4:5 ratio of the contiguous 3rds. After that, filling in your 4ths & 5ths and fine adjusting them to make all of the RBI's smoothly progress and having no SBI's that stick out or are too pure will be relatively easy. You will also be doing the most you can to avoid the Reverse Well error, even if you are not sure what Reverse Well is or what it sounds like. When you have the F3-A4 to your satisfaction, fill out the rest of the low tenor by estimating the same kind of octave, then check both the SBI's and the RBI's making the exact compromise which best satisfies each one along the way. Do the same up to the end of the midrange. You will find that 3rds above A4 begin to not be discernible but you still have some 6ths and 10ths to satisfy. I think it may be possible for you to achieve the same Equal Beating effect that many HT's have, notably the 1/7 Comma Meantone and my own Equal Beating Victorian if you get a sufficiently narrow octave to produce sufficiently tempered 5ths. What you will want to try to achieve is for every major triad in second inversion, the 3rd on top will beat exactly the same as the 6th. Example: The C4-E4 3rd beats exactly the same as the G3-C4 6th. In my mind, if you can get this to happen throughout the entire midrange where the audible RBI's lie, from F2 to C5, for instance, you would want to make this compromise take precedence over all others. In the regular and stretched out versions of ET, the 3rd on top is always a little faster than the 6th. This causes a kind of muddying effect and conflict. I think many will be surprised at how beautifully clear and harmonious the sound of a piano can be tuned in ET but with deliberately more narrowed 5ths than are common practice. After C5, you can begin to make the octave a bit wider. If you have done the deliberately narrow octave, then by now, you can start to make it pure. When you get to F5, you can start the comparison of the double octave and octave and a 5th. Making both be equally compromised is yet another example of the Equal Beating concept. I highly recommend it as the very best octave tuning compromise. It can easily be done aurally with the same precision as the SAT provides. If in the beginning at F5, it makes the double octave too obviously wide, then compromise the octave until it sounds acceptable to you. You must realize however, that this will make the octave and a 5th beat more strongly. Pulling the tempering out of your octave and 5ths as soon as you can will help you avoid a flat and dull sounding treble. If you have started with a very narrow octave however, you need to pull yourself out of it more gradually to avoid losing your very harmonious and possibly even the Equal Beating effect. I recommend tuning to the top first, then doing the bass, for several reasons. The top is always more difficult and tedious. Get this part out of your way first, then "it will all be downhill from then on" so to speak. The bass is much easier and more relaxing to tune, so do it as you are winding down. Tune down the same kind of octaves you started in the temperament to at least C3. Then you can start to widen them but continue to listen to all 4ths & 5ths and also 3rds, 6ths and 10ths at least down to F2. Below that, just some single octaves will do, anything that sounds like it is right in the sweet spot to you will be correct. You may tune back and forth to see what sounds too flat and what is obviously too sharp if it helps you to find just the right compromise. Have fun exploring different tuning concepts. Sincerely, Bill Bremmer RPT Madison, Wisconsin
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