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I don't think there is a magic sequence that will solve the temperament puzzle. I sometimes think of it as sculpting, and ponder the Michelangelo quote: "Every block of stone has a statue inside it and it is the task of the sculptor to discover it."
A temperament is discovered not imposed. You have to get to know the harmonic characteristics of the particular piano and then by nudging things around find the most appropriate compromise. It is at the microcosm level of fine tuning where the limitations of the particular instrument and its scale show up. At this level the small discrepancies become apparent - especially (as mentioned earlier) due to the nature of open unisons where every interval is a composite of 6 stings sounding together.
The problem with a sequence is it is always building off previously tuned notes so, even though the intervals tuned early on will be acceptable, the compounding of discrepancies creates a limit to the level of accuracy.
I agree with you that the unique characteristics of a piano (iH, resonances, voicing, etc.) must be considered when creating a temperament. In theory, there should be only one optimum solution for an Equal Temperament on a given piano. (It could be argued that Entropy Tuner, in theory, could produce that solution.)
The discussion has considered what might create the best compromise for beat rates and progressions of the various intervals - M3s, P4s, P5s, as well as other interval tests.
My question is - When creating an Unequal Temperament, is it best to use these same compromises to come close to the theoretical UT, or stick to the best M3 set, which is generally how UTs are perceived?
I find that I prefer, since it is easier, to set P5s and accept the resulting M3s, as long as they are close to the 'quality' of sound desired, since my ears are more tolerant of very narrow P5s than badly set M3s.
Prout: Oh please, where how can we possible get to the crux of the matter of tuning ET if we start looking at the priorities of UT sequences???
Ryan: You talk about there not being magical sequences, but you can sculpt a temperament (sounds magical to me...); that a temperament can only be "discovered" as the most minute discrepencies appear when tuning open unisons (what's THAT got to do with anything?!?!?); but then stating that any sequence just compounds discrepencies, anyway, and will always be a reason for ET inaccuracies. Sorry, this all sounds like a big excuse for not bothering to really tune ET.
Not that progressive M3s and M6s are often tuned, but with a decent piano, I can't blame the scaling. I have looked at quite a few. It is only when there is a jump in iH that the RBIs can't both be progressive. There are other reasons ET, as defined by progressive RBIs, is not often tuned than it needing to be "discovered", perhaps with open unisons!
Jeff Deutschle Part-Time Tuner Who taught the first chicken how to peck?
Prout: Oh please, where how can we possible get to the crux of the matter of tuning ET if we start looking at the priorities of UT sequences???
Well, given that it is impossible to tune ET on a piano, every tuning is an Unequal Temperament, therefore there is no 'crux of the matter'. You guys are going around in circles (pun intended).
If you are willing to accept that fact, then any chosen temperament must accommodate the vagaries of a given piano. It is simply a matter of choosing which set of intervals you prioritize in order to give the listener that which they expected.
Nothing can be perfect, only within specifications. No 2x4 used for constructing a house is exactly 1-1/2 inches by 3-1/2 inches, but they are close enough to use to build a square and plumb house. Or as BDB puts it, if you can't tell a difference, there is no difference.
Your post is odd in another way. It is true what you say: "...any chosen temperament must accommodate the vagaries of a given piano." This has nothing to do being willing to accept that ET is impossible to tune or not. Sure, some combinations of intervals will have different ratios on some pianos than on others, and jumps in scaling requires jumps in beatrates, although not everyone agrees on this. So what? That is just playing the hand you are dealt. Again, it seems to be more of an excuse for not bothering to really tune ET.
Imagine a carpenter that built a house where the doors wouldn't all close, there were some leaks in the roof and the floors were so uneven in spots that the chairs would wobble. When you asked him about it he talks silliness about how being square and plumb is immpossible to achieve, so it is better to "accomodate the vagaries" of the materials and building site. It is simply a matter of priotitizing what is square and plumb!
Jeff Deutschle Part-Time Tuner Who taught the first chicken how to peck?
Prout: Oh please, where how can we possible get to the crux of the matter of tuning ET if we start looking at the priorities of UT sequences???
Ryan: You talk about there not being magical sequences, but you can sculpt a temperament (sounds magical to me...); that a temperament can only be "discovered" as the most minute discrepencies appear when tuning open unisons (what's THAT got to do with anything?!?!?); but then stating that any sequence just compounds discrepencies, anyway, and will always be a reason for ET inaccuracies. Sorry, this all sounds like a big excuse for not bothering to really tune ET.
Not that progressive M3s and M6s are often tuned, but with a decent piano, I can't blame the scaling. I have looked at quite a few. It is only when there is a jump in iH that the RBIs can't both be progressive. There are other reasons ET, as defined by progressive RBIs, is not often tuned than it needing to be "discovered", perhaps with open unisons!
My father was a sculptor. A few years ago he made a bust of my daughter and it was interesting to watch the progression from a rough shaped blob of clay to a life-like depiction. Sculptors don't get the eye perfect and then move to the nose, and then the ears, trying to make everything life-like as they go along. I found a youtube video that has a good example:
The reason tuning seems like sculpting to me is you start out with a rough overall shape - a decent quasi equal temperament. That becomes the raw material from which you get to the next level of refinement. The second stage of refinement becomes the raw material to get to the third level of refinement, at which point most all of the intervals will fit into narrow windows. If the piano is to be used for a PTG master tuning, things will be nitpicked even more - because there are not the usual time constraints of retail work.
As the temperament begins to expand outwards, it is often possible to find small improvements in the temperament due to the increased information available from the newly tuned notes. There is nothing "magical" about it - it is really just a practical analogy of how I experience the process.
I'll try to clarify my statement about the best temperament is "discovered" and not imposed. It is discovered because the exact position of each string is unknown at the start. ETD's make an educated guess at the proper frequencies of select partials by analyzing the inharmonicity of several sample strings and then making an educated guess regarding the positions of the rest of the strings.
However, the inharmonicity of each string is unique, and how that string interacts with another string to form an interval is also unique. Also the sound of 3 strings together is not exactly the same as the sound of one string alone. The interval created by 9 strings sounding together is not quite the same as the interval created by 2 strings.
So you can only know the best position for a note through a certain amount of trial and error.
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Sorry, this all sounds like a big excuse for not bothering to really tune ET.
Not at all! Quite the opposite: It is about how to tune the closest approximation to ET as possible. My point was that the sequence will only get you so far, because a sequence is always assuming a previously tuned note is correct. Yet, early in the tuning process, it is not possible to know the ideal position. These early assumptions will sometimes have to be revisited during the refining process.
Ryan Sowers, Pianova Piano Service Olympia, WA www.pianova.net
Great post Ryan. "Well sculpted" I might even say.
If you are tuning a piano, because of inharmonicity you are not tuning mathematical perfect ET. But you should be trying to divide the octaves into intervals whose relative beat speeds allow for an equal sound quality to be experienced by the listeners.
So if you want to learn to temper the whole compass of a piano you must start by studying how all the octaves fit together, then divide all the octaves by your chosen procedure to tune and test intonation of all the intervals.
Once enough notes are in the range of acceptable equality, you can refine things by identifying the worst interval of each class. Find the worst 5th and then test to see which of the notes can be moved to improve the 5th without ruining the 4th, M3, M6, m3 and m6 connected to that note. After you go through these iterative processes enough you will begin to almost know exactly what should be done with each interval when you first begin to tune a piano.
In a seemingly infinite universe-infinite human creativity is-seemingly possible. According to NASA, 93% of the earth like planets possible in the known universe have yet to be formed. Contact: toneman1@me.com
So Ryan, if I were to construct a square, would I "discover" four right angles, or would I draw them?
Actually, I know just what you mean and I think that is part of the problem. Typically we don't really tune the notes, we arrange them. And since we tend to like our own creations, we fool ourselves into thinking our "baby" is beautiful.
I think a solution is to use a chromatic mini-temperament as small as possible and expand the tuning from there. There are fewer relationship to get wrong and we are dealing with chromatic beatrates rather than trying to fit intervals between intervals and "arranging" a tuning. The interval needs to be large enough to assure that both M3s and m3s are progressive and to assure that the ratio of 4th and 5ths beatrates are correct. Eight notes spanning a P5 will do it. You can check it out on the mini-temperament Topic.
Jeff Deutschle Part-Time Tuner Who taught the first chicken how to peck?
Ed, I think there is plenty of wiggle room when expanding a temperament that it is not necessary to fix the size of the octaves first. Dr. White's sequence doesn't!
Jeff Deutschle Part-Time Tuner Who taught the first chicken how to peck?
So Ryan, if I were to construct a square, would I "discover" four right angles, or would I draw them?
In a simplified way, "yes". But a square would be the equivalent of tuning 4 notes like an octave with an inside 4th and 5th. It is very easy to get this "perfect": A clean sounding octave with a slightly narrow fifth and a slightly wide forth in the middle. No problem at all.
The problem is trying to make all 27 intervals within an octave fit their strict definitions. And to do it with open unisons. This means each of those 27 intervals is made of 6 strings for a total of 192 strings. And we are arranging these intervals, not with our eyes, but with our ears!
I think your square analogy, is a bit oversimplified!
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Actually, I know just what you mean and I think that is part of the problem. Typically we don't really tune the notes, we arrange them. And since we tend to like our own creations, we fool ourselves into thinking our "baby" is beautiful.
You are saying there is a difference between tuning and arranging. I don't believe that there is. An ETD user perhaps may tune a note because they are matching the strings to a specific number. But that number is only an educated guess, and experience shows that ETDs aren't always the best "arrangers" of notes and that their solutions to the equal temperament puzzle can often be improved upon. A real tuning is an arrangement that is created by listening to the piano and finding that specific piano's solution.
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I think a solution is to use a chromatic mini-temperament as small as possible and expand the tuning from there. There are fewer relationship to get wrong and we are dealing with chromatic beatrates rather than trying to fit intervals between intervals and "arranging" a tuning. The interval needs to be large enough to assure that both M3s and m3s are progressive and to assure that the ratio of 4th and 5ths beatrates are correct. Eight notes spanning a P5 will do it. You can check it out on the mini-temperament Topic.
Great! I look forward to hearing the recording!
Ryan Sowers, Pianova Piano Service Olympia, WA www.pianova.net
Thanks for sticking with this. It is the type of tuning "philosophical" discussion I need to work out some apparent contradictions in what I read.
Most published tuning sequences do not have a refining step, but when tuners write about tuning, that is often not just a step but the entire process. You seem to be one of those. I think we could look at this as a spectrum of tuning philosophies. Dr. White's step-by-step, fix it as it occurs sequence would be at one end; Mr. Swafford's Every-Which-Way at the other. I think the word "elegant" would apply here as when used in science. Elegent: (of a scientific theory or solution to a problem) pleasingly ingenious and simple.
We should also to agree to disagree.
Your mention of tuning an octave and a 4th and 5th within as being something that can easily be done perfectly struck me. Until recently, I thought this was true, but no longer. If just one note is a 1/3 cent off from ideal, it will probably be impossible to have all the RBIs be progressive. As you say, when adding more and more notes (and strings if open tuning) it is harder and harder to pinpoint where an error comes from. I think the tendency is to blame the victim, in this case, the piano!
An elegant solution is to have only necessary steps, and ones where there are defined go/no-go decisions to be made. This is far different than roughing out a tuning and then refining until you don't know what else to do or just get sick and tired of it and move on.
Yes, I do plan on making some kind of video/recording of the 8 note chromatic temperament. Hopefully in time for Christmas. I am interested in having it analyzed!
Jeff Deutschle Part-Time Tuner Who taught the first chicken how to peck?
Most published tuning sequences do not have a refining step, because I believed that it is assumed that refining is necessary. Of course a tuner may decide that the the results are good enough and not bother to refine. But if you are trying for "high end" results, it is a forgone conclusion that refinements will be made.
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Your mention of tuning an octave and a 4th and 5th within as being something that can easily be done perfectly struck me. Until recently, I thought this was true, but no longer. If just one note is a 1/3 cent off from ideal, it will probably be impossible to have all the RBIs be progressive. As you say, when adding more and more notes (and strings if open tuning) it is harder and harder to pinpoint where an error comes from. I think the tendency is to blame the victim, in this case, the piano!
You say "if just one note is 1/3 cent off from ideal, it will probably be impossible to have all the RBI's be progressive." However, how do you now what the ideal is? In the real world there is no ideal place for a note in and of itself. A note is ideal when it effectively fits in with all the intervals it may be used with.
Maybe if you had a theoretically perfect piano your logic would work. But they don't exist. One of the reasons tuners hate tuning spinet and console pianos is that the intervals don't follow the theoretical beat rates. We all know this is because of the excessive inharmonicity caused by the short strings. We also know that the longer the strings the closer to the theoretical beat rates the intervals turn out to be. But not exactly. The problems that spinets have show up on larger pianos, but in smaller amounts.
Ryan Sowers, Pianova Piano Service Olympia, WA www.pianova.net
Ryan, I think you are waffling. So do many authors of tuning sequences. There will be some "anchor notes" that once tuned should not be adjusted, or intervals that can be tuned "perfectly." Then later there are foggier and foggier explanations (excuses...) that go something like this: You can't know that any notes are right until all are tuned. As more intervals are tuned it becomes more complicated to determine errors. No piano is perfect, so ET cannot be tuned anyway. If these were all really true, why not just start with the last one, and forget the excuses?
But let me answer your question as an illustration of this.
Originally Posted by rysowers
...
You say "if just one note is 1/3 cent off from ideal, it will probably be impossible to have all the RBI's be progressive." However, how do you now what the ideal is? In the real world there is no ideal place for a note in and of itself. A note is ideal when it effectively fits in with all the intervals it may be used with.
...
The answer is obvious. We know a note, or rather all the notes, are in the ideal place when all the RBIs are progressive.
But we must start somewhere. We can sketch a temperament together and then try to fix all the errors. I know from what many write that this gets to be very close to having all RBIs progressive (a practical definition of ET), and I agree. It will get you close, but you always end up either needing to move on because of the clock or your ego. You just don't know where the individual imperceptible errors are that add up to a perceptual error that seems can only be fixed by making other things worse. So we blame the victim - the piano.
OR, we can recognize the true nature of the challenge. We need to start with just one interval, two notes, and tune all other intervals to agree with this beginning. It is like the old saying. A man with one watch always knows what time it is, a man with two watches, is never quite sure.
As long as you are not trying to tune a temperament across a break in scaling, there is no reason to blame the piano for unprogressive RBIs. When expanding a temperament and crossing a break in scaling, well, that is a different subject and I will not look for any consensus.
Jeff Deutschle Part-Time Tuner Who taught the first chicken how to peck?
Another problem that makes it practically impossible to tune equal temperament aurally without a refinement process is this. Strings don't behave perfectly with all of their partial frequencies matching what the inharmonicity constant and our formulas would predict. Instead, the imperfect string might have a second partial that's a little flat and a third partial a little sharp and so on.
This fact can readily be seen when trying to tune a single string using the Entropy Piano Tuner's strobe display. On well behaved strings, the strobe moves in the same direction for all partials when you are sharp or flat. On badly behaved strings, when you are close to in tune, you may have some partials moving one way in the display and some moving the other, indicating (as I understand it) that some are sharp and some are flat of theoretical numbers at the same time on the same string.
Another problem that makes it practically impossible to tune equal temperament aurally without a refinement process is this. Strings don't behave perfectly with all of their partial frequencies matching what the inharmonicity constant and our formulas would predict. Instead, the imperfect string might have a second partial that's a little flat and a third partial a little sharp and so on.
This fact can readily be seen when trying to tune a single string using the Entropy Piano Tuner's strobe display. On well behaved strings, the strobe moves in the same direction for all partials when you are sharp or flat. On badly behaved strings, when you are close to in tune, you may have some partials moving one way in the display and some moving the other, indicating (as I understand it) that some are sharp and some are flat of theoretical numbers at the same time on the same string.
Good point, but is it enough to put a tuning out of tolerance? If you measure anything close enough, you will find it is not perfect. Bob, have you are anyone else determined if the observed phenomena is actually large enough "... that makes it practically impossible to tune equal temperament aurally without a refinement process ..." ?
But then I don't understand what a refining process would have to do with it, unless of course a sequence sketches in a temperament to begin with. Otherwise, all these partials would be taken into account while setting the temperament.
And another thing... Ok, so with some strings the theoretical numbers do not match reality. So is the string "wrong" or the theory? Hmmm?
Jeff Deutschle Part-Time Tuner Who taught the first chicken how to peck?
This discussion is becoming so surreal. Ryan, Prout, Ed and Bob have all given the correct answers, some really very eloquent. Nobody is waffling in my opinion and the answers have been very coherent.
The theory is an oversimplified model, and following it will affect tuning in a very real and audible way. If you bang pianos out with a machine all day or tune poor scales all the time, you will never hear it.
Focused tuning on a good scale (result of strength and quantity of partials) will reveal the unpredictable distribution of energy necessitating a 'sculpting' method, or refinement, in any temperament sequence. 'Immutable' notes are relative and can only emerge in relation to their primary intervals.
This is the entire reason behind open string techniques and Virgil's parallel note tuning sequences. Many great tuners have been aware of the phenomena but only knew to describe the technique around it.
If you don't listen carefully enough (or are not self-critical enough), you will never hear it. Our ears can be remarkably selective in ignoring things.
This discussion is becoming so surreal. Ryan, Prout, Ed and Bob have all given the correct answers, some really very eloquent. Nobody is waffling in my opinion and the answers have been very coherent.
The theory is an oversimplified model, and following it will affect tuning in a very real and audible way. If you bang pianos out with a machine all day or tune poor scales all the time, you will never hear it.
Focused tuning on a good scale (result of strength and quantity of partials) will reveal the unpredictable distribution of energy necessitating a 'sculpting' method, or refinement, in any temperament sequence. 'Immutable' notes are relative and can only emerge in relation to their primary intervals.
This is the entire reason behind open string techniques and Virgil's parallel note tuning sequences. Many great tuners have been aware of the phenomena but only knew to describe the technique around it.
If you don't listen carefully enough (or are not self-critical enough), you will never hear it. Our ears can be remarkably selective in ignoring things.
// end rant //
Rant ignored.
Jeff Deutschle Part-Time Tuner Who taught the first chicken how to peck?
Now, is there some specific point you want to talk about?
No, otherwise I would have asked. You bring up points of contentious debate as if these concepts hadn't already been laid to rest in the literature. It's strange to observe when educated techs are offering great explanations.
The concept of mini-temperament is a solid one, and one we can also see many tuners gravitating towards at the end of their careers. The tone-cluster concept is basically the same.
The most interesting point I could bring up for discussion is how this concept brings into question the 'standard' definition of superior ET as progressive 3rds by its very design.