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I came up with an interesting tuning sequence, which I've tested in simulations. It is of no practical use, but since I find it interesting, perhaps others will also.
First tune A3 from A4 (from fork) and leave it there. The temperament octave is F3-E4.
Select a random note (excluding A3) from the temperament octave. Now play a P4 and a P5 from that note within the temperament octave, except when you selected A#3 or B3, then play two P4s. Now tune the note you randomly selected to make both intervals equal beating.
Repeat.
Eventually all notes will end up in ET with less than 1 cent error. If all notes were randomly detuned by as much as 50 cents you will need to do this about 600-1500 times.
Continuing beyond this will not improve matters substantially, as in true ET they are of course not equal beating. (You can get the error down to about 0.6 cents if your persist, but not less than that.)
If you refine the sequence by making the intervals beat in the correct proportion (by 4/3 or 3/2) you can continue and will end up with an error below 0.2 cents if you persist for 1000-2000 repeats.
Kees
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Ah, funny you should come up with this. I was toying with something similar last year and thought of it as an iterative convergence. My thought was to start with an untuned span of a 5th, then tune the next span of a 5th above by tuning each note with an equal beating lower 4th and 5th chromatically up to the top of the span. Then do the same in reverse down the lower span. Then up the upper span again, down the lower etc etc. My fuzzy logic tells me it should eventually converge towards equal spaced semitones i.e. ET. The thought came to me because of the fact that tuning up the treble with equal beating extended 5ths and 8ths in theory smoothes out an unequal temperament towards the extremity. So why not use this property to tune an equal tempered middle section. Edit - I tried it on a client's piano and it did not work very well 
Last edited by Chris Leslie; 07/13/14 03:02 AM.
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Ah! Interesting!
Equal beating intervals. I guess Bill Bremmer has designed his EBVT III in such a way.
Here is another exercise on tuning ET, not by using equal beating intervals, but by just intonation fifths, fourths, major thirds and octaves.
Tuning equal temperament by using just intonation techniques was discovered/invented in 1807 by John Farey.
He discovered that five just fourths, minus two just fifths, minus one just major third create a fifth that is almost identical to an equal temperament fifth. Today this type of fifth is known as a "schisma fifth".
schisma fifth ratio = 4/3*4/3*4/3*4/3*4/3*2/3*2/3*4/5 = 16384/10935 = 1.498308185
ET fifth ratio = 2^(7/12) = 1.498307077
The error is = 1200*log2(1.498308185/1.498307077) = 0.00128 cents!
And he designed a bearing plan (tuning sequence) to tune equal temperament by tuning only pure, beatless, intervals.
Tune each note from the previous one, everything without exception is to be tuned in just intonation (characters in bold indicate ET tuned notes):
C4, Ab3, Db4, Gb3/F#3, B3, E4, A3, D4, G3
G3, Eb3, Ab3, Db4, Gb3/F#3, B3,E3, A3, D4
D4, Bb3, Eb4, Ab3, Db4, Gb4/F#4, B3, E4, A3
A3, F3, Bb3, Eb4, Ab3, Db4, Gb3/F#3, B3, E4
E4, E3, C3, F3, Bb3, Eb4, Ab3, Db4, Gb3/F#3, B3
B3, B2, G2, C3, F3, Bb3, Eb3, Ab3, Db3, Gb3/F#3
F#3, D3, G2, C3, F3, Bb2, Eb3, Ab3, Db4/C#4
C#4, C#3, A2, D3, G2, C3, F3, Bb3, Eb3, Ab3/G#3
G#3, E3, A2, D3, G2, C3, F3, Bb2, Eb3, Eb4/D#4
D#4, D#3, B2, E3, A2, D3, G2, C3, F3, Bb3/A#3
A#3, A#2, F#2, B2, E3, A2, D3, G2, C3, F3, F4
Of course, this is only a theoretical exercise as the errors involved in tuning such a long sequence, 97 notes to tune, are far greater than is acceptable for a correct ET setting.
But it is interesting to know he has discovered a just intonation technique to temper an ET fifth.
Last edited by Gadzar; 07/13/14 06:53 AM.
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I am intrigued that other tuners are still experimenting with these ideas. It seems to imply that the current sequences available today are still considered inefficient by some technicians. I believe this to be so.
That is why I am working on a sequence and method that I will soon publish as a book. In my opinion, a highly accurate and efficient sequence/method would have to have the following elements:
1. The sequence should require as little iteration as possible. I.e. each step should place each note as accurately as possible. (My sequence has one iteration, using the upper skeleton to iterate F4)
2. The sequence should use windows as much as possible, instead of equalities. (My sequence uses windows for every note except one, which uses an equality.)
3. The method should emphasize and encourage clean unisons and good stability.
4. Obviously, the sequence should use as many checks as possible to ensure accuracy from P4/P5 and m3/M3/M6 relationships.
The real beauty of this approach is a mating of method with sequence. I've tried to explain some of the elements of this method, but without the big picture, the benefits of each element are not clear. That's why I'm not being specific regarding the details.
The book is almost half done and I'll probably post the introduction when I'm finished.
If anyone would like to proof read it before it's published, let me know.
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...Here is another exercise on tuning ET, not by using equal beating intervals, but by just intonation fifths, fourths, major thirds and octaves...
...Tuning equal temperament by using just intonation techniques was discovered/invented in 1807 by John Farey.
But it is interesting to know he has discovered a just intonation technique to temper an ET fifth.
This may be slightly off-topic, but there is also a similar method to get an absolute beat rate of a RBI by tuning perfect just intervals. It is simple: If you tune the beginning of a 5th/4th sequence from A3 (A3-E4-B3-F#4) using perfect intervals, then the beat rate of the A3-F#4 6th is very close to the M3 beat rate of F4-A4. This could be used as a seed to begin a Contiguous M3rd sequence using something absolute to start with. With no iH, the beat rate of the M6th is 13.8, and the M3rd is also 13.8. With strong inharmonicity, the beat rate of the M6th is about 13.5, and the M3rd is about 13.0. The difference with strong iH makes an error of only about 0.5 cents.
Last edited by Chris Leslie; 07/13/14 05:46 PM.
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This is amazing. There are several combinations of pure, equal beating and/or proportional beating intervals which give the correct tempering of ET intervals.
The sequence of Bill Bremmer, ET via Marpurg is a good example, it combines proportional tuning (for the setting of CM3s), pure tuning of P4s and P5s and equal beating P4s and P5s to get an almost perfect ET.
Also, if you are good to judge the tempering of M3s then you can try this: tune an octave, say for example A3A4, now tune pure P4s from A3 up to D4 and G4 and from A4 down to E4 and B3. Then you tune D#4 by equally tempering both M3s B3D#4 and D#4G4 (proportional tuning). The resulting D#4 is quasi perfect ET, the theoretical error is 0.000000000000384 cents.
From there, you can tune two chains of CM3s: F3, A3, C#4, F4, A4 and G3,B3, D#4, G4 just by proportional tuning of M3s. And you have tuned every other note between F3 and A4.
Last edited by Gadzar; 07/13/14 07:53 PM.
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Rafael, I have done my math on the D#4 method and I get something different. If I make equal beating M3rds either side of D#4 (if that is what you mean) the D#4 ends up sharp by 1 cent. If the D#4 is at proper ET pitch then the two M3rds have a 4/5 beat speed ratio, just like in a CM3rd progression but at wrong pitch.
Last edited by Chris Leslie; 07/14/14 04:16 AM.
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Rafael,
Very interesting shortcut to D#4.
I'm trying to figure out how one would then proceed to tune the ladder of CM3s: G3-B3-D#4-G4.
The other CM3 ladder is different, because both A3 and A4 are fixed. But with this ladder on G3, the only reference is D#4, and there is no outside M3 for verification. I suppose the logical way is to tune D#3 as an auxiliary note?
Autodidact interested in piano technology.
1970 44" Ibach, daily music maker.
1977 "Ortega" 8' + 8' harpsichord (Rainer Schütze, Heidelberg)
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Chris, here are the maths. I hope I'm not wrong.
From A3=220 we tune up a pure fourth D4=4/3×220=293.33 and up a pure fourth to G4=4/3×293.33=391.11
From A4 we tune down a pure fourth to E4=3/4×440=330 and down a pure fourth to B3= 3/4×330=247.5
Now to proportionally tune D#4 to B3 and G4 we must have D#4/B3=G4/D#4 so D#4=(B3×G4)^(1/2)=(247.5×391.11)^(1/2)=311.13
Wich compared to ET theoretical frequence of D#4=440×2^(-6/12)=311.13
results in no error at all.
Last edited by Gadzar; 07/14/14 05:44 AM.
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Mark,
If you are good at tempering proportional M3s, all you need is one note of a CM3s chain.
Here we tune
A4 to fork
A3 to A4, the octave size you like.
D4 to A3, as a pure fourth
G4 to D4, as a pure fourth
E4 to A4, as a pure fourth
B3 to E4, as a pure fourth
D#4 to B3/G4, as a proportional M3. That means D#4/B3 = G4/D#4.
to aurally tune such M3s you must be good on tempering M3s, and place D#4 at the spot where B3D#4 sounds as tempered as D#4G4. For this, G4D#4 should beat faster than D#4B3, in a approximate ratio of 5 to 4, but I usually do not count beats, nor estimate beat rates by themselves, but I estimate the amount of tempering (harshness or roughness) in the major thirds.
Now you have A and D# accurately tuned, you can tune the rest of the CM3s involved with your usual techniques, for example tuning:
-F3 to A3 at about 7 bps
-F4 to F3 as a clean octave, the same size A3A4 or slightly wider.
-C#4 to A3/F4, as a proportional M3, exactly the same as for D#4, from here refine the tuning of F3, C#4 and F4 to have a nice progression of the beat rates of the CM3s.
From there you have several ways to continue, one is what you said tuning
-D#3 to D#4 as a clean octave, the same size of F3F4 or slightly wider.
and then tune G3, B3 between D#3 and D#4 to have a nice progression of CM3s.
Another would be to directly tune B3 to have A3C#4 < B3D#4 < C#4F4.
And then tune G3 to have F3A3 < G3B3 < A3C#4.
Now you can refine the tuning of D#3, G3, B3 by checking the progression of the CM3s.
By using the same proportional technique you can now tune the remaining two chains of CM3s.
For example tune A#3 inbetween F3 and D#4. The fourths F3A#3 and A#3D#4 must have the same amount of tempering.
And C4 can be tuned inbetween G3 and F4 the same way, using the fourths G3C4 and C4F4.
This is a sequence that is very accurate and self correcting.
Some may not like it because it doesn't use P5s. But it can be modified to include P5s.
Last edited by Gadzar; 07/14/14 06:44 AM.
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Chris, here are the maths. I hope I'm not wrong.
From A3=220 we tune up a pure fourth D4=4/3×220=293.33 and up a pure fourth to G4=4/3×293.33=391.11
From A4 we tune down a pure fourth to E4=3/4×440=330 and down a pure fourth to B3= 3/4×330=247.5
Now to proportionally tune D#4 to B3 and G4 we must have D#4/B3=G4/D#4 so D#4=(B3×G4)^(1/2)=(247.5×391.11)^(1/2)=311.13
Wich compared to ET theoretical frequence of D#4=440×2^(-6/12)=311.13
results in no error at all. I confused equal beating and proportional M3rds. But as you explained to Mark, to tune proportional M3rds either side of D#4 I don't think you can do it without expanding out to a whole chain of CM3rds.
Last edited by Chris Leslie; 07/14/14 07:07 AM.
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Thanks for this, Rafael.
I've been using ET via Marpurg, and I see some possible shortcuts here, to get more quickly to the F3-F4 whole-tone scale, having to correct fewer notes afterwards.
Autodidact interested in piano technology.
1970 44" Ibach, daily music maker.
1977 "Ortega" 8' + 8' harpsichord (Rainer Schütze, Heidelberg)
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I don't see this as a tuning sequence for my dayly work. I see it as a curiosity to tinker around.
For my dayly work I tune:
A4, A3, F3, F4, C#4 (CM3s), A#3/F#3/B3, G#3/C4/G3 (setting the P4s tempering with G3B3, Sanderson/Baldassin), then expanding the mini temperament F3-C#4 to D4, D#4, E4.
I tune this as quick as I can, leaving imperfections untouched.
In the fine tuning, I retune every note with all tests I judge necessary. Kent Swafford has a very nice essay on correcting a temperament.
For me it is useless to pursue perfection in the first pass, you are going to move everything in the fine tuning, so why bother?
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Great work, Doel and others! I got the idea of equal beating intervals from Owen Jorgensen's little known and little used second publication, The Handbook of Equal Beating Temperaments. The usefulness at fist seemed only to be that one was not guessing at anything. Whatever the piano's scaling and inharmonicity may be, it is incorporated automatically.
In the EBVT, I was looking for a way to provide a sequence that I could reproduce reliably more than anything else and that perhaps others who may take an interest could also. But I discovered something else that was far more profound: When two intervals, either slowly or rapidly beating, have the very same beat rate, those beats tend to cancel each other.
This is what makes the EBVT (or EBVT III) sound so much purer than it is when music is played in the simple keys. It often sounds like a much stronger temperament that would have too much harshness in the remote keys to be useful. Yet the acoustic trick allows for moderated harshness in the remote keys that is not only tolerable but often just enough to actually be titillating. Who said that width of Rapidly Beating Intervals (RBI) has no effect on emotions? I certainly know that it does!
For more than 20 years, I refused to tune ET for anyone for any reason. Yet, what I found as a pleasant surprise from the ET via Marpurg was that same canceling effect due to the equally beating intervals. Now, if I choose to tune a temperament with no key color or someone insists upon ET because of ultra sensitivity to RBI's any wider than what ET would provide for, I have a solution that provides for a much "cleaner" sound overall than a true ET could provide.
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Kees:
Just some general comments, because you know what you are doing and what you are talking about. That is, what I am going to mention, I am sure you have already thought about.
You have described an algorithm. It favors the 2nd, 3rd and 4th partials when iH exists. I expect that in the case of a jump in iH, the RBIs would not be progressive.
I will often use the relationship of the SBI beatrates to smooth out a tuning, or when touching up a tuning. It is the same idea.
I had been toying with the idea of a sequence that is based on the M3/M6 test. Of course this requires a temperament span of greater than an octave. It may also work well as an algorithm, I think... Something like:
A3 = 220hz
A3-C#3 = 9bps
A3-E4 = -0.5bps
G3-E4 = 9bps
G3-D4/A3-D4 = -1/2 ratio
Of course, so far these are just rule-of thumb numbers. As the algorithm progresses, any errors would be corrected only part way. I would think three times thru the sequence would be sufficient. I'll have to play with it on my simulator.
This would continue with something like:
B3-E4/E4-A4 = 2/3 ratio AND
G3-B3/A3-C#3 = 7/8 ratio (both won't happen, so the error is averaged between the two)
F3-D4 = G3-B3
Then next M3/M6 test can produce a A3-D#4-A4 contiguous tritone, but the m3/M3 inversion of the M3/M6 test would probably be used.
Then three more M3/M6 tests should complete the temperament. Of course then the errors are checked and corrected by averaging. I suppose the averaging could be weighted according to a preference...
I'll see if I can find the time to play with this. Not as an aural tuning sequence, but as an algorithm.
Jeff Deutschle
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Who taught the first chicken how to peck?
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...But I discovered something else that was far more profound: When two intervals, either slowly or rapidly beating, have the very same beat rate, those beats tend to cancel each other.
This is what makes the EBVT (or EBVT III) sound so much purer than it is when music is played in the simple keys. It often sounds like a much stronger temperament that would have too much harshness in the remote keys to be useful. Yet the acoustic trick allows for moderated harshness in the remote keys that is not only tolerable but often just enough to actually be titillating. Who said that width of Rapidly Beating Intervals (RBI) has no effect on emotions? I certainly know that it does!
For more than 20 years, I refused to tune ET for anyone for any reason. Yet, what I found as a pleasant surprise from the ET via Marpurg was that same canceling effect due to the equally beating intervals. Now, if I choose to tune a temperament with no key color or someone insists upon ET because of ultra sensitivity to RBI's any wider than what ET would provide for, I have a solution that provides for a much "cleaner" sound overall than a true ET could provide. With my research i found the opposite to be true: Equal beating increases (worsens) beat intensities and does not cancel them.
Last edited by Bernhard Stopper; 07/14/14 10:13 AM.
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Mark,
If you are good at tempering proportional M3s, all you need is one note of a CM3s chain.
Here we tune
A4 to fork
A3 to A4, the octave size you like.
D4 to A3, as a pure fourth
G4 to D4, as a pure fourth
E4 to A4, as a pure fourth
B3 to E4, as a pure fourth
D#4 to B3/G4, as a proportional M3. That means D#4/B3 = G4/D#4.
to aurally tune such M3s you must be good on tempering M3s, and place D#4 at the spot where B3D#4 sounds as tempered as D#4G4. For this, G4D#4 should beat faster than D#4B3, in a approximate ratio of 5 to 4, but I usually do not count beats, nor estimate beat rates by themselves, but I estimate the amount of tempering (harshness or roughness) in the major thirds.
Now you have A and D# accurately tuned, you can tune the rest of the CM3s involved with your usual techniques, for example tuning:
-F3 to A3 at about 7 bps
-F4 to F3 as a clean octave, the same size A3A4 or slightly wider.
-C#4 to A3/F4, as a proportional M3, exactly the same as for D#4, from here refine the tuning of F3, C#4 and F4 to have a nice progression of the beat rates of the CM3s.
From there you have several ways to continue, one is what you said tuning
-D#3 to D#4 as a clean octave, the same size of F3F4 or slightly wider.
and then tune G3, B3 between D#3 and D#4 to have a nice progression of CM3s.
Another would be to directly tune B3 to have A3C#4 < B3D#4 < C#4F4.
And then tune G3 to have F3A3 < G3B3 < A3C#4.
Now you can refine the tuning of D#3, G3, B3 by checking the progression of the CM3s.
By using the same proportional technique you can now tune the remaining two chains of CM3s.
For example tune A#3 inbetween F3 and D#4. The fourths F3A#3 and A#3D#4 must have the same amount of tempering.
And C4 can be tuned inbetween G3 and F4 the same way, using the fourths G3C4 and C4F4.
This is a sequence that is very accurate and self correcting.
Some may not like it because it doesn't use P5s. But it can be modified to include P5s.
But, tuning a pure interval in ET is a waste of time, because you will have to correct it later on.
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But, tuning a pure interval in ET is a waste of time, because you will have to correct it later on. I guess you have not read my earlier post: I don't see this as a tuning sequence for my dayly work. I see it as a curiosity to tinker around.
For my dayly work I tune:
A4, A3, F3, F4, C#4 (CM3s), A#3/F#3/B3, G#3/C4/G3 (setting the P4s tempering with G3B3, Sanderson/Baldassin), then expanding the mini temperament F3-C#4 to D4, D#4, E4.
I tune this as quick as I can, leaving imperfections untouched.
In the fine tuning, I retune every note with all tests I judge necessary. Kent Swafford has a very nice essay on correcting a temperament.
For me it is useless to pursue perfection in the first pass, you are going to move everything in the fine tuning, so why bother? And if you did read this post, then you have to understand the objective of the exercise is to precisely locate the tuning of D#4, nothing more. I am not tuning the piano. And about what you say of waste of time, let me tell you that I never tune a piano in a single pass. never! I always do at least two passes, even if the piano is at pitch. So, all what I do in the first pass is to be moved in the fine tuning. It is in this second pass, the fine tuning pass, where I strive for stability. In this pass, more than tuning I am setting the pins and rendering the strings. Maybe you'll say it's a waste of time, but it's the way I do it.
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...But I discovered something else that was far more profound: When two intervals, either slowly or rapidly beating, have the very same beat rate, those beats tend to cancel each other.
This is what makes the EBVT (or EBVT III) sound so much purer than it is when music is played in the simple keys. It often sounds like a much stronger temperament that would have too much harshness in the remote keys to be useful. Yet the acoustic trick allows for moderated harshness in the remote keys that is not only tolerable but often just enough to actually be titillating. Who said that width of Rapidly Beating Intervals (RBI) has no effect on emotions? I certainly know that it does!
For more than 20 years, I refused to tune ET for anyone for any reason. Yet, what I found as a pleasant surprise from the ET via Marpurg was that same canceling effect due to the equally beating intervals. Now, if I choose to tune a temperament with no key color or someone insists upon ET because of ultra sensitivity to RBI's any wider than what ET would provide for, I have a solution that provides for a much "cleaner" sound overall than a true ET could provide. With my research i found the opposite to be true: Equal beating increases (worsens) beat intensities and does not cancel them. AHA! YES! This is crazy isn't it? But the truth is, sometimes equal beating cancels and sometimes it increases sound. The scientific terms are Constructive and Destructive Interference. Equal beating wave forms have identical periods, i.e. identical distances between peaks and valleys. When peaks line up with peaks, and valleys line up with valleys, we have constructive interference, and that means a louder beat. Also called "being in phase". When peaks line up with valleys, and valleys line up with peaks, we have destructive interference, and that means cancellation. (Also called "being out of phase") See https://www.youtube.com/watch?v=OKsmqzRFFsk (Just the beginning) https://www.youtube.com/watch?v=YuveKkmeFWg (Describes cancelling) That's another reason why ETD's can't tune unisons; they don't measure phase. Also, this can explain one reason why dead on unisons are not appropriate for a piano, even though intuitively, one might think a "dead on" unison is preferable. A "dead on" unison might be in phase (louder) or out of phase (quieter), and a piano with all "dead on" unisons would have uneven tone.
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Phase differences are location-specific. You can hear this with a tuning fork. Just twirl the fork near your ear, and it will sound louder or softer, because more of the sound will be in phase or out of phase.
(Of course, everyone who is interested in tuning should have at least one high-quality tuning fork!)
Last edited by BDB; 07/14/14 12:39 PM.
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