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I cannot imagine what would be difficult to understand about contiguous thirds. If you can tune a tempered fifth by the method given, you should be able to tune a tempered third by adapting that method to a wide, fast-beating interval. The nice thing is that there are only three major thirds in an octave, so that once you have done a couple, you are at a point where you can see how far off you are. So what exactly is the problem? Can people with this problem not hear the beats? Do they have trouble counting that fast? Just saying "Most people I instruct have trouble with this" could mean the problem is with the instructor.
The area between C3 and C4 probably has the lowest inharmonicity in the piano. There are other issues with that octave, particularly if some of the strings are overwound, but inharmonicity is not the problem.
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Originally posted by BDB: I cannot imagine what would be difficult to understand about contiguous thirds. If you can tune a tempered fifth by the method given, you should be able to tune a tempered third by adapting that method to a wide, fast-beating interval. The nice thing is that there are only three major thirds in an octave, so that once you have done a couple, you are at a point where you can see how far off you are. So what exactly is the problem? Can people with this problem not hear the beats? Do they have trouble counting that fast?
... BDB: I don’t like tuning with CM3. I can do it, but I experience problems with them. First, I don’t trust the accuracy. Within an octave, the other two notes can be over 1 cent off and still sound progressive. Considering that one of the final test is chromatic, progressive M3s, then depending on M3s that are each an M3 apart for the rest of the temperament does not seem accurate. My experience with them is that I can get them progressive, but unable to get them to have a beat rate that is exactly half-way between the other two. Second, since I can’t trust them too much, as I refine the temperment I find myself in the dilemma of which of the two notes tuned as CM3s is correct, or are they both incorrect? The tuning sequence contains branches, and I am unsure which branch to take to correct an error. Third, when I play a M3 and tune one of the notes, it is not obvious to me when the string first starts to render. I can tune SBIs by smaller amounts and tune them with more stability because I can hear better what the string is doing. Fourth, for challenging pianos, I may not want the CM3 to have a strict 4:5 or 3.95:5, or whatever beat speed ratio for the pianos iH and octave stretch. I still want all M3 to beat progressively faster, but sometimes not linearly. Last is the accuracy of the other intervals. That is one of the questions in the first post of this topic. M3s can only tune more M3s. Other intervals must be used to tune the other 8 notes of the scale. I hear how great CM3s are for creating intervals adjusted for the pianos iH and octave stretch (which sometimes is nonlinear), but little about how this accuracy (which I am unable to obtain) helps to adjust the other intervals for the pianos iH and the octave stretch. I tried to word all this to show that I have a problem with CM3s, not that CM3s are faulty in themselves. How could they be? For that matter how could any interval be faulty? However, any interval can be used in a faulty manner. It depends on the tuner. I am unable to use CM3s in a way that other tuners apparently can. Regards,
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Upright, maybe BDB should answer your question, he seems to know all the answers. From what I know, however, I strongly disagree that CM3s can be over a cent off and still sound progressive. That simply is not possible. In the sequence the way I wrote it, where the F3-F4 octave is already set, you cannot have more than a 0.5 cent error and have the CM3s sound plausible. It is easier to have the A3 and C#4 absolutely spot on than it is to have either one off by any amount at all.
Regardless of whether the piano has quirky inharmonicity or not, you can still place the A3 and C#4 accurately within the F3-F4 octave. Whatever inharmonicity is there, it will be incorporated as you find the slower/faster/faster combination. Forget about 4:5, *don't* *count* any beats, only compare them.
As you tune the 4th and 5th from each of these, you will reach a compromise between what you tune from one of the notes of the set of CM3s and another. That will again incorporate the inharmonicity between two notes which have been tuned from two other verified and reliable notes rather than estimates upon unverified estimates.
For example, as you tune D4 from A3, you must reconcile G4 tuned from C4, the latter being the most reliable note of all since it is the starting note. This is far and away more certain than tuning a sequence starting at C4, then F3, then G3 then D3, then A3. In that scenario, the D3 is tuned from a note that cannot be verified by any means at all, the A3 is then tuned from that unverified note. When the A3 is compared to the F3, it is also compared to a note which had no possible way to control the amount of tempering other than a mere estimate of what it should sound like.
The resultant F3-A3 M3 and the A3-D4 M6 are both constructed upon estimates upon estimates, none of which have any possible control. If the F3-A3 M3 and the F3-D4 M6 don't agree, which of the notes leading up to them is incorrect and by how much? If you back up through those previously tuned notes to try to reach a compromise, it is likely that you will leave a 4th or 5th either too much or too little tempered which will in turn lead to a starting point that is faulty, by as much as 2 cents for the notes tuned from them afterwards. That will lead to RBIs which cannot be reconciled.
But perhaps BDB can explain why this does not, in fact occur. Perhaps any of the other participants in this Forum who have never heard of anyone making that kind of error and claim that I am just making it up through sophistry can explain why it never happens. Let anyone explain how a skilled and experienced tuner solves the inharmonicity problem and tunes so accurately using the Braide-White system that a score of 100 on the Temperament section of the PTG Tuning Exam is achieved.
A vague description is not adequate. Counting theoretical beat rates doesn't work. Backing up to find a single error doesn't work. Just exactly how is it done and how extremely complex of a process is it really and how many years of experience and what level of skill does it really take to solve that kind of problem?
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Bill: Thank you for the reply. Originally posted by Bill Bremmer RPT: ...
The resultant F3-A3 M3 and the A3-D4 M6 are both constructed upon estimates upon estimates, none of which have any possible control. If the F3-A3 M3 and the F3-D4 M6 don't agree, which of the notes leading up to them is incorrect and by how much?
... By saying that the M3 and M6 do not agree, do you mean that the M6 does not beat a bit faster? If that is what you mean, the answer is that this test is for the A3-D4 interval, the pitch of F3 does not matter, within reason. I read you sequence carefully, and enjoyed it. Here are some comments. There seems to be another typo. (It is tough to catch them all.) Step 13 mentions F#3-B4. You probably meant F#3- B3. I liked very much the comment in step 9, “…but no note can be considered infallible.†Step 3 and 5 mention a the beat rate of a 5th stacked above a 4th as being “proportional†with the 4th beating faster. I believe, theoretically, they would beat at the same speed and this would be an octave test. Maybe I’m not following your intent with this check. In step 13 you mention that the contiguous 4th should beat similarly. I always look for a slight progression in SBIs that are a 4ths or more apart, unless I doing something for a compromise. Maybe this would be ultra-fine tuning. Now, I find it just as interesting how people look at temperament sequences, as the sequences themselves. I hope you could explain why you used the A#3 as you did. I was expecting you to create the first CM3s with it. Regards,
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Thanks for the typo info, Upright. I will fix it. Yes, the M3-M6 is actually a test for a pure 4th, if both intervals beat the same, the 4th is pure. In ET, the M6 is supposed to be slightly faster. But that is the point, the M3 could sound plausible and the M6 could beat a little faster but you don't have anything to tell you if the F3-A3 M3 is right or not. If you had tuned F3-A3-C#4-F4 CM3s, you would be able to trust the F3-A3 M3 much more. If the F3-D4 M6 beats a little faster and also agrees with both the G3 and A3, you have reasonable assurance that all of these are correct.
In my latest sequence the F3 would have been tuned from the F4 which agreed with both the C4 and C5. In the BW sequence, you don't tune F4 until last to find out if the octave is "pure". You could change BW's sequence but then you'd be doing what I said to do, not what BW said to do.
In the BW sequence, the F3 is an estimate and the A3 is tuned by an estimate from an estimate from an estimate. You have to be more or less lucky to get them all exactly right. None of those estimates refers back to another reliable note, so inharmonicity isn't accounted for, it's only guessed at. If the F3-A3 M3 sounds too fast or slow, you still are making yet another unverifiable estimate if you correct it.
Why not estimate it right in the beginning and prove it with just two more notes which will also become just as reliable as a result of doing so? That is the reason for starting with the CM3s and Owen Jorgensen confirmed to me the value and reliability of starting a temperament that way. That's the reason why Coleman, Potter, Baldassin, Sanderson, NBS School, Chicago School and others all use that method today. I've never seen anyone but you claim it is not a reliable way to start a temperament, although I have seen many say they don't know how to do it and don't understand it well.
In my discussion with Owen Jorgensen, he suggested that the word, "similar" was the best choice for what I was trying to do with this latest sequence. Any two contiguous 4ths within F3-F4 (there are only 3 of them) are SBIs. There is hardly enough distinction between them to really ponder whether the top one is a little faster than the bottom. When you have them both beating what is apparently the same amount, you are close enough. Any further fine distinction would be pointed out by an RBI check.
What you'd really be trying to do is rule out one being beatless and the other beating or one beating as would seem correct and the other approaching an RBI. So, when they both beat about the same, that is really about the best you can do with them, otherwise, you'd be concentrating on using the wrong type of interval to make a very fine correction.
Many tuning books and manuals I've seen suggest that the SBIs are the coarse adjustment and the RBIs are the fine adjustment and should really be used that way. Tuning a set of 3 CM3s within an octave first does run contrary to that suggestion, yes, and that is why it takes a little more time with those first 3 intervals to get them exactly right. But thereafter, you can do, at every step, a coarse adjustment (SBI) and have a fine adjustment (RBI) to prove it correct or refine it to the very limits of perceptibility.
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Bill:
Thanks for the response. Apparently, I am very sensitive to the beat rate of SBIs. I sometimes use a tempo check on the progressiveness of SBI beat rates. If a SBI is played for one beat, and then the next for one beat, and so on, the steadiness of the tempo indicates the progressiveness of the beat rate. It is very difficult to get them completely even, because they are so close to being just, but the test shows when any are much different than the others.
You mentioned in another current Topic that Mr. Jorgensen used additional checks with the BW sequence. That is one of the questions in the subject of this Topic. Can you tell us what additional tests he uses?
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Originally posted by BDB: The area between C3 and C4 probably has the lowest inharmonicity in the piano. There are other issues with that octave, particularly if some of the strings are overwound, but inharmonicity is not the problem. I have to disagree, provided we're talking about pianos on which this area includes the wound string/plain wire break. The problem is inconsistency of inharmonicity. It's understood that, on a typical spinet for example, the highest wound strings have lower inharmonicity than the lowest plain wire strings. That's why it can be difficult to get all the intervals to work across that break. All the intervals don't work because the relationships between all the partials don't progress evenly across that break. As I understand it, changes in the way partials stack up are essentially changes in inharmonicity. Jeff
Jeff A. Smith Registered Piano Technician Indiana, USA
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I have made changes in scales that have increased the inharmonicity especially at the break, which have made the piano easier to tune. Inharmonicity is not the problem. Lack of evenness in the strength of the notes is probably more the reason for difficulty in tuning small pianos.
For a long time, most pianos had 26 or fewer notes in the bass, so that from B3 up are plain wire. That still holds true for a substantial number of pianos today. So most of the time, the difference between wound and plain strings is not an issue when laying the temper. (Perhaps only the most skillful tuners should tackle spinets. I am still feeling a glow of self-satisfaction from that CP 60 last week!) But even so, one has to be able to tune those intervals anywhere on any piano.
I have been working on putting together my tuning methods and theories to post, but it takes some time and effort, and my tuning jobs take precedence.
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Jeff:
I notice that compromises are necessary whenever there are wound and unwound strings on the same bridge of any piano, including mid-sized grands. I wonder if it is done on larger pianos for tonal more than scaling reasons.
BDB:
Please post your tuning methods and especially your theories when you can. I am genuinely interested.
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The next post is the sequence for tuning an ET temperament that I've been using lately.
I am not as experienced in writing temperament sequence as others, and can only hope that those reading the sequence will understand what I am doing. If not hopefully they will ask for clarification and not view it as nonsense because it was not written in a way that they could understand.
More than anything, I hope it will encourage other members to post how they deal with the questions raised at the beginning of this Topic.
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E-G# Sequence Comments
The sequence spans a tenth so that all M3s can be checked for progressively faster beat rates. This also gives a number of octaves so that the beat rate ratio of the P4s and P5s can be determined. And, the four sets of CM3s each span an octave for the best placement of the notes. Both the SBIs and the CM3s within these octaves are adjusted for the pianos iH and desired octave stretch.
E3 is included because on small pianos it is often in the bass section. This way, the difference in the beat rate of intervals when crossing the break can be determined and adjustments to the temperament can be made if desired.
The order that the notes are tuned follows the circle of fifths, but the last four notes are rough tuned earlier in the sequence to complete the four sets of CM3s for checking the tempering of the P4s and P5s. And as mentioned above, there are also the additional octaves that are tuned.
There are two ways to look at how the sequence forms the temperament. It can be looked at as a string of SBI’s whose beat rate is checked and corrected by sets of CM3s. Or as the four sets of CM3s being tuned after one note of each has been tuned and correctly spaced from each other. Then they are refined by SBIs.
There are three additional checks for tuning the SBIs that are not included, as such, in classic SBI sequences.
The first is the octave check. When making adjustments to SBIs, it must be decided whether only the P4s, only the P5s or both need adjustment. By having the first SBIs tuned within an octave, when a P4 or P5 is adjusted, the beat rate of the P5 or P4 within the same octave can be used to check the other SBIs of the same interval.
The second check helps determine the difference in the iH of the two notes that form any SBI. This can help make the decision on compromises before any RBIs are tuned. When tuning F3-C4 (or any P5) listen to the difference in beat rates between G#3-C4 and F-G#3. (G#3 does not need to be tuned accurately.) On a well scaled piano, the difference in beat rate will be about the same as the difference between two M3s that are a whole step apart when the F3-C4 is 2 cents narrow. For a challenging piano, with F3 being unwound, the compromise of making the F3-F4 octave, the F3-C4 P5 and the yet-to-be-tuned F3-G#3 m3 wider; and the yet-to-be-tuned G#3-C4 M3 narrower; can be estimated. With the F3 being wound, the compromises are reversed. This technique allows a compromise to be made without going back and forth through the sequence to see how it will work out. When tuning G3-C4 the test note is D#4.
The third check is the CM3s check. Regardless of the sequence used, contiguous major thirds will beat at a rate of about 4:5 from each other. Since they also repeat, they can be used to check each other. This sequence waits to tune any set of CM3s until the first M3 of each of the 4 sets of CM3s is tuned. It also uses them more as a check of the M3 that was tuned, rather than tune it directly. This allows more flexibility in tuning challenging pianos when the octave stretch may need to change within the temperament octave. The sets of CM3s should still have progressive beat rates. Otherwise, the chromatic M3s will not beat progressively either. But on challenging pianos, it may be best to have the progression nonlinear.
E-G# Sequence Steps
The 7:8 ratio check is the same beat rate change as M3rds a whole step apart.
The 15:16 ratio check is the same beat rate change as M3rds a half step apart.
If a check does not prove the interval, either go back and find the error, or slightly over-correct the note and work backwards to adjust the previous intervals.
1. Tune C4 to a pitch source.
Tune C3 to C4 with desired octave stretch.
2.Tune F3 to C3 and C4.
Check with m3-M3 (F3-G#3 beats 8 times for every 7 that G#3-C4 beats. If the beat rate is too fast or too slow to judge, adjust G#3.)
Tune F4 to F3 with desired octave stretch.
3. Tune G3 to C3 and C4.
Check with m3-m6 (C4-D#4 beats 8 times for every 7 that G3-D#4 beat. If the beat rate is too fast or too slow to judge, adjust D#4.)
Check that G3-C4 beats no faster than C4-F4.
Tune G4 to G3 with desired octave stretch.
Check F3-C4 beats no faster than C4-G4
4. Tune D4 to G3.
Check with m3-M3 test.
(F3-D4 should beat about 8bps)
5. Tune A3 to D4.
Check with m3-m6 test. (If the D4-F4 beats too fast to hear, and you don’t want to retune F4, the F3 can be used for a M3-M6 test instead. However, this is testing lower partials, and will give different results on a challenging piano.)
Rough Check F3-A3 by rough tuning C#4 to A3 and F4. (CM3s check: There should be about a 4:5 beat speed ratio between F3-A3 and A3-C#4, and between A3-C#4 and C#4-F4.)
Check that A3-D4 beats no faster than D4-G4.
6. Tune E4 to A3.
Check with m3-M3 test.
Tune E3 to E4 with desired octave stretch.
Check that E3-A3 beats no faster than A3-D4.
Rough Check C4-E4 by rough tuning G#3 to C#4, E3 and C4. (CM3s check)
Rough Check G#3-C4 beats 15 times for every 16 times A3-C#4.
*Time spent on getting A3 and E4 correct will be worth it to avoid large errors later. G#3 and C#4 can be used to shim each other and the beat speed checked with other 4ths. But the purpose is to prove that A3 and E4 can fit in their CM3s, not to tune G#3 and C#4. Until they are tuned to a SBI, it is not definite where they belong, especially on a small piano.*
7. Tune B3 to E3 and E4.
Check beat rate with other 4ths and 5ths. The m3-m6 test is no longer needed.
Rough Check G3-B3 by rough tuning D#4 to G#3, G4 and B3. (CM3s check)
Rough Check G3-B3 beats 15 times for every 16 times G#3-C4 beats.
Check that G3-B3 beats no faster than F3-D4. (The more octave stretch, the more difference in beat rates.)
8. Tune F#3 to B3.
Check beat rate with other 4ths.
Check that F#3-B3 beats no faster than B3-E4.
Tune F#4 to F#3 with desired octave stretch.
Rough Check D4-F#4 by rough tuning A#3 to D#4, F#3 and D4. (CM3s check)
Rough Check that C#4-F4 beats 15 times for every 16 times D4-F#4 beats.
Rough Check D4-F#4 beats 15 times for every 16 times D#4-G4 beats.
Or if the above two rough checks beat too fast to hear, then:
Check G3-E4 beats 7 at least 7 times for every 8 times F#3-A3 beats.
Check F#3-A3 beats no more than 15 times for every 16 times C4-E4 beats.
Check D4-F#4 beats no slower than A3-C4.
*These last four notes have already been rough tuned and complete the 4 CM3s. Tuning these last 4 notes should really be a final adjustment. Any tests can be used, but will not all be noted here.*
9. Tune C#4 to F#3 and F#4.
Checks.
10. Tune G#3 to C#4.
Checks
Tune G#4 to G#3.
Checks.
11. Tune D#3 to G#3 and G#4.
Checks.
12. Tune A#3 to D#4, F3 and F4.
Checks, Checks, Checks.
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I'm following this thread. It is really interesting and I think I can learn so much trying everyone's sequences.
This is the first time I see the 7:8 and 15:16 ratios, I am not sure I can estimate them. It is also the first time I hear about m3/M3, m3/m6 tests.
I suppose that is why this sequence seems to me a bit complicated. But I'm curious and I'll give it a try and I will refer you the results. I have two pianos to practice: a horrible Melodigrand 40" tall console made in 1968, and a fine brand new Petrof 46" tall studio.
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Upright, I have a doubt. In step 8. Tune F#3 to B3, and then F#4 to F#3 you give several checks: Check F#3-A4 beats no more than 15 times for every 16 times C4-E4 beats.
I don't understand this check. Can you please explain more on this? Maybe there is a typo...
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Gadzar:
I suppose the checks in this sequence are complicated, although the order that the notes are tuned is not. Please understand that I am not advocating that anyone else use it. It just happens to be what I use. I’ll try to explain the concepts you asked about more thoroughly. These concepts, rather than the individual steps, could be useful in any sequence.
When you play chromatic M3s, the beat rate increases by the factor of the twelfth root of two, or about 1/16 faster for each note. This is the basis of what I call the 15:16 test. For M3s played a whole step apart, the beat rates increase by the square of the twelfth root of two, or about 1/8 faster for each whole step. This is the basis of what I call the 7:8 test. I am used to listening to these beat rate ratios when I am checking RBIs.
The Melodigrand might be a good choice to understand the m3/M3 and m3/m6 checks. Try tuning the lowest unwould string a just fifth from the correct note above. Now listen to the difference in the beat rate of the test. (You may have to adjust the test note to get a beat rate that is easy to hear.) Do the same with the next fifth chromatically lower. I expect there will be quite a difference between the tests of the two intervals. Of course these intervals aren’t tuned just. When they are narrowed by 2 cents, the beat rate of the test will change also. If the iH of the notes are close to each other, the test will have a 7:8 ratio. Otherwise, a compromise is necessary.
Yes I made a typo in step 8. Thank You for pointing it out. If that’s the only mistake, I’ll be surprised. I’ll make the correction. The check should read “Check F#3-A3 beats no more than 15 times for every 16 times C4-E4 beats.â€
Regards,
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So: For the 16:15 ratio: the 12th root of 2 is 1.05946309 and the ratio 16/15 equals 1.06666666 which is near enough to be considered the same. So for crhomatic intervalls the speed of the beat rates increases by an amount of 16 to 15 approximatly. What is less obvious is the 8:7 ratio. To test a 5th you use m3-M3 test. The m3 (coincident partials 6:5) should beat faster than the contiguous M3 (coincident partials 5:4) by a ratio of 8:7. You are comparing the 6th partial of the lower note to the 4th partial of the higher note of the 5th. If the m3-M3 were to beat at the same rate it would give a 6:4 5th and by making the m3 beat slightly faster you get a slightly wider 5th, meant to be a 3:2 5th. To test a 4th you use the m6-m3 test. The m6 (coincident partials 8:5)should beat faster than the m3 (coincident partials 6:5) by a ratio of 8:7. If the m6-m3 were to beat at the same speed it would give a 8:6 4th and by making the m3 beat slightly faster you tune a slightly narrower 4th, meant to give a 4:3 4th. I used the words "slightly faster". Why a 8:7 ratio? I don't know. But I don't care neither, as far as it works. What I don't understand yet is the test: The check should read “Check F#3-A3 beats no more than 15 times for every 16 times C4-E4 beats.†It is a m3-M3 test, but the 3rds are not contiguous and they don't build up a 5th.
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Gadzar:
The check you mention is a variation of the M3 inside M6 outside check. I think of it as the Dominant 7 check, because it is made from the notes of a dominant 7 chord. One inversion of the M3 inside M6 outside check is when a m3s beat rate is compared to a M3 beat rate with the M3 being a P4 above the m3 (Edit- the lower notes of the two intervals are a P4 apart). Since the B3-D#4 interval is not available as a check, the C4-E4 interval can be used with the variation that the F#3-A3 beats no more than 15 for every 16 times that the C4-E4 beats. Something to keep in mind when using this inversion of the M3 inside M6 outside test is the m3 (being narrow of just) will beat slower for pianos with greater iH and for octaves that have greater stretch. That is why I say “no more thanâ€.
I think it is good that you understand the concepts and using them with your own terms!
Regards,
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Posts: 9,230 |
Congratulations for your descriptive detailled, , Upright. That is some job !
Professional of the profession. Foo Foo specialist I wish to add some kind and sensitive phrase but nothing comes to mind.!
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Joined: Dec 2007
Posts: 839
500 Post Club Member
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OP
500 Post Club Member
Joined: Dec 2007
Posts: 839 |
Kamin:
Thank you for the nice comment.
Gadzar:
I made an error in my last post, that I have corrected. I said dim5 when I should have said P4. Sorry.
Part-time tuner
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Joined: Dec 2006
Posts: 2,758
2000 Post Club Member
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2000 Post Club Member
Joined: Dec 2006
Posts: 2,758 |
Upright:
I'm finished with the Melodygrand console. I have tried your sequence. It was a disaster.
I use to have a bad time when tuning SBIs. But with some patience I can tune a very acceptable ET. I usually use the M6-M10 test for 3:2 5ths and the M3-M6 for the 4:3 4ths. As you know I have a verituner and therefore I can check my aural tunings against it.
But I'am not used to the m3-M3 and m6-m3 tests, so the 4ths and 5ths I have tuned today didn't work at all.
Can the iH of the piano change the 8:7 ratio for these tests? If not I guess it's me that just can't estimate an accurate 8:7 ratio.
Plain wire strings in this piano start at F3. From F3 to G#3 there are bi-chords. Three-chords start at A3.
The break is between E3 (last wound bi-chord) and F3.
When tuning pure 5ths E3-B3 and F3-C4 there was a difference in the m3-M3 test as you said.
Sorry... I´ll keep trying.
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