
#1246071  08/09/09 06:14 AM
Re: Modern tuning theory from a mathematical persp
[Re: Silverwood Pianos]

5000 Post Club Member
Registered: 11/13/08
Posts: 5295
Loc: Bradford County, PA

".....The point here is that once we have calculated A3 = 219.819514 hz, which gives us a 2.00164 ratio with A4, we can not and should not take the 12th root of 2.00164 as the semitone ratio to calculate the notes between A3A4. By doing so we are imposing the iH of only those two notes to all of the remaing notes in the octave, ignoring the iH of each individual note. In fact there is no constant semitone ratio. This ratio changes at each subsequent note......"
I agree that both the semitone ratio and the octave ratio are not constant with a fixed octave type. But I do not know of a practical way to show a calculated a variable semitone ratio to post here, and noted this in an earlier post. But understanding that the octave ratio is variable should bring understanding that beat rate ratios are also variable. Why would only CM3s have a fixed beat rate ratio? Wouldn’t all beat ratios be fixed then, including the semitone ratio? But we have already agreed that the semitone ratio is not fixed. Or how about the beat ratio of 5ths that is generally understood not to be fixed either. There has not been shown any reason to believe that one beat rate ratio would be fixed and others not.
Of course any fixed beat rate ratio, including CM3s, can be used to tune across a break in scaling. I would not expect other beat rates to be smooth as a result, though. Arbitrarily choosing one interval to be perfectly smooth at the cost of others is not compromising.
_________________________
Jeff Deutschle PartTime Tuner Who taught the first chicken how to peck?

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#1246148  08/09/09 11:26 AM
Re: Modern tuning theory from a mathematical persp
[Re: UnrightTooner]

2000 Post Club Member
Registered: 04/02/08
Posts: 2481
Loc: Niagara Region, On. Canada

Jeff, quite often a compromise is defined as a surrendering of objectives which leaves no one happy in the end. A customer may in fact favour leaning towards perfection of one intervals' smooth progression of beat rates at a higher than normal cost of another. This would be rare I admit, but consider that 2 different tuners could reach a compromise that varies from each other because of their personal preferences of perfection (eg. SBI's vs RBI's). It would be nice to see a topic posted in which tuners could express their opinions on WHY they prefer to lean one way or the other (if they do). It could lead back to a mathematical relationship or, one that is based on a musical cause. An example would be the fact that M3rds, trichords/tetrachords and are not incorporated into the left hand chords down into the lower tenorbass as much as 5ths and larger spanse intervals. If these are compromised more and you end up with smoother rates on the intervals more prevalently used by musicians, a compromise is achieved that has the most relevant effect.
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Piano Technician George Brown College /85 Niagara Region

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#1246395  08/09/09 08:07 PM
Re: Modern tuning theory from a mathematical persp
[Re: daniokeeper]

2000 Post Club Member
Registered: 12/15/06
Posts: 2180
Loc: Mexico City

The practical way to calculate the frequencies that meet your specifications, be it slow or fast beating intervals, is using a computer, cycling through an algorithm which leads you to the correct values.
Take for example the pendular movement, there is no mathematical solution to describe this movement. Mathematics have not come to a solution, but that doesn't prevent watchmakers from making good and accurate pendulus clocks!
In the same way, even if there is no mathematical solution to the problems posed by E.T. tuning! That doesn't mean tuners can not calculate accurate partials to divide the octave in 12 semitones that meet their specifications. Computers are here to make the hard work and can do thousands of iterations through an algorithm in a few miliseconds.
That is why we must welcome technology advances, as ETDs, in our dayly work and take profit of the ressources they provide.
Of course you can calculate a partial or a beat rate using a simple calculator or even by hand. But you can not calculate the 4 CM3s as I did here with this same calculator. Only a computer can do this.
You can not estimate actual partials by listening to beats! You have to measure actual partials with an ETD and from that numerical data then you calculate your tuning.
I did it for the Knabe upright I am repairing, the partials are:
A4 = 0, 1.50, 5.43, 9.72 F4 = 0, 0.38, 2.33, 5.68, 10.18 C#4 = 0, 0.87, 1.06, 3.57, 6.32 A3 = 0, 0.05, 1.79, 3.16, 5.4, 7.62 F3 = 0, 0.16, 1.36, 2.79, 4.42, 6.69
And taking 4:2 octaves A3A4, F3F4 we obtain the following set of CM3s:
A4 = 440 hz F4 = 349.218652 hz C#4 = 277.139621 hz A3 = 219.789153 hz F3 = 174.366426 hz
F4A4 = 13.518418 bps C#4F4 = 10.699191 bps A3C#4 = 8.467906 bps F3A3 = 6.701949 bps
CM3 common beat rates ratio = 1.2634991
We confirm that for tuning CM3s in this particular piano, the ratio is not 1.25 but 1.26, and it is posible to have the same ratio for all the four CM3s, even if iH of strings has lots of jumps and is not in a smooth progression.
If we take a close look at iH data we can see that:
The second partial of C#4 is negatif, 0.87 cents. It constitutes a jump in iH distribution. In theory it should be positif and hihger than the second partial of A3 which is 0.05 cents.
We can also see that contrary to expected second partial of F3=0.16 cents is higher than second partial of A3=0.05 cents.
The third partial of C#4=1.06 cents is lower than the third partial of A3=1.79 cents, in theory it should be higher.
There is evidently a problem with C#4's center string. Trying to solve this I measured left and right strings of C#4 and I've got similar results. So the problem is not in the wires but somewhere else. Maybe in the bridge or in the V bar or even in the scale of the piano.
All these irregularities must be taken into account when tuning this piano or any other piano. We can not measure two or four or any given number of partials and from there calculate the hole temperament! We must measure all of them! That's what we do in aural tuning, we hear at all the partials off each and every string.
The more partials are used to calculate the temperament, the more accurate and "Equal" it will be.

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#1246578  08/10/09 07:09 AM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

5000 Post Club Member
Registered: 11/13/08
Posts: 5295
Loc: Bradford County, PA

Emmery:
Sounds like a great idea for a Topic. I have tried to discuss the compromises across a break and it tends to be a train wreck. If you start one I will surely join in.
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Jeff Deutschle PartTime Tuner Who taught the first chicken how to peck?

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#1246580  08/10/09 07:13 AM
Re: Modern tuning theory from a mathematical persp
[Re: UnrightTooner]

5000 Post Club Member
Registered: 11/13/08
Posts: 5295
Loc: Bradford County, PA

Joe:
Thanks for sharing that quote. This may point out the difference between RBI and SBI tuners. I particularly like the how old and new reverses itself through generations. Long and short hair on men seems to be another example. Do they still call Classical music “longhaired” music like they did in the fifties?
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Jeff Deutschle PartTime Tuner Who taught the first chicken how to peck?

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#1246584  08/10/09 07:49 AM
Re: Modern tuning theory from a mathematical persp
[Re: Silverwood Pianos]

5000 Post Club Member
Registered: 11/13/08
Posts: 5295
Loc: Bradford County, PA

Gadzar:
It is very nice to be having productive exchanges with you and sincerely hope that we can continue them.
Yes computers can do amazing things. I prefer do work out what I can on a calculator, or even with a pencil if it is not too complicated, because it helps me to grasp concepts better.
When considering empirical data and theoretical data, what can be significant for one type can be insignificant to the other and visa versa. For example, in your earlier post, you stated that the CM3 ratio was 1.250. On closer examination, when including a few more decimal places, the ratio is slightly more than this and is larger for the upper intervals than the lower intervals. Now of course this is insignificant when tuning aurally on a piano with this scale, but for understanding whether the CM3 ratio is theoretically fixed or variable it is significant.
I am not sure when a mathematical procedure must have an algorithm and when it does not. For instance, the CHAS algorithm seems unnecessary. Delta is an interim value. The algorithm could be reduced to an equation. So I am not convinced that determining theoretical iH frequencies must be done with an algorithm (although I suppose that anything dealing with roots that produce irrational numbers has algorithms already.)
But let’s look more closely at the CM3 algorithm. First it only works for CM3s. It cannot be used to calculate the other 9 notes of the chromatic scale. Second, it requires that a fixed CM3 ratio be used. If you remember, the SAT manual mentions that on challengingly scaled pianos the CM3 ratio must vary so that the octaves work out. But also, consider different tuning styles. If the a fixed CM3 ratio works on a particular piano with 4:2 octaves, would you expect it also to work for a compromise of 4:2 and 6:3 for the temperament octave, a 6:3 octave below the temperament and a 4:2 above the temperament? Actually, I wouldn’t be surprised if on some pianos a fixed CM3 ratio would only be possible with a variable octave type. I suppose that the CM3 algorithm can be altered to allow a progressive CM3 ratio (as can be done aurally), but it still leaves the other 9 notes undefined.
I guess it seems inefficient to work out the frequencies using beat rates when the octave ratios and cent deiviations are already available.
_________________________
Jeff Deutschle PartTime Tuner Who taught the first chicken how to peck?

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#1246748  08/10/09 01:33 PM
Re: Modern tuning theory from a mathematical persp
[Re: UnrightTooner]

2000 Post Club Member
Registered: 12/15/06
Posts: 2180
Loc: Mexico City

In my opinion the CM3's ratio is not important. This algorithm is not designed to find a CM3's ratio. It is designed to find 4 CM3's that build up a given common octave width for A3A4 and F3F4.
So why divide the octave in major thirds? Can not we calculate fifths or fourths or sixths? Yes we can, but it is not practical.
There are only two intervals that build up an octave: major thirds and minor thirds, i.e. F3A3C#4F4 and F3G#3B3D4F4.
Building up an octave using another type of intervals involves the use of two different intervalls, i.e. fifths and fourths: F3C4G3D4A3E4B3F#3C#4G#3D#4A#3F4, and it is much more complicated to find an algorithm that can calculate all this fifths and fourths too meet our specifications (beat rates).
So the simplest way to divide the octave is by using major thirds.
Just the same problem as when tuning aurally. It is much more easier to find 3 major thirds that fit in a correct octave, than finding twelve fifths/fourths that fit the same octave. Actually no tuner goes from the 1st to the twelfth fifth/fourth without using and checking other intervals.
The CM3s ratio is not fixed. What is fixed is the width of the octave A3A4. From there you divide the octave in CM3s, the algorithm does calculate the proper CM3s ratio which build up the specified octave's width. So the algorithm will work fine on a compromise of a 6:3 and 4:2 octave or whatever compromise you prefer.
To find out the other 9 notes I can imagine an algorithm that assumes a common width for the fourths in the temperament octave and then adjust it by multiple iteration to fit with the already calculated notes. For example: taking the 9 note minitemperament sequence of Dr Albert Sanderson F3A#3F#3B3, C#4G#3C4G3 which gives G3B3 as a check interval to adjust the width of the previously tuned fourths. This way we are using partials: 3, 4 and 5.
Another possibility is designing an algorithm that divides the octave in Contiguous minor 3rds, starting with a estimated beat rates ratio of 6:5. It will give you G#3, B3 and D4, from where you can calculate the missing chaines of CM3s: F#3A#3D4, G3B3D#4, G#3C4E4. With the latter algorithm we will be using partials 4, 5 and 6.
In my opinion the best would be to find an algorithm that uses all of the predominant partials in the temperament octave: partials 2, 3, 4, 5 and 6, to establish a compromise between octaves (6:3 and 4:2), fifths (3:2 and 6:4), fourths (4:3), major thirds (5:4), minor thirds (6:5), and major sixths (5:3) all at the same time.
Edited by Gadzar (08/10/09 01:36 PM)

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#1246915  08/10/09 07:14 PM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

3000 Post Club Member
Registered: 08/21/02
Posts: 3646
Loc: Madison, WI USA

That is VERY good thinking, Rafael. I had never considered it before but I think it would be possible to build a very accurate temperament between F3 and F4 first by constructing the three CM3s within it (with the outside F4A4 M3 as the controller), then a chain of m3s similarly. F3G#3, G#3B3, B3D4 and D4F4. That would leave: F#3, G3, A#3, C4 and D#4.
Tune first the CM3s, F3A3, A3C#4, C#4 F4 and F4A4. Tune first the octave A3A4 to whichever octave type is chosen, using the appropriate test. Next, estimate the F3A3 M3. Tune F4 from F3 to the same type octave as A3A4, using the appropriate test. Tune C#4 from A3 temporarily so that the F3A3 and A3 C#4 M3s beat exactly the same. Then sharpen C#4 so that the F3A3 and A3 C#4 CM3s beat proportionately at a 4:5 ratio. Check the 4:5 progression of the CM3s, F3A3, A3C#4, C#4 F4 and F4A4. If the relationship is found to be improper, adjust F3, F4 and C#4 again, all slightly sharper or flatter, each the same direction by about the same amount. Be sure to check that the F3F4 octave is again the same size as the A3A4 octave.
The four notes, F3, A3, C#4 and F4 will then be highly reliable but not necessarily 100%. If at any time during the process of constructing the rest of the temperament, it is deemed necessary to adjust any of these notes slightly, it should and must be done, bearing in mind that any such adjustment will require the reevaluation of all four pitches, F3, A3, C#4 and F4. Each one, if any should require an extremely small adjustment. The A3 should be considered the most reliable of all, the F3 and F4 second and the C#4 the least yet each should still be within a very small margin of error if the first set of CM3s were properly set according to a reasonable standard of aural beat rate perception.
The problem would be how to estimate the m3s without going between them excessively until the sequence of m3s produces smoothly ascending m3s at an approximate ratio of 5:6. Listen first to the C#4F4 M3 (10.9 theoretically) then tune G#3 temporarily so that the F3G#3 m3 is equal beating to the C#4F4 M3, then sharpen G#3 slightly so that the F3G#3 m3 beats slightly slower (9.4) than the C#4F4 M3 (10.9); that will be a good starting estimate. Now, listen to the F4A4 M3 (13.8). Temporarily tune the D4F4 m3 equal beating with F4A4, then flatten D4 slightly so that the D4F4 m3 beats slightly faster (15.8) than the F4A4 M3 (13.8).
The G#3 now forms a 4th with C#4. Does the G#3C#4 4th have the appropriate beat (0.9 theoretically but probably very close to 1 beat per second)? If the G#3C4 4th sounds either too pure or too tempered, flatten or sharpen it accordingly and compare again the F3G#3 m3 with the C#4F4 M3 for the slightly slower/faster relationship. When both the tempering of the G#3C#4 4th and the relationship of the F3G#3 m3 and C#4F4 M3 agree, you now have a reliable F3G#3 m3 and a reliable G#3C#4 4th and thus a reliable G#3.
The D4 now forms a 4th with A3. Listen for the proper amount of tempering of the A3D4 4th. It's theoretical rate is almost exactly 1.0, so it should sound very similar if not virtually equal beating to the G#3C#4 4th. If the A3D4 4th sounds either too pure or too tempered, flatten or sharpen D4 accordingly and compare again the D4F4 m3 and the F3A3 M3 for the slightly faster/slower relationship. When both the tempering of the A3D4 4th and the relationship of the D4F4 m3 and F4A4 M3 agree, you now have a reliable F3G#3 m3 and a reliable G#3C#4 4th and thus a reliable G#3 and D4.
Now, tune G#3 so that the G#3B3 m3 and the F3G#3 m3s are equal beating. Then, flatten B3 so that the G#3B3 m3 beats slightly faster than the F3G#3 m3 (an approximate 5:6 ratio). Now, tune the B3 so that the B3D4 and D4F4 m3s are equal beating, then sharpen B3 so that the B3D4 m3 beats slightly slower than the F4F4 m3 (an approximate 5:6 ratio).
Listen to all four contiguous m3s (Cm3s) for an evenly and slight progression of beat rate between all four (a 5:6 ratio, a smaller difference from the 4:5 ratio of the CM3s). If there is an apparent error, remember that the A3 and C#4 can be good references for the two 4ths, G#3C#4 and A3D4. Only the very slightest adjustments should be necessary.
Now, go back to the remaining untuned notes, F#3, G3, A#3, C4 and D#4. Tune F#3 temporarily to C#4 as a pure 5th, then compare it to the 4th, F#3B3 which will beat strongly. Sharpen F#3 so that the F#3C#4 5th beats very slightly slower than the F#3B3 4th (0.6 vs. 0.8 theoretically, a 3:4 ratio; a very small difference). Check the newly formed CM3s, F#3A#3 and A#3D4. If the 4:5 ratio of CM3s seems improper, sharpen of flatten F#3 accordingly and then check again the corresponding 4th and 5th formed from F#3 (F#3B3 and F#3C#4). Adjust F#3 very slightly until all three interval checks agree: 4th, 5th and CM3s.
Skip G3 for the moment. Both A#3 and C4 can be tuned from F3 and F4 with good available checks. Temporarily tune A#3 between F3 and F4 so that both the resultant 4th and 5th sound apparently pure. Temporarily tune C4 from both F3 and F4 so that both the resultant 5th and 4th sound apparently pure. (The choice of octave size may affect the purity or beatlessness of the 4ths and 5ths. A wider choice of octave such a 6:3 may result in a very slight beat when both the 4th and 5th are made as beatless as possible between F3 and F4).
Sharpen A#3 so that the F3A#3 4th and the A#3F4 5th both beat at slightly less than 1 beat per second theoretically (0.8) but depending on the type of octave chosen, the 5th will beat slightly slower than the 4th in a wider type such as 6:3. The proof of correct tempering will be now to compare the F#3A#3 and A#3D4 CM3s. If the relationship seems improper, very slightly sharpen or flatten A#3 until all three interval checks agree: 4th, 5th and CM3s.
Flatten C4 so that the F3A#3 4th and A3C4 5th both beat proportionately, the 4th slightly faster than the 5th, (1.9 vs. 0.6 theoretically). Compare the G#3C4 and C4E4 CM3s. If the relationship seems improper, adjust C4 so that the two CM3s, G#3C4 and C4E4 have the proper 4:5 ratio. Compare the 4th and 5th again. A wider choice of octave may result in the 4th beating twice as fast as the 5th (a 2:1 ratio, about 2 beats per second vs. 1 beat per second in this area of the scale). The CM3 test should prevail.
Go to G3. G3 can now be tuned as a 4th and from C4 as a 4th and D4 as a 5th. Temporarily tune G4 from C4 as a pure 4th. Compare with the G3D4 5th which will beat strongly. Flatten G4 until the G4 4th beats slightly faster than the G3D4 5th (0.9 vs. 0.7 theoretically). Depending on the octave size, the 5th may beat about half the speed as the 4th. Listen to the G3B3 and G3E4 M6. The M6 should beat slightly faster than the M3 (7.8 vs. 8.9 theoretically). Also, the "inside M3" G3B3 M3 should beat about the same (quasi equal beating, 7.8 vs. 7.9 theoretically) as the "outside" F3D4 M6. If either test seems improper, adjust G3 very slightly sharper or flatter until the 4th, 5th and both M6 tests agree.
Finally, the D#4 may be tuned similarly to the way G3 was tuned but there is an additional CM3 test. Temporarily tune D#3 from A#3 as a pure 4th. Compare the G#3D#4 5th which will beat strongly. Sharpen D#4 until the A#4D#4 4th beats slightly faster than the G#3D#4 5th (1.1 vs. 0.7 theoretically). Depending on the octave type chosen, the 4th may beat nearly twice as fast as the 5th. Compare the CM3s G3B3 and B3D#4. Also compare the F#3A#3 M3 and F#3D#4 M6 for a slightly slower faster relationship (7.3 vs. 8.4 theoretically) and for quasi equal beating, the "inside" G#3C4 M3 and the "outside" F#3D#4 M6 (8.2 vs. 8.4 theoretically). Adjust D#4 slightly sharper or flatter so that all interval relationships agree.
Since no ET sequence is ever perfect, a final check for smooth progressions of all intervals must be done. When a slight error is found, check first the CM3s on either side of a suspected pitch. Then check 4ths and 5ths. Remember that the 4th will beat slightly faster than the 5th and may be about twice as fast with a wider octave. Check M3M6 intervals as available and the quasi equal beating "inside M3, outside M6" tests (example: G3B3 and F3D4 ((7.8 vs. 7.9)) when available). All 4ths should apparently sound similar and all 5ths should apparently sound similar. All M3s. m3s and M6s should progress evenly to within a very small distinction chromatically.
The theoretical beat rates to the nearest tenth were obtained from Owen Jorgensen's book, "Tuning...". The suggestion of tuning a 4th or 5th pure first then tempering it, as well as tuning a rapidly beating interval first as equal beating to another was obtained from both Jorgensen publications, "Tuning..." and the "Handbook of Equal Beating Temperaments..". To his credit, BraideWhile also suggested the technique of tuning a 4th or 5th pure first as a reference point in his 1946 edition of "Tuning and Allied Arts".
Upon Rafael's suggestion, I came up with the above idea spontaneously. It still seems awkward compared to other simpler ideas. However, I will keep it in my files and upon reflection and comments from others, I may publish it in the future as yet another approach to tuning and equal temperament which keeps errors to a minimum as each step progresses. All comments are invited.

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#1247112  08/11/09 07:46 AM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

5000 Post Club Member
Registered: 11/13/08
Posts: 5295
Loc: Bradford County, PA

Gadzar:
Something that may help in constructing a ladder of Cm3s is remembering the outside M6 inside M3 test and its inversion: the m3 noncontiguous M3 test. These both form a dominant seventh chord. So the FD M6 should beat the same as the GB M3, but since the M3 is not available, the FD M6 can be tuned to beat between the beat rate of FA and AC#. Likewise, the FG# m3 should beat the same as the A#D M3, which again is not available, so instead it should beat between the beat rate of AC# and C#F.
You would think that this second test would produce a m3 that is too narrow and beats too fast, but unless a 6:3 octave or wider is tuned, the m3 will beat faster than the M6 in the test’s inversion, in which case the M6 would also beat faster than its M3. As long as a typical stretch is used, on a well scaled piano, these beat speed relationships will be correct. Remembering these concepts may help if individual steps get confusing. Also, if additional octaves are tuned with the Cm3s, then a ladder of M6s can be checked for progression, also.
I have been looking at different methods of nonlinear interpolation and noticed that Excel has a LOGINT function. It might be worth looking at.
You posted:
” In my opinion the best would be to find an algorithm that uses all of the predominant partials in the temperament octave: partials 2, 3, 4, 5 and 6, to establish a compromise between octaves (6:3 and 4:2), fifths (3:2 and 6:4), fourths (4:3), major thirds (5:4), minor thirds (6:5), and major sixths (5:3) all at the same time.”
My tuning sequence does this, but you would not care for its origin. (I say this ironically, and hope you can enjoy the humor in it.)
_________________________
Jeff Deutschle PartTime Tuner Who taught the first chicken how to peck?

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#1247151  08/11/09 09:19 AM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

2000 Post Club Member
Registered: 12/15/06
Posts: 2180
Loc: Mexico City

I tried the "Contiguous Major/Minor Thirds Temperament" in a Knabe full uprigth. It worked out wonderfully! Quick and easy!
I did not follow exactly the sequence suggested by Bill Bremmer, I came out with a sequence to tune a span of a thenth, from F3 to A4, which seemed to make more sense, tuning contiguous minor thirds and checking with 4ths, 5ths and other intervals where available.
It has a great level of accuraty, not compounding errors and no guessing, only two estimations for the notes G#3 and D4, which can be fine tuned three and one steps forward.
I’ve found that hearing at the contiguous minor thirds is not that difficult. In fact it seems to be even more comfortable than hearing at major 3rds. The beats seem to be more distinct in minor thirds. It is easy to appreciate their increasing beat rate if we take a chain of 4 or 5 contiguous minor thirds and play them successively in a paused way, one per second approx. That is why I tuned a tenth instead of an octave, to better appreciate the progression.
Here is the sequence:
First I tune CM3s F3A3C#4F4A4 as always. Then I tune the Cm3s beggining with G#3 as Bill suggested. It forms a 4th with the already tuned C#4 and a m3 with F3 also tuned before.
Steps 1 to 5 are not detailed here, Bill Bremmer has explained them in great detail in numerous places.
Contiguous MajorMinor Thirds Temperament
1. Tune A4 to fork. Test F2Fork = F2A4.
2. Tune A3 as a 6:3/4:2 octave from A4. Test M10M3, m3M6.
3. Estimate F3 as a M3 form A3, beating at approximately 7 bps.
4. Tune F4 as a 6:3/4:2 octave from F3, the same width than A3A4. Test M3M10 and m3M6.
5. Tune C#4 as a M3 from A3 beating in a 4:5 ratio with F3A3. If C#4F4 and F4A4 M3 don’t fit the progression of the other CM3s adjust F3, F4 and C#4 consequently to have a smooth progression in the beat rates of CM3s: F3A3C#4F4A4.
From this point we begin to tune chains of Cm3’s, starting with F3G#3B3D4F4G#4.
6. Estimate G#3. Minor third F3G#3 has a theoretical beat rate of 9.4 bps, it should then beat faster than A3C#4 (8.7 bps) and slower than F4A4 (11.0 bps), so it is really easy to tune it by first making a beatless fourth G#3C#4 and then flattening G#3 until F3G#3 beats at a mean rate between the beat rates of F4A4 and C#4F4 M3s. In addition we can check the tempering of G#3C#4 fourth to beat at near 1 bps.
The tuning of G#3 will be refined in step 9.
7. Tune G#4 as an octave compatible with A3A4 and F3F4 octaves tuned before. (M3M10, m3M6 tests). F4G#4 should beat twice the beat rate of F3G#3. Check C#4G#4 fifth.
8. Estimate D4. We can first tune a beatless fourth A3D4 and then tempering by sharpening D4 to beat at nearly 1 bps, almost equal beating with G#3C#4. Furthermore D4F4 m3 should beat faster than F4A4 M3. In fact this is the m3M3 test for a 6:4 fifth D4A4.
The tuning of D4 may be refined in the following step.
9. Tune B3. All we have to do is tune B3 to make G3#B3 m3 beat slower than B3D4 m3, but faster than F3G#3 m3. For that we can tune first a pure m3 G3B3 and then flatten B3 until G#3B3 beats faster than F3G#3 but slower than B3D4. Check for a smooth progression between the Cm3s F3G#3B3D4G#4. Theoretical ratio 5:6.
If the beat rate of B3D4 can not be fitted with the other minor thirds progression (F3G#3, G#B3, D4F4), then it will be necessary to adjust the tuning of G#3 and D4 and then retune G#4 and B3.
For example if B3D4 beats too fast, that means B3 is too sharp and we must flatten G#3, G#4 and B3. Or D4 is too flat and we must sharppen it. Checking the fourths G#3C#4 and A3D4 will give us some light about which of these two notes is in fault. Maybe we should fine tune the two of them.
From here there are many ways to continue because there are plenty of intervals available (m3s, M3s, 4ths and 5ths) to test with the notes already tuned. We choose to tune a chain of Contiguous minor thirds (Cm3s): F#3A3C4D#4F#4A4, striving for a smooth progression in their beat rates (theoretical ratio 5:6).
10. Tune F#3. First tune a beatless fourth F#3B3 and then flatten F#3 to make F#3A3 m3 beat slightly faster than F3G#3 m3. Test F#3B3 fourth (M3M6 test) and F#3C#4 fifth (m3M3 test).
11. Tune F#4 as an octave compatible with octaves tuned before (M3M10 and m3M6 tests). F#4A4 m3 should beat twice as fast as F#3A3.
12. Tune C4. First tune a beatless 5th F3C4 and then flatten C4 to makeA3C4 beat at the 5:6 ratio with F#3A3, slightly faster than G#3B3. Major third G#3C4 should beat slower than A3C#4 M3. Test F3C4 fifth and C4F4 fourth.
13. Tune D#4, first as a beatless fifth with G#3 then flatten D#4 to make C4D#4 m3 beat at the proper ratio (5:6) with A3C4. Compare C4D#4 m3 slightly faster than B3D4 and B3D#4 M3 faster than A3C#4. Test fifth G#3D#4, fourth D#4G#4.Major sixth A3F#4 must beat almost the same as B3D#4 M3 (M6inside M3 test)
At this point we can verify the tuning of F#3. Minor thirds D#4F#4 and F#4A4 must fit the progression of the Cm3s F#3A3C4D#4. If they don't we must adjust F#3, then F#4, C4 and D#4.
If for example F#4A4 beats too fast, that means F#4 is too sharp, so F#3 is also too sharp, we must flatten F#3 and repeat steps 11, 12 and 13.
Now we will tune the last chain of Cm3s: G3A#3C#4E4G4
14. Tune A#3, first as a beatless 4th from F3, then sharpen A#3 to make A#3C#4 m3 beat faster than A3C4 m3 but slower than B3D4 m3. Check F#3A#3 M3 beats faster than F3A3 M3. Test F3A#3 4th, A#3F4 fifth. Major sixth G#3F4 must beat almost the same as A#3D4 M3 (M6inside M3 test)
15. Tune G3 as a beatless 4th from C4, then flatten G3 to make G3A#3 m3 beat at a 5:6 ratio with A#3C#4. It must beat faster than F#3A3 m3, but slower than G#3B3. G3Compare G3B3 M3 with G#3C4 and F#3A#3 M3s. Check G3C4 4th and G3D4 fifth.
16. Tune G4 as an octave compatible with octaves tuned before (M10M3 and m3M6 tests). Compare D#4G4 M3 with D4F#4. Check C4G4 5th, D4G4 4th..
17. Tune E4 as a beatless 4th from A4, then flatten E4 to make C#4E4 beat at a nearly 5;6 ratio with A#3C#4. Check all possible intervals with E4 (m3, M3, 4th, 5th, 6th, up and down).A#3G4 must beat almost the same as C4E4 M3 (M6inside M3 test)
If beat rate of E4G4 doesn’t fit the progression of Cm3s, we must retune A#3 consequently and repeat from step 15.
Play runs of m3s, M3s, 4ths, 5ths, 6ths, 8ves, M6insideM3’s tests, etc. and fixe any uneveness found.
For easy comparison between M3s and m3s I give the following theoretical beat rates.
I must insist that these are theoretical beat rates which are not actually tuned on any piano. They serve exclusively for comparison purposes, between given intervals.
For example we can see that F3G#3 minor third has a beat rate of 9.4 bps and , A3C#4 major third has a beat rate of 8.7 bps and C#4F4 major third has a beat rate of 11.0 bps, so we can conclude that F3G#3 should beat slightly faster than A3C#4, but slower than C#4F4 and we can tune G#3 to do so, but the actual beat rates would be different from 9.4, 8.7 and 11.0 bps.
Major 3rds beat rates in bps (A4 = 440 hz).
F3A3 = 6.9 F#3A#3 = 7.3 G3B3 = 7.8 G#3C4 = 8.2 A3C#4 = 8.7 A#3D4 = 9.2 B3D#4 = 9.8 C4E4 = 10.4 C#4F4 = 11.0 D4F#4 = 11.7 D#4G4 = 12.3 E4G#4 = 13.1 F4A4 = 13.9
The minus sign in the beat rates of minor thirds is to show that they are narrow from pure, i.e. 6th partial of lower note has a greater frequency than 5th partial of upper note.
minor thirds beat rates in bps (A4 = 440 hz).
F3G#3 = 9.4 F#3A3 = 10.0 G3A#3 = 10.6 G#3B3 = 11.2 A3C4 = 11.9 A#3C#4 = 12.6 B3D4 = 13.3 C4D#4 = 14.1 C#4E4 = 15.0 D4F4 = 15.8 D#4F#4 = 16.8 E4G4 = 17.8 F4G#4 = 18.8 F#4A4 = 20.0
Edited by Gadzar (08/11/09 09:44 AM)

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#1247162  08/11/09 09:44 AM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

5000 Post Club Member
Registered: 11/13/08
Posts: 5295
Loc: Bradford County, PA

”For example we can see that B3D#4 major third beats at 9.8 bps and , F3G#3 minor third beats at 9.4 bps, so we can conclude that B3D#4 should beat slightly faster than F3G#3 and we can tune them that way, but the actual beat rates would be different from 9.4 and 9.8 bps.”
You should have a different opinion on this if you include iH and octave type in your calculations. Oddly, the ratio of M3 beat rates to m3 beat rates change in an unexpected way.
_________________________
Jeff Deutschle PartTime Tuner Who taught the first chicken how to peck?

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#1247171  08/11/09 09:59 AM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

2000 Post Club Member
Registered: 12/15/06
Posts: 2180
Loc: Mexico City

Once more Tooner: we are dividing the temperament octave into Major and minor thirds, listenning to actual partials, so iH is already taken into account!
We are checking not only thirds, but also 4ths, 5ths, 6ths and 8ves, ALL OF THEM. So partials 2, 3, 4, 5 and 6 of each note are tuned in a convenient, compromised, balanced way!
CAN'T YOU UNDERSTAND WHAT WE ARE DOING HERE?!!!!!
Il m'agasse ce gars! Il m'énerve!

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#1247178  08/11/09 10:09 AM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

3000 Post Club Member
Registered: 08/21/02
Posts: 3646
Loc: Madison, WI USA

Quite excellent, Rafael! I want to keep this idea going and eventually publish it in the PTG Journal. I will surely keep your name attached to it as the person who initiated the idea. It certainly does make sense to expand the temperament all the way to A4. This idea is so refined that I believe it would be useful for constructing a master tuning for a PTG tuning exam. It is obviously a fine tuning sequence. When a master tuning committee constructs a master tuning, they begin with a tuning which is already almost perfect. The m3s serve to split the F3F4 octave very nicely and control the tempering of the 4ths and 5ths which beat so slowly that anything from a pure interval to 2 beats per second can sound ok as it stands alone.
It was sad to hear of Owen Jorgensen's passing. His very last article emphasized the value of m3s. In previous articles I have written, I have said that they are useful all the way down to about F2. When expanding the temperament downward into the Bass, the chromatic and Cm3s can be used to provide the ultimate refinement. Other great technicians such as Jim Coleman, Sr. have also suggested this. Whatever irregularity there is in scaling does not matter. When CM3s and Cm3s are used, the 4ths and 5ths will also follow suit and a perfect ET can be constructed upon any scale.

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#1247181  08/11/09 10:12 AM
Re: Modern tuning theory from a mathematical persp
[Re: Bill Bremmer RPT]

5000 Post Club Member
Registered: 11/13/08
Posts: 5295
Loc: Bradford County, PA

Gadzar:
What I am saying is that iH affects beat rates in unexpected ways. Include iH in your math and see what happens.
_________________________
Jeff Deutschle PartTime Tuner Who taught the first chicken how to peck?

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#1247186  08/11/09 10:23 AM
Re: Modern tuning theory from a mathematical persp
[Re: Bill Bremmer RPT]

3000 Post Club Member
Registered: 08/21/02
Posts: 3646
Loc: Madison, WI USA

Rafael, if you haven't figured it out by now, whatever anyone else says is wrong according to Tooner and he can prove it! If you should happen to agree with what he says, then he will change his mind, find new math to prove once again that you are wrong and he is right. His basic premise is that the kind of pianos he tunes, all the crummy types because he is not really a working piano technician who earns his living by tuning pianos, cannot be tuned. He has proven that to himself and desires to prove it to everyone else too. It doesn't matter to him that other technicians routinely do what he says cannot be done because he has already proven that it cannot, therefore, they are all wrong.
PTG is wrong, all of the exam master tunings are wrong, all of the modern methods which use CM3s are wrong, all of the books are wrong except Braide White who is the only one who was ever right. Owen Jorgensen was wrong too. He should never have written anything about unequal temperaments. They make Tooner's skin crawl. Only Tooner's version of ET, the one he has to alter from ET for the pianos he tunes is right. You won't ever be able to tell him anything, only he will be able to tell you that whatever you think you know is wrong, he is right and can prove it.

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#1247201  08/11/09 10:47 AM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

2000 Post Club Member
Registered: 12/15/06
Posts: 2180
Loc: Mexico City

I don't have to include iH in my maths to see what happens.
Tooner, I tune pianos in the way I want them to sound. To speak clearly:
In theoretical E.T. F3G#3 beats slower than F4A4 and faster than C#4F4, right?
So, I tune pianos to sound that way. With F3G#3 beating faster than C#4F4 and slower than F4A4. The actual beat rates? I don't care. But the relationship between them? Yes! I do care! I also tune narrow fifths, beating slower than fourths, wide fourths of course, and progressing (wide) major thirds, tenths and major sixths. With narrow minor thirds, and...etc, etc, etc. That's what I tune in pianos. I like to call it E.T. I repeat for you: If it is calculated as E.T., and sounds as E.T. then it is E.T.
Happy?

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#1247208  08/11/09 10:59 AM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

5000 Post Club Member
Registered: 11/13/08
Posts: 5295
Loc: Bradford County, PA

Gadzar:
Theoretically F3G#3 beats at nearly the same rate as A#3D4. When a typical stretch is used on a well scaled piano, F3G#3 will beat more nearly like B3D#4.
I question everything I think, do or say when it comes to tuning. I continue to look for those that I can discuss the math with. I keep hoping that you may be one, but I guess not.
_________________________
Jeff Deutschle PartTime Tuner Who taught the first chicken how to peck?

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#1247219  08/11/09 11:13 AM
Re: Modern tuning theory from a mathematical persp
[Re: UnrightTooner]

2000 Post Club Member
Registered: 12/15/06
Posts: 2180
Loc: Mexico City

Are you saying that in a well scaled piano the beat reat of F3G#3 won't be between those of C#4F4 and F4A4? Or maybe that in a well scaled piano F3G#3 woul beat faster than F4A4? And, in a well scaled piano, C#4F4 would beat faster than F4A4? I don't get your point. You say: What I am saying is that iH affects beat rates in unexpected ways. Include iH in your math and see what happens. Please, tell me what iH makes to beat rates of F3G#3, C#4F4 and F4A4? Are their beat rates affected in a way that couterdicts what I have said? i.e.: C#4F4 < F3G#3 < F4A4 Please answer that!
Edited by Gadzar (08/11/09 01:13 PM)

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#1247231  08/11/09 11:32 AM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

5000 Post Club Member
Registered: 11/13/08
Posts: 5295
Loc: Bradford County, PA

Gadzar:
My last post said exactly what I meant, including an ackonwledgement that we cannot discuss the math. Your last post confirms this.
_________________________
Jeff Deutschle PartTime Tuner Who taught the first chicken how to peck?

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#1247357  08/11/09 03:04 PM
Re: Modern tuning theory from a mathematical persp
[Re: UnrightTooner]

2000 Post Club Member
Registered: 12/15/06
Posts: 2180
Loc: Mexico City

Tooner,
Would you please include iH in calculating beat rates of M3s and m3s and post the results here so I can work with them? Can you?
I don't know how to make such calculations. (you say iH affects beat rates in an unexpected way).
I don't know of a formula that calculates accurately iH of piano's strings. I know a lot of formulas that intend to do it, but none of them, to my knowledge is accurate enough to actually tune a piano with its predictions.
I don't know a single ETD which can calculate a decent tuning without intervention and adjusting of stretch at different points by the human tuner. (I am not sure of this one, in regard of Mr. Stopper's Onlypure software, I have heard the user makes an adjustment at the beginning of the tuning but I don’t know what he adjusts for sure.)
Some ETD programmers take empirical data directly measured in pianos and put them into their software.
For example, in Tunelab, they use an empirical formula, with an empirical table of data to calculate iH constants for a given set of notes and from there calculate a tuning curve.
I don't know how Accutuner calculates its FAC tunings, but they say they only measure three notes and from there they calculate the entire tuning curve.
Verituner takes readings of all partials as we tune each string, so it works on actual partials and thus has not to use an expression of iH in its calculations. (I guess)
I confess I really don't know how they make their calculations.
Maybe Young? Have I to measure string's diameters, lengths, tensions, mass, rigidity, etc...before tuning the piano? Do partials duplicate in frequence every eight notes? In every piano?
This will be usefull maybe to design a piano, but can you calculate iH beat rates of a real piano with it?
Do you really want me to put here the inharmonic beat rates of M3s and m3s? Anything else?
Are you really aware of what you are asking me?
But even if I could include here an accurate expression of iH and calculate "inharmonic" beat rates of any given piano (not an easy task!), the resulting tuning would be exactly the same I already described.
The only difference would be that this tuning will be expressed numerically in terms of Hertz and cents, instead of "aurally" expressed: i.e.: in terms of uncalculated beat rates ratios and progressions, which in spite of not being "calculated" they are tunable for sure.
It is the tuner which tunes the piano and he imposes the beat rates. Math's only express numerically what is done by the tuner.
The fact that beat rates are affected by iH is not a recent discovery. Even Braid White knew it 100 years ago.
But you focus your attention in the fact that I took theoretical beat rates without iH, which surely will turn out being valid in this particular case. And you say nothing about the sequence I propose.
For a long time I have been searching a sequence to set the temperament in a piano, where I have no guesses to make. Where the piano has no surprises reserved when I went to tune it. A sequence where I am in absolute control over the tuning, knowing at each step where I am and where I am going to.
I think I have finally found such a sequence. All I have to decide by myself is the width of the temperament octave, the remaining is only appreciating and adjusting beat rates, no more decisions to make. I divide the initial octave A3A4 in equal M3s of whatever width is needed, then I divide it again in equal m3s of the exact width to build up that octave. All this while having absolute control over the other kind of intervals involved in the tuning: fifths, fourths, sixths and octaves, which are the result of the only one decision I took at the beggining, namely: the width of the initial A3A4 octave.
This allows me to harmoniously balance all of the intervals in the temperament. There are no surprises, there is no guessing, there are no subjective ways of tempering fifths and fourths. It is all pure, simple and objective beat rate appreciating and setting.
I thank Bill Bremmer for being the one who found the way to successfully divide the octave in Cm3s: tuning G#3 and D4 in the first place and wrote here the way of doing it. “El resto es pan comido” as we say in spanish.
As I said, I have tried to do it in several ocasions but I failed to find the right way to tune the Cm3s.
Edited by Gadzar (08/11/09 03:37 PM)

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#1247470  08/11/09 07:34 PM
Re: Modern tuning theory from a mathematical persp
[Re: Gadzar]

1000 Post Club Member
Registered: 02/01/09
Posts: 1176
Loc: PA

Rafael, I hope I'm not making comment that is too stupid or obvious, but since we're leaving the "math" and going back to "aural" for now, I've been thinking about how to decide on the correct octave width for the initial temperament. To me, the ideal temperament really is an 88note temperament, not a 13note or 17note temperament. (Of course, typing it and accomplishing it are two very different things. ) All intervals compromised as far as what is reasonable, including octaves. When deciding about the A4 to A3 octave, why not also temporarily tune the other A's (A1, A2 A5, A6) and temper them all to achieve your best compromise between each other (A4:A2, A1:A3, A3:A5, etc.) and let that be your guide on the width of A4:A3? You could also do the same on the F3:F4. Then use your CM3's and Cm3's. Again, I'm sorry if this seems obvious. It's just that I never seem to read about tempering octaves. I do read about stretch. I do read about octave types. I do read about using other intervals to check octaves. But, I never seem to read much about using other octaves and octave multiples to temper octaves while tuning aurally (in addition to the other checks).
Edited by daniokeeper (08/11/09 08:20 PM)

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#1247497  08/11/09 08:26 PM
Re: Modern tuning theory from a mathematical persp
[Re: daniokeeper]

3000 Post Club Member
Registered: 08/21/02
Posts: 3646
Loc: Madison, WI USA

Joe, I wrote about the idea of tempered octaves long ago: http://web.archive.org/web/20031203194817/www.billbremmer.com/TemperedOctaves.htmIt's actually very simple but the very idea that octaves are not pure freaks some people out. I agree completely with you however that the entire range of the piano should reflect the temperament. None of the ETD programs view the outer octaves as a compromise between the double octave and octave and 5th (12th). In ET, they may very well approximate it but by default, not design. When the temperament is slightly unequal as is my usual practice, the ETD programs miss the boat entirely. I often see ideas that want to create a "temperament" over a larger area than one octave. However, I have never seen that as having any particular advantage. Once the temperament octave is refined, expanding it beyond one octave, replicating it both higher and lower is simply the next task. No serious tuner ever tunes a temperament octave and merely tunes octaves up and down from that. Changing inharmonicity in various parts of the scale make what would be theoretically possible quite impossible. Rafael: you have reached the usual impasse with Tooner. Once you do not agree that he is right, there is no further basis for discussion. He contradicts himself: inharmonicity has no significant effect on beat rates in one post but in another, he demands to know what effects it has and demands that you show how much effect there is in all of the infinite possibilities there are. It is a game. It is a game that only he wins. It has nothing to do with any practical application, it is only a game designed to prove that no piano can actually ever be tuned according to theory. We already all know that. Only BraideWhite's book has the answers, no others do and no other method is valid. CM3s or Cm3s cannot be accurate. They were not in BraideWhite's book, so they cannot work. Tooner and only Tooner knows how to tune a piano properly. You must agree with that or there is no basis for discussion.

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#1247599  08/11/09 10:45 PM
Re: Modern tuning theory from a mathematical persp
[Re: Bill Bremmer RPT]

2000 Post Club Member
Registered: 12/15/06
Posts: 2180
Loc: Mexico City

Yes, I suppose I would better not get upset by what Tooner says. In this same thread at the beggining he was saying that changes in beat rates due to iH were insignificant. Erus:
On the one hand, if an interval does not sound long enough to tell the difference, there is no difference. On the other hand if you do not tune to the proper beat rates, the tuning will be in error.
But if there is a significant difference that cannot be heard, sometimes this difference will be additive and other times subtractive. The error in the final result can be expected to be insignificant anyway. I am sure you understand statistics. As I have mentioned in other Topics these effects are largely selfcorrecting when it comes to comparing iH to noniH theoretical beat rates. . If someone is interested I can show the mathematical analysis that I use to understand this. M3s may very well be the most selfcorrecting interval. This gives us actual beat rates of:
A3D4: 1.001 bps F3A3: 6.959 bps
Comparing this to the theoretical beat rates of:
A3D4: 0.992 bps F3A3: 6.930 bps
This gives us differences in beat rates between noniH theoretical pitches and those calculated with iH of:
A3D4: 0.009 bps F3A3: 0.029 bps
I was going to determine just how much of a cent difference this is, but do not see a point to it. Since the nearly coincident partial is about 880, each cent will be equal to about 0.5 bps. The difference is much less than what could be perceived, let alone actually tuned. I consider this to be insignificant. He uses theoretical beat rates: Likewise, the FG# m3 should beat the same as the A#D M3, which again is not available, so instead it should beat between the beat rate of AC# and C#F. But, when I do use theoretical beat rates, in the very same intervals, in the very same way he did, then he says: You should have a different opinion on this if you include iH and octave type in your calculations. Oddly, the ratio of M3 beat rates to m3 beat rates change in an unexpected way. And finally he doesn't want to discuss: ... including an ackonwledgement that we cannot discuss the math... Well, I hope next time I will resist the temptation of playing his game again. I have tuned the two pianos I have mentioned before. A spinet and a Baby Grand. In both of them I made a pitch raise with my Verituner and then a second pass fine tuning using the CM3Cm3 Sequence. In setting the temperament I had no problem at all. It worked out as expected, with no surprises. For the baby grand, I did not like the tuning calculated by Verituner in the bass section, so I tweaked it a little. I think I am on the correct way. About the hole piano being an 88 notes temperament: I agree, that is the ideal. When starting to tune aurally, I used to tune from A3A4 out by octaves up to the top and down to the bottom. And testing each octave with fifth and fourth (where audible). Then I played a run of 10ths and/or 17ths looking for uneveness in the progression of the beat rates. Now, I've abandoned the tuning of octaves and I tune, ASAP, a compromise between double octaves and 12ths, the "mindless octaves" of Bill Bremmer in both ways going up to the treble and down to the bottom. I follow this procedure: First I play the octave, say F4F5 and tune the upper note first to a beatless octave, then I go sharp from beatless just a little and then I play the 15th (F3F5) and depress the sustain pedal while flattening and playing the upper note (F5). Then I compare with 12th (A#3F5), depress the pedal and adjust F5 until I have equal beating between the 15th F3F5 and the 12th A#3F5. I believe one must compare and tune bass and treble notes to the central octave, using wider intervals as one reaches the extremes of the keyboard, using 10ths, 17ths, 19ths, etc... But, I confess I don't tune this way! Until now, what has worked better for me is the "mindless" 15th12th equal beating compromise. It is really "mindless". I mean: if you check your tuning going down to the bottom by playing a run of 10ths, then 17ths, etc. there is a progression of beat rates, which I've found often ends with pure, beatless intervals. And when checking you look for uneveness in that progression to correct the tuning of notes in fault. But when you tune the bass or the treble with an equal beating 15th12th, then there is no progression, you do not bother of beat rates accelerating or decelerating, you only strive for equal beating intervals, it becomes "mindless". And the most amazing of that is when you are finished. The bottom meets (or fits?) the treble perfectly as if you were tuning directly the one to the other by means of double, triple or quadruple octaves! I have never tried to tune A2A3 or A4A5 octaves as if it was the temperament octave. Except when tuning the double octave A2A4 temperament of Dr. Albert Sanderson. I don't know if it is possible or desirable at all.
Edited by Gadzar (08/11/09 11:33 PM)

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#1247754  08/12/09 07:22 AM
Re: Modern tuning theory from a mathematical persp
[Re: daniokeeper]

5000 Post Club Member
Registered: 11/13/08
Posts: 5295
Loc: Bradford County, PA

Actually, I am starting to realize that the change in beat rates due to iH, separate from additional stretch, may be significant after all (for me that is 0.2 cents or more.) But this is through mathematical investigation. A blanket statement with no support is not convincing. But if it is greater than the error from tuning an initial set of CM3s may have is another question.
It is easy to imagine the effects of iH as stretching a rubber band with marks on it. This analogy is true for additional octave stretch when the iH does not change, such as the difference between tuning 4:2 or 6:3 octaves. It does not hold true when going from harmonic to inharmonic tones nor from a piano with less iH to one with more.
Gadzar, you have shown that you have the tools and knowledge to investigate this but may not realize it. Consider the difference between the theoretical beat rate of F3A3 and what your algorithm calculated. It was a little bit slower, not faster as would be expected. Best of luck.
_________________________
Jeff Deutschle PartTime Tuner Who taught the first chicken how to peck?

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