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#2303917 - 07/19/14 02:24 AM Re: Theoretical tuning sequence [Re: DoelKees]
Gadzar Offline
1000 Post Club Member

Registered: 12/15/06
Posts: 1726
Loc: Mexico City
Kees,

You deserve a free copy of your own book. As the M3/M6 test you propose is not Mark Cerisano's test.

You see, what Cerisano proposes is a m3/M3 test, i.e. FG#/BD#.

I've seen his video. He checks his temperament with the normal M3/M6 test.

He plays:

F3D4/G3B3
F#3D#4/G#3C4
G3E4/A3C#4
G#3F4/A#3D4

and here, as his temperament is set from F3 to F4, he can not check A3F#4/B3D4, because F#4 is not yet tuned. So instead of playing A3F#4 he plays the inversion F#3A3 and continues as follows:

F#3A3/B3D#4
G3A#3/C4E4
G#3B3/C#4F4

Of course this is wrong, because the 6:3 octave is narrow and the m3 beats faster than its inversion, the M6.

I don't know when nor how but since he made this video, he has discovered his error and now he lowers a half step the m3 hopping it would beat the same as the M3, so instead of testing F#3A3/B3D#4 he tests F3G#3/B3D#4.

In sum what he tests is:

F3G#3/B3D#4
F#3A3/C4E4
G3A#3/C#4F4

For the Steinway D this gives:

F3G#3/B3D#4 10.1/9.8 0.3 bps.
F#3A3/C4E4 10.8/10.3 0.5 bps.
G3A#3/C#4F4 11.4/10.9 0.5 bps.

The normal M3/M6 test gives

A3F#4/B3D#4 10.1/9.8 0.3 bps.
A#3G4/C4E4 10.7/10.3 0.4 bps.
B3G#4/C#4F4 11.3/10.9 0.4 bps.

So the Cerisano's proposal is no better than the usual M3/M6 test.

But if we lower the m3 a half step again, as I suggested in my post

Originally Posted By: Gadzar
There is no "mathematical proof". It all depends on how much you stretch (contract) the 6:3 octaves.

If you have a pure 6:3 octave F#3F#4 then F#3A3=A3F#4 and with the inside M3 outside M6, we have B3D#4=A3F#4=F#3A3

But if you tune slightly narrow 6:3 octaves then you can have B3D#4=F3G#3
Or even B3D#4=E3G3.



Then we have:

E3G3/B3D#4 9.5/9.8 0.3
F3G#3/C4E4 10.1/10.3 0.2
F#3A3/C#4F4 10.7/10.9 0.2

This is better than the normal M3/M6 test.


What you propose is to modify the normal M3/M6 test by lowering the M6 a half step, without inverting it to a m3, which of course takes out of play the contracted 6:3 octave, and you compare the beat rates of:

B3D#4/G#3F4 9.8/9.5 0.3 bps.
C4E4/A3F#4 10.3/10.1 0.2 bps.
C#4F4/A#3G4 10.9/10.7 0.2 bps.

which is better than the normal M3/M6 test.


Anyway, as we can see here, this tests are not accurate, there is not a true equal beating, so I do not use them to set my temperament. I can use them to check and detect erros, but not for tuning.



Edited by Gadzar (07/19/14 02:43 AM)
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#2303923 - 07/19/14 02:37 AM Re: Theoretical tuning sequence [Re: Hakki]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1723
Loc: Vancouver, Canada
Originally Posted By: Hakki
Nice work Kees.

But whether you get a free copy depends on the accuracy of your inharmonicity model.

Did you measure all the ih values? Or what equation are you using for the ih values? Did you verify the accuracy of your ih model?

Right, I left that job for you.

Kees

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#2303924 - 07/19/14 02:39 AM Re: Theoretical tuning sequence [Re: Hakki]
Hakki Offline
2000 Post Club Member

Registered: 05/26/01
Posts: 2559
Originally Posted By: Hakki
Nice work Kees.

But whether you get a free copy depends on the accuracy of your inharmonicity model.

Did you measure all the ih values? Or what equation are you using for the ih values? Did you verify the accuracy of your ih model?


Oh btw, how accurate the Tunelab model can also make your results invalid.
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#2304114 - 07/19/14 04:59 PM Re: Theoretical tuning sequence [Re: Bernhard Stopper]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1723
Loc: Vancouver, Canada
Originally Posted By: Bernhard Stopper
Originally Posted By: Bill Bremmer RPT
Originally Posted By: Chris Leslie
Originally Posted By: Bill Bremmer RPT
Here are two short videos that show how equally beating intervals, when played together, tend to suppress the beating sound and reduce slightly the overall volume of sound.

https://www.youtube.com/watch?v=qiCDrkPzCnI

https://www.youtube.com/watch?v=qnwnA1dXkJM


In the lower CEG sample, the clarity of the beating in the triad is lessened and muddied but I still hear the same beating.


I hear the beating completely disappear after a few seconds.


There is a certain degree of beat masking blending in after some time, caused by the fact that the intervals are NOT equally beating. Maybe Kees can provide us the beat rate differences of your examples?

CE 6.4bps
GE 6.6bps
GC 1.2bps

That's equal beating within the accuracy of my reading of the beat rates from the spectrogram.

In ET GE would beat about 1bps slower than CE.

So if in the example we'd raise G a tiny beat to make GE beat at 5.4bps it should sound better according to you, and worse according to Bill. (The P4 will of course improve a bit from 1.2bps (which is correct for EBVT) to about 1bps which hopefully does not affect our perception of the rapid beats.)

That should be easy to demonstrate.

Kees

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#2304235 - 07/19/14 11:55 PM Re: Theoretical tuning sequence [Re: DoelKees]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1258
Loc: Montreal, Quebec, Canada
Originally Posted By: DoelKees
Originally Posted By: Mark Cerisano, RPT
What makes the relationship unique is that x>y for a different reason that z>y. That's why it is not as trivial as you say. Does that make sense?

Not really. No matter what the reason, it still doesn't follow that x=z unless you assume x=z (i.e., the beat rate differences (x-y and x-z are the same).

Why would they be the same in your example?

Somehow you have to mention that not only x>y and z>y, but also that both x and z are very close to y, in math speak |x-y|<=epsilon and |z-y|<=epsilon with epsilon the typical beat rate difference of your example.

In that case you can conclude |x-z|<epsilon, i.e., x is closer to z than the maximum of |x-y| and |z-y| but x could be larger or smaller than z.

To tighten the bound you need another assumption, namely that the difference between the beat rate differences d=||x-y|-|z-y|| is small, smaller than epsilon. (In your writeup you assumed this to be zero.) Suppose you know that d<q*epsilon with 0<=q<1.

The equations are now:
x>y
z>y
x-y<=epsilon
z-y<=epsilon
|(x-y)-(z-y)|<q*epsilon

Then from the last equation it follows that
|x-z|<q*epsilon

Kees


Touché
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#2304247 - 07/20/14 12:35 AM Re: Theoretical tuning sequence [Re: DoelKees]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1258
Loc: Montreal, Quebec, Canada
Wait, the Cerisano M3/M6 test? What is that? I did what?

My test is the m7b5 test, which is based on the usual 3/6 test being accurate, and the 6:3 octave being narrow by the smallest amount.

But you are doing exactly what I was hoping to find out. In fact, my next question to you was going to be how to calculate partial frequencies based on iH models, for the purpose of calculating beat rates, and there you go and do the whole thing already!

Ok, so if I use your numbers for my m7b5 equality:

Does F3G#3 = B3D#4?

(Each number refers to the four charts you made.)
F3 m3=-9.4/-10.1/-10.1/-10.6
B3 M3=9.8/9.8/9.6/9.6
Differences 0.4/-0.3/-0.5/-1.0 (Where "-" means slower on top)

That does NOT prove my m7b5 test, but shows that the relationship is affected by iH.

How narrow are the 6:3's on each piano?
For A3A4.
A3 m3=-11.9/-13.1/-13.1/-13.9
C4 M6=11.9/12.0/12.0/12.1
Diff = 0/1.1/1.1/1.8

It's funny, because when I tune A3A4, I fit it between 4:2 and 6:3 by making 4:2 wide and 6:3 narrow. I assess the wideness and narrowness by the difference in the beat speed tests.

But, on some pianos, while I hear definite differences between the tests, on others, the differences are almost zero, (as for as I can tell with my current beat speed difference sensitivity), which I conclude means, on those pianos, I can tune 4:2 OR 6:3 and be pretty close to a good octave each time. BUT, whether or not this window is large or zero (and once inverted!?) doesn't seem to depend on the length of the strings, and by inference, the iH of the piano, as your numbers infer.

Comments?

Also, notice how all the P4 increase. That's my final test. (Where SBI is actually more precise than RBI, IMHO. Whodathought?)

BTW, you earned the book a long time ago!
_________________________
Mark Cerisano, RPT
www.howtotunepianos.com

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#2304255 - 07/20/14 01:04 AM Re: Theoretical tuning sequence [Re: Gadzar]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1258
Loc: Montreal, Quebec, Canada
Originally Posted By: Gadzar
Kees,

You deserve a free copy of your own book. As the M3/M6 test you propose is not Mark Cerisano's test.

You see, what Cerisano proposes is a m3/M3 test, i.e. FG#/BD#.

I've seen his video. He checks his temperament with the normal M3/M6 test.

He plays:

F3D4/G3B3
F#3D#4/G#3C4
G3E4/A3C#4
G#3F4/A#3D4

and here, as his temperament is set from F3 to F4, he can not check A3F#4/B3D4, because F#4 is not yet tuned. So instead of playing A3F#4 he plays the inversion F#3A3 and continues as follows:

F#3A3/B3D#4
G3A#3/C4E4
G#3B3/C#4F4

Of course this is wrong, because the 6:3 octave is narrow and the m3 beats faster than its inversion, the M6.

I don't know when nor how but since he made this video, he has discovered his error and now he lowers a half step the m3 hopping it would beat the same as the M3, so instead of testing F#3A3/B3D#4 he tests F3G#3/B3D#4.

In sum what he tests is:

F3G#3/B3D#4
F#3A3/C4E4
G3A#3/C#4F4

For the Steinway D this gives:

F3G#3/B3D#4 10.1/9.8 0.3 bps.
F#3A3/C4E4 10.8/10.3 0.5 bps.
G3A#3/C#4F4 11.4/10.9 0.5 bps.

The normal M3/M6 test gives

A3F#4/B3D#4 10.1/9.8 0.3 bps.
A#3G4/C4E4 10.7/10.3 0.4 bps.
B3G#4/C#4F4 11.3/10.9 0.4 bps.

So the Cerisano's proposal is no better than the usual M3/M6 test.

But if we lower the m3 a half step again, as I suggested in my post

Originally Posted By: Gadzar
There is no "mathematical proof". It all depends on how much you stretch (contract) the 6:3 octaves.

If you have a pure 6:3 octave F#3F#4 then F#3A3=A3F#4 and with the inside M3 outside M6, we have B3D#4=A3F#4=F#3A3

But if you tune slightly narrow 6:3 octaves then you can have B3D#4=F3G#3
Or even B3D#4=E3G3.



Then we have:

E3G3/B3D#4 9.5/9.8 0.3
F3G#3/C4E4 10.1/10.3 0.2
F#3A3/C#4F4 10.7/10.9 0.2

This is better than the normal M3/M6 test.


What you propose is to modify the normal M3/M6 test by lowering the M6 a half step, without inverting it to a m3, which of course takes out of play the contracted 6:3 octave, and you compare the beat rates of:

B3D#4/G#3F4 9.8/9.5 0.3 bps.
C4E4/A3F#4 10.3/10.1 0.2 bps.
C#4F4/A#3G4 10.9/10.7 0.2 bps.

which is better than the normal M3/M6 test.


Anyway, as we can see here, this tests are not accurate, there is not a true equal beating, so I do not use them to set my temperament. I can use them to check and detect erros, but not for tuning.



Hi Gadzar,

I am flattered that you analyzed my video. Thank you. That is what the academic community calls necessary and why I posted it and refer to it on PW because I KNOW nobody is going to let me get away with anything. You guys are great.

However, with my quick reading, I think I see some typos/errors.

After the highest 3/6 test, I used
F3G#3/B3D#4
F#3A3/C4E4
G3A#3/C#4F4

The tests you quoted are wrong, and not mine. I never discovered those tests and have never used those tests. Go back and check the video again. Unless I had a seizure and blacked out (I won't discount that possibility) you won't find them.

"So the Cerisano's proposal [m7b5] is no better than the usual M3/M6 test."

I'll accept that. The 3/6 test is very popular.

But, I am intrigued by your even narrower 6:3 conclusion. In fact, I may devise multiple bisecting window sequences that depend on the conclusion of the 6:3 when tuning A3A4 and F3F4. As you see above, I have found some pianos to have almost 6:3=4:2 and others; a very wide difference, although the degree of narrowness is not easy to assess.

"Anyway, as we can see here, this tests are not accurate, there is not a true equal beating,"

But the question is, how close are they? Within how many cents of perfect ET, defined by Kees' criteria?

"I do not use them to set my temperament. I can use them to check and detect erros, but not for tuning."

If they can detect errors after the tuning, aren't they more accurate than the methods you are now using to tune the temperament?

I don't mean to be facetious. Actually, for me, I did not use these to tune either before I developed the bisecting window sequence. Because, as Bill put it correctly one time, with the 3/6 test, if it doesn't work, you've got four possible notes to correct. I.e. it is only useful when all the notes are tuned to a very high accuracy.

With the bisecting window method, each note is tuned to a very high accuracy right from the beginning. After the tuning there is little or no refining because the technique used to check the tuning, is the same as the one used to make the tuning.

The P4 tests have a smaller tolerance in my experience, so I use them at the end.

The bisecting window method gives a CM3 accuracy for each note as you tune it. In fact, it is more accurate than the CM3 method because the windows get smaller as you tune.

I promise to post it once I have finalized the best presentation format.


Edited by Mark Cerisano, RPT (07/20/14 01:05 AM)
_________________________
Mark Cerisano, RPT
www.howtotunepianos.com

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#2304262 - 07/20/14 02:18 AM Re: Theoretical tuning sequence [Re: DoelKees]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1723
Loc: Vancouver, Canada
I guess everybody better draw there own conclusions from those beatrate tables.

What I get is that usual M3/M6 test should be replaced by (for example):

Not
"G3B3 = F3D4"
but
"E3C#4 < G3B3 < F3D4 and G3B3 should be closer to the leftmost M6."

This of course assumes the M6's are tuned progressive but nobody can apparently get all the M3 and M6 progressive by ear, so I'm not sure if this is of any practical use.

Tests involving m3's have the problem that the m3 is more sensitive to inharmonicity than the M3/M6 as the 6th partial is less stable.

Kees

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#2304267 - 07/20/14 02:52 AM Re: Theoretical tuning sequence [Re: DoelKees]
Gadzar Offline
1000 Post Club Member

Registered: 12/15/06
Posts: 1726
Loc: Mexico City

Originally Posted By: Mark Cerisano, RPT
If they can detect errors after the tuning, aren't they more accurate than the methods you are now using to tune the temperament?


No! The methods I use to tune the temperament are the same used in the M3/M6 test. The methods consist in adjusting the beat rates of M3s, M6s, P4s, P5s and m3s in even smooth progressions.

I do not use the M3/M6 test to tune my temperament because:

1.- The notes involved are not available until the last steps of my sequence. I tune: A4 A3 F3 F4 C#4 A#3 F#3 B3 G#3 C4 G3 D4 D#4 E4 the first available M3/M6 tests are F3D4/G3B3 and G#3F4/A#3D4 and D4 is tuned at step 12 of 14.

2.- If a M3/M6 test proves wrong, there may be from 1 to as much as 4 notes mistuned and it does not informs us which note or notes is/are in fault, nor if they are flat or sharp, nor by how much. We only know that the M3 doesn't beat at the same speed than the M6, but we have no clue on how to correct the error. In order to correct it we must do other tests to identify which notes in which direction and by how much are to be retuned.

For all the above this test is better suited to check but not for setting the temperament.
_________________________
Rafael Melo
Piano Technician
rafaelmelo@afinacionpianos.com.mx

Serving Mexico City and suburbs.

http://www.afinacionpianos.com.mx

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#2304278 - 07/20/14 04:43 AM Re: Theoretical tuning sequence [Re: DoelKees]
Chris Leslie Offline
500 Post Club Member

Registered: 01/01/11
Posts: 625
Loc: Canberra, ACT, Australia
Originally Posted By: DoelKees
I guess everybody better draw there own conclusions from those beatrate tables.

What I get is that usual M3/M6 test should be replaced by (for example):

Not
"G3B3 = F3D4"
but
"E3C#4 < G3B3 < F3D4 and G3B3 should be closer to the leftmost M6."

This of course assumes the M6's are tuned progressive but nobody can apparently get all the M3 and M6 progressive by ear, so I'm not sure if this is of any practical use.

Tests involving m3's have the problem that the m3 is more sensitive to inharmonicity than the M3/M6 as the 6th partial is less stable.

Kees

For high iH instruments, and with a generous stretch in the mid-range, E3C#4 is very close to, or even faster than, G3B3. Given that, the M3/M6 inside/outside test is not consistent enough to use as a temperament setter, or as in a beat rate window.


Edited by Chris Leslie (07/20/14 08:07 AM)
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Piano technician
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#2304293 - 07/20/14 07:08 AM Re: Theoretical tuning sequence [Re: DoelKees]
Bernhard Stopper Offline
Full Member

Registered: 09/22/08
Posts: 211
Loc: Germany
Originally Posted By: DoelKees
Originally Posted By: Bernhard Stopper
Originally Posted By: Bill Bremmer RPT
Originally Posted By: Chris Leslie
Originally Posted By: Bill Bremmer RPT
Here are two short videos that show how equally beating intervals, when played together, tend to suppress the beating sound and reduce slightly the overall volume of sound.

https://www.youtube.com/watch?v=qiCDrkPzCnI

https://www.youtube.com/watch?v=qnwnA1dXkJM


In the lower CEG sample, the clarity of the beating in the triad is lessened and muddied but I still hear the same beating.


I hear the beating completely disappear after a few seconds.


There is a certain degree of beat masking blending in after some time, caused by the fact that the intervals are NOT equally beating. Maybe Kees can provide us the beat rate differences of your examples?

CE 6.4bps
GE 6.6bps
GC 1.2bps

That's equal beating within the accuracy of my reading of the beat rates from the spectrogram.

In ET GE would beat about 1bps slower than CE.

So if in the example we'd raise G a tiny beat to make GE beat at 5.4bps it should sound better according to you, and worse according to Bill. (The P4 will of course improve a bit from 1.2bps (which is correct for EBVT) to about 1bps which hopefully does not affect our perception of the rapid beats.)

That should be easy to demonstrate.

Kees


No, i did not say that if you raise G to beat at 5.4 bps it must sound necessarily better. What i did say is that the masking that was blending in after some time in this example, comes from the tiny difference of bps of the two intervals. And that the masking effect would not blend in, if the beating is exactly the same.



Edited by Bernhard Stopper (07/20/14 07:13 AM)
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www.piano-stopper.de

Salieri: "Mediocrities everywhere, now and to come: I absolve you all! Amen! Amen! Amen!"
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#2304384 - 07/20/14 12:18 PM Re: Theoretical tuning sequence [Re: DoelKees]
Olek Offline
7000 Post Club Member

Registered: 03/14/08
Posts: 7540
Loc: France
So what is more prone to happen with similar beating ?
My gut feeling is still there is an "additive" way and a "substractive one"

WHat is it related to ? phasing, or frequency coupling ?
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#2304415 - 07/20/14 01:38 PM Re: Theoretical tuning sequence [Re: DoelKees]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1258
Loc: Montreal, Quebec, Canada
Re: the 3/6 test. I always heard it wasn't exact. Now I have more motivation to analyze the 6:3 A3A4 and F3F4 at the beginning, to determine the best sequence if I will be using the 3/6 and m7b5 or neighbouring tests to tune the temperament.

However, using the sequence as it is now, allows me to produce an ET temperament that is within my beat speed sensitivity tolerance, meaning I don't have to refine as much.

Again, the real elephant in the room is beat speed sensitivity. If you can't tell if F#3A#3 is between FA and GB, you won't be able to get good accuracy on ET, IMHO.

Does anyone else feel this way? What have you done, what is your approach to developing this sensitivity to the highest level possible. IMHO, this is the information beginning techs really need.

Please don't reply "Tune 1000 pianos". I think we should be beyond that with all the technology we have that can speed up learning.
_________________________
Mark Cerisano, RPT
www.howtotunepianos.com

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#2304472 - 07/20/14 04:11 PM Re: Theoretical tuning sequence [Re: Bernhard Stopper]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1723
Loc: Vancouver, Canada
Originally Posted By: Bernhard Stopper
Originally Posted By: DoelKees
Originally Posted By: Bernhard Stopper
Originally Posted By: Bill Bremmer RPT
Originally Posted By: Chris Leslie
Originally Posted By: Bill Bremmer RPT
Here are two short videos that show how equally beating intervals, when played together, tend to suppress the beating sound and reduce slightly the overall volume of sound.

https://www.youtube.com/watch?v=qiCDrkPzCnI

https://www.youtube.com/watch?v=qnwnA1dXkJM


In the lower CEG sample, the clarity of the beating in the triad is lessened and muddied but I still hear the same beating.


I hear the beating completely disappear after a few seconds.


There is a certain degree of beat masking blending in after some time, caused by the fact that the intervals are NOT equally beating. Maybe Kees can provide us the beat rate differences of your examples?

CE 6.4bps
GE 6.6bps
GC 1.2bps

That's equal beating within the accuracy of my reading of the beat rates from the spectrogram.

In ET GE would beat about 1bps slower than CE.

So if in the example we'd raise G a tiny beat to make GE beat at 5.4bps it should sound better according to you, and worse according to Bill. (The P4 will of course improve a bit from 1.2bps (which is correct for EBVT) to about 1bps which hopefully does not affect our perception of the rapid beats.)

That should be easy to demonstrate.

Kees


No, i did not say that if you raise G to beat at 5.4 bps it must sound necessarily better. What i did say is that the masking that was blending in after some time in this example, comes from the tiny difference of bps of the two intervals. And that the masking effect would not blend in, if the beating is exactly the same.

No beats will ever be exactly the same, making that a tautology.
Or did you have some tolerance (smaller than 0.2) in mind for the equality?

Kees


Edited by DoelKees (07/20/14 04:11 PM)

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#2304495 - 07/20/14 05:33 PM Re: Theoretical tuning sequence [Re: DoelKees]
Olek Offline
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Registered: 03/14/08
Posts: 7540
Loc: France
Why wanting to compare beats only at one particular partial match.?

How efficient is it?
Comparing beats of similar intervals can be precise but from 2 different ones I always find that nebulous.
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#2304513 - 07/20/14 06:46 PM Re: Theoretical tuning sequence [Re: Bernhard Stopper]
alfredo capurso Offline
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Registered: 07/10/07
Posts: 1061
Loc: Sicily - Italy
Originally Posted By: Bernhard Stopper
Originally Posted By: Olek
Accepting beats an not fighting them makes them more discrete in the end, and they also goes well with the piano tone, and the shape it takes.

The consonances I hear/tune are an addition of slow beats (within the 2 octaves framework for instance.


A reduction of beats generally comes along with an increase of consonance, not the other way around.




Hi Bernhard,

I cannot agree with the above statement of yours, perhaps because I relate '..increase of consonance..' to (frequencies and) beats_order and proportion. Infinite (ordered) ways in which beats can blend and add color.

Perhaps I am misinterpreting your statement, would you like to expand?

Regards, a.c.
.
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#2304516 - 07/20/14 06:57 PM Re: Theoretical tuning sequence [Re: DoelKees]
DoelKees Offline
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Registered: 05/01/10
Posts: 1723
Loc: Vancouver, Canada
Originally Posted By: DoelKees

No beats will ever be exactly the same, making that a tautology.
Or did you have some tolerance (smaller than 0.2) in mind for the equality?

I tried an experiment with A3D4F#4 by moving the A3 by 0.5 a cent.
Here are the beat rates:
D4F#4 9.0 bps
A3: -0.5 0 +.5 +1
A3F#4 9.0 8.7 8.4 8.1

Here's the chord for those 4 cases (left to right), the first beating "exactly" equal beating. http://persianney.com/misc/all4.mp3

Can more sensitive ears than mine hear any of the beat masking/cancellation effects discussed?

Kees

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#2304612 - 07/21/14 12:16 AM Re: Theoretical tuning sequence [Re: DoelKees]
Chris Leslie Offline
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Registered: 01/01/11
Posts: 625
Loc: Canberra, ACT, Australia
Originally Posted By: DoelKees
Originally Posted By: DoelKees

No beats will ever be exactly the same, making that a tautology.
Or did you have some tolerance (smaller than 0.2) in mind for the equality?

I tried an experiment with A3D4F#4 by moving the A3 by 0.5 a cent.
Here are the beat rates:
D4F#4 9.0 bps
A3: -0.5 0 +.5 +1
A3F#4 9.0 8.7 8.4 8.1

Here's the chord for those 4 cases (left to right), the first beating "exactly" equal beating. http://persianney.com/misc/all4.mp3

Can more sensitive ears than mine hear any of the beat masking/cancellation effects discussed?

Kees



Well, not claims about having more sensitive ears, but can hardly tell any difference at all with the RBIs. They all beat obviously. If anything, the first chord could be very slightly more clear but that may be just an illusion because it came first. The SBI seems to perturb slightly differently for each chord, but hardly by much.
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#2304757 - 07/21/14 09:43 AM Re: Theoretical tuning sequence [Re: DoelKees]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4939
Loc: Bradford County, PA
Originally Posted By: DoelKees
Originally Posted By: UnrightTooner
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.

I fooled around a bit with this idea, modified to suit my own interest.

Temperament range F3-C#5 so you have all possible M3/M6 tests. Then pick a random M3/M6 test and adjust the upper note to make them equal beating. Unless it's an A. This does not work: it gets worse at every iteration. Picking the lower note is no better.

Not sure why.

Kees


I played around with it some, too. I decided on a different goal - the same as yours with 4ths and 5ths: Given random errors, produce a refining algorithm. I ran into the same sort of problems, things getting worse, at least for some notes.

I tried correcting two notes at once, one from each interval, in hopes that it would be self-correcting. Maybe I'll try a different pair...
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#2304773 - 07/21/14 10:25 AM Re: Theoretical tuning sequence [Re: Mark Cerisano, RPT]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4939
Loc: Bradford County, PA
Originally Posted By: Mark Cerisano, RPT
.....

I know it works in my sequence, but is there any theoretical proof that it is valid?


The M6/M3 equal beating test and its m3/M3 inversion work just because of happenstance. But there are "theoretical proofs" that show where this comes from.

Take the M6/M3 test with the intervals of F3-D4 and G3-B3. And also consider two other intervals: F3-A3, A3-D4. This gives us the following approximate beatrates (disregarding iH and stretch):

F3-A3 7bps
F3-D4 8bps
G3-B3 8bps
A3-D4 1bps

Hmmm, 8-7 = 1 and that is the M3-M6 test for a P4. In otherwords, the reason there is a 1bps difference between F3-A3 and F3-D4 is because A3-D4 is 1 bps.

And hmmm, the ratio of the two M3s, G3-B3 and F3-A3 is 8/7. This is because they are two semitones apart. Since beatrates double each octave, they follow the same pattern as theoretical frequencies. The ratio of frequencies (and beats) two semitones apart is the square of the twelfth-root-of-two, or about 8/7.

See, it is just a happenstance with theoretical 12-TET with 2:1 octaves. It would not happen with 11-TET nor with 2.5:1 octaves.
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Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#2305092 - 07/21/14 09:38 PM Re: Theoretical tuning sequence [Re: Mark Cerisano, RPT]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1723
Loc: Vancouver, Canada
Originally Posted By: Mark Cerisano, RPT

Again, the real elephant in the room is beat speed sensitivity. If you can't tell if F#3A#3 is between FA and GB, you won't be able to get good accuracy on ET, IMHO.

Does anyone else feel this way? What have you done, what is your approach to developing this sensitivity to the highest level possible. IMHO, this is the information beginning techs really need.

Please don't reply "Tune 1000 pianos". I think we should be beyond that with all the technology we have that can speed up learning.

It think it has two components:

1) developing beat speed sensitivity when the beats are clearly audible. Some training app with synthetic tones of even with metronome clicks would be a great learning tool for this I think.

2) actually being able to focus on listening to the correct beat on a real piano where there is interference from higher partials and other distractions. Developing training software for this would probably have to work with recorded samples of a real piano.

I think such training tools would be a vast improvement over the traditional "tune 1000 pianos" method.

Kees

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#2305151 - 07/22/14 12:28 AM Re: Theoretical tuning sequence [Re: DoelKees]
Gadzar Offline
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Registered: 12/15/06
Posts: 1726
Loc: Mexico City
I have been focusing my ears into estimating the amount of tempering in the M3s, instead of counting beats.

For example play two distant M3s and tell which one is more tempered. Obviously the upper third will beat faster but that doesn't mean it is more tempered.

Another exercise I like to practice is to tune CM3s over any note and hear if the resulting octave is good. If it isn't repeat the exercise.

After some time and practice, I have a good idea of how the M3s must sound on that piano, without counting beats. Once I know it I am able to set the temperament in this piano quickly and accurately at once.

So I never really strive to hear which M3 is faster but which is "more tempered" or "harsher".



Edited by Gadzar (07/22/14 12:30 AM)
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#2305253 - 07/22/14 08:26 AM Re: Theoretical tuning sequence [Re: DoelKees]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4939
Loc: Bradford County, PA
Originally Posted By: Mark Cerisano, RPT

Again, the real elephant in the room is beat speed sensitivity. If you can't tell if F#3A#3 is between FA and GB, you won't be able to get good accuracy on ET, IMHO.

Does anyone else feel this way? What have you done, what is your approach to developing this sensitivity to the highest level possible. IMHO, this is the information beginning techs really need.

Please don't reply "Tune 1000 pianos". I think we should be beyond that with all the technology we have that can speed up learning.


Let's say the goal is to have M3s and M6s progressive. That requires an accuracy of +/- 0.2 cents. When tuning aurally, it requires the discernment of the difference of a beat ratio of +/- 1/16 (difference between 16 bps and 17 bps or 8 bps and 8.5 bps). The scaling also has to make it possible to have both M3s and M6s progressive.

Considering that none of these tolerances will be zero, the stability needed is actually greater than 0.2 cents.

Given a particular piano and tuner any of these factors may be the weak link. Like does it really matter if the tuner can discern the difference between 8 bps and 8.5 bps if the pins are horribly jumpy?
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Part-Time Tuner
Who taught the first chicken how to peck?

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#2305460 - 07/22/14 03:20 PM Re: Theoretical tuning sequence [Re: Olek]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1258
Loc: Montreal, Quebec, Canada
Originally Posted By: Olek
Why wanting to compare beats only at one particular partial match.?

How efficient is it?
Comparing beats of similar intervals can be precise but from 2 different ones I always find that nebulous.


Nebulous? Perhaps. Impossible? No. Accurate? Heck ya.

But, I know what you are saying.

Basically, the big difference between tuning with partials using check notes, and just tuning intervals so they sound good, is that the tuner is using an indirect judgement, instead of a direct judgement.

An indirect judgement is when the tuner makes a conclusion about the quality of an interval, by playing test intervals, or check intervals created using the check note, instead of just listening to the interval directly.

Tuning an interval like an octave, so it sounds good directly, means playing the octave, and adjusting one note, while listening to the octave, and making a judgement about when it sounds best.

Tuning an interval indirectly means playing an interval, testing it with check notes, making a decision about which way one of the notes needs to go, making that change (and possibly listening to the direct interval to confirm), then checking the interval with check notes again. Ending with a direct assessment by playing the interval itself.

I can't tell you how many times I've tuned an interval directly, thinking it sounded good, then checked it indirectly with check notes, adjusted it, then re-listened directly to the interval, and it sounds better to me.

It has never happened the other way around. I.e. tuned an interval directly, adjusted it with check notes, and the interval sounds worse.

The only exception I can think of are the low bass strings that have poor iH, and unisons on mismatched strings. Those need to be tuned directly so that all the beating partials can be reduced as much as possible. Perhaps some beat masking is helping here too.

The direct relationship between check notes and the quality of the interval is - if the check intervals are equal, there is no beating at the coincidental partial. If there is only one coincidental partial in hearing range, this is the only one that needs to be considered.

Sometimes the fifth's 2nd coincidental partial can reek havoc when trying to tune fifths using the M6/M10 test.

Octaves have many coincidental partials that we need to be concerned with.

By identifying a certain sound with a certain check interval relationship, you can accurately reproduce the same interval sound. That's precision. Example, tuning midrange octaves between a 4:2 and a 6:3.

Tuning intervals can be done by ear, using a direct approach, but that is the same as tuning A4 to the fork directly, instead of using F2. Here, the check note gives us more accuracy and precision.

It's the same with other intervals, if the check interval relationships have tight tolerances. I.e., trying to just make an octave a wide 4:2 without checking the 6:3 can lead to a wide 6:3 and an octave that could sound better. Also, the precision of octaves produced with only 4:2 checks is low.
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#2305464 - 07/22/14 03:33 PM Re: Theoretical tuning sequence [Re: Gadzar]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1258
Loc: Montreal, Quebec, Canada
Originally Posted By: Gadzar

Originally Posted By: Mark Cerisano, RPT
If they can detect errors after the tuning, aren't they more accurate than the methods you are now using to tune the temperament?


No! The methods I use to tune the temperament are the same used in the M3/M6 test. The methods consist in adjusting the beat rates of M3s, M6s, P4s, P5s and m3s in even smooth progressions.

I do not use the M3/M6 test to tune my temperament because:

1.- The notes involved are not available until the last steps of my sequence. I tune: A4 A3 F3 F4 C#4 A#3 F#3 B3 G#3 C4 G3 D4 D#4 E4 the first available M3/M6 tests are F3D4/G3B3 and G#3F4/A#3D4 and D4 is tuned at step 12 of 14.

2.- If a M3/M6 test proves wrong, there may be from 1 to as much as 4 notes mistuned and it does not informs us which note or notes is/are in fault, nor if they are flat or sharp, nor by how much. We only know that the M3 doesn't beat at the same speed than the M6, but we have no clue on how to correct the error. In order to correct it we must do other tests to identify which notes in which direction and by how much are to be retuned.

For all the above this test is better suited to check but not for setting the temperament.



Hi Rafael,

I assume you tune the CM3's by creating an accurate C# first, then following with an accurate F4? That's what I do.

But then for A#, what do you have?
A# from F? C#3F3<C#3A#3 and C#3F4<C#3A#3? That leaves A#3 on the same side of both tests. I.e. A#3 can be too sharp and still confirm the tests.

The level of accuracy of tuning A#3, IMHO, is much lower than that of the CM3's. But then again it depends on a tuner's aural beat difference sensitivity. If it's poor, his CM3's won't be of much help.

But by using the 3/6 equality indirectly, where GB = FD, we can tune D so that FA<FD<AC# and D must be tuned so that FD is exactly between FA and AC#.

High accuracy, early on in the temperament, using the 3/6 equality, not the test.


Edited by Mark Cerisano, RPT (07/22/14 03:34 PM)
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#2305474 - 07/22/14 03:41 PM Re: Theoretical tuning sequence [Re: UnrightTooner]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1258
Loc: Montreal, Quebec, Canada
Originally Posted By: UnrightTooner
Originally Posted By: Mark Cerisano, RPT
.....

I know it works in my sequence, but is there any theoretical proof that it is valid?


The M6/M3 equal beating test and its m3/M3 inversion work just because of happenstance. But there are "theoretical proofs" that show where this comes from.

Take the M6/M3 test with the intervals of F3-D4 and G3-B3. And also consider two other intervals: F3-A3, A3-D4. This gives us the following approximate beatrates (disregarding iH and stretch):

F3-A3 7bps
F3-D4 8bps
G3-B3 8bps
A3-D4 1bps

Hmmm, 8-7 = 1 and that is the M3-M6 test for a P4. In otherwords, the reason there is a 1bps difference between F3-A3 and F3-D4 is because A3-D4 is 1 bps.

And hmmm, the ratio of the two M3s, G3-B3 and F3-A3 is 8/7. This is because they are two semitones apart. Since beatrates double each octave, they follow the same pattern as theoretical frequencies. The ratio of frequencies (and beats) two semitones apart is the square of the twelfth-root-of-two, or about 8/7.

See, it is just a happenstance with theoretical 12-TET with 2:1 octaves. It would not happen with 11-TET nor with 2.5:1 octaves.


Maybe not, but these are wide intervals. Stretch would increase all their beat rates. Perhaps not identically, but my work is to create a method and sequence that beginners can use to get repeatable results, once they improve their beat speed difference sensitivity.

Continuation of these relationships with close to high accuracy across multiple iH, makes it very useful in determining the best sequence with windows.
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#2305497 - 07/22/14 04:09 PM Re: Theoretical tuning sequence [Re: DoelKees]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1258
Loc: Montreal, Quebec, Canada
Originally Posted By: DoelKees
Originally Posted By: Mark Cerisano, RPT

Again, the real elephant in the room is beat speed sensitivity. If you can't tell if F#3A#3 is between FA and GB, you won't be able to get good accuracy on ET, IMHO.

Does anyone else feel this way? What have you done, what is your approach to developing this sensitivity to the highest level possible. IMHO, this is the information beginning techs really need.

Please don't reply "Tune 1000 pianos". I think we should be beyond that with all the technology we have that can speed up learning.

It think it has two components:

1) developing beat speed sensitivity when the beats are clearly audible. Some training app with synthetic tones of even with metronome clicks would be a great learning tool for this I think.

2) actually being able to focus on listening to the correct beat on a real piano where there is interference from higher partials and other distractions. Developing training software for this would probably have to work with recorded samples of a real piano.

I think such training tools would be a vast improvement over the traditional "tune 1000 pianos" method.

Kees


Thanks Kees,

1. I hope to soon create an app that references actual interval recordings from a piano. Each interval is recorded multiple times, at different sizes.

You start with one set of check intervals and have to decide which way to change the note to be tuned. Based on that choice, the recorded interval set changes.

When satisfied, you move onto another note to tune, with a new set of recorded check intervals, which are based on the choice you made to tune the previous note.

In this way, there is really only one "path" to the correct end. It's like passing through multiple rooms, each room with multiple doors. Only one set of doors (tunings) leads to the correct room at the other end.

But, depending on which room you do end up in, you would get a score that would rate your beat speed difference sensitivity.

2. I have an idea for a unit that would focus in on the coincidental partial by filtering out the extra noise in a real piano using a band pass filter. This could help beginners train their ears faster.

The deluxe model listens to the interval and determines the coincidental partial based on the Fourier Transform calculation. If it returns the two lowest frequencies as less than an octave, that means we are listening to an interval, and not one string. Based on those frequencies, the unit creates a band pass filter around that coincidental partial.

Example. We play 400 and 505 Hz. A complex waveform is produced. The unit returns the lowest two frequencies that are contained in the waveform as 400 and 505. Some common ratios are stored in the algorithm and 5/4 is the closest, hence it is a M3. A bandpass filter is created around 2000Hz, perhaps something like -100bD and 100Hz wide.

Anybody know how doable that Is?

The regular model, which is more analog, could just have two knobs; a pass frequency, and a width. The tuner has to "tune" tyne unit.

The basic model could just have a high pass filter set at 880Hz. (Most coincidental partials are above that, and all the temperament fundamentals are below that.)

Anybody know enough about electronics and DSP to confirm or criticize if this could even be done? I've had the idea for about 8 years now.
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www.howtotunepianos.com

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#2305512 - 07/22/14 04:28 PM Re: Theoretical tuning sequence [Re: UnrightTooner]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1258
Loc: Montreal, Quebec, Canada
Originally Posted By: UnrightTooner
Originally Posted By: Mark Cerisano, RPT

Again, the real elephant in the room is beat speed sensitivity. If you can't tell if F#3A#3 is between FA and GB, you won't be able to get good accuracy on ET, IMHO.

Does anyone else feel this way? What have you done, what is your approach to developing this sensitivity to the highest level possible. IMHO, this is the information beginning techs really need.

Please don't reply "Tune 1000 pianos". I think we should be beyond that with all the technology we have that can speed up learning.


Let's say the goal is to have M3s and M6s progressive. That requires an accuracy of +/- 0.2 cents. When tuning aurally, it requires the discernment of the difference of a beat ratio of +/- 1/16 (difference between 16 bps and 17 bps or 8 bps and 8.5 bps). The scaling also has to make it possible to have both M3s and M6s progressive.

Considering that none of these tolerances will be zero, the stability needed is actually greater than 0.2 cents.

Given a particular piano and tuner any of these factors may be the weak link. Like does it really matter if the tuner can discern the difference between 8 bps and 8.5 bps if the pins are horribly jumpy?


Stability is the first and last skill a tuner will learn. As their beat speed difference sensitivity improves, they will demand better of their stability technique.

So, if a tuner can tell the difference between 8 and 8.5 bps, then it certainly will matter if the pins are jumpy.

Let's not forget that an attempt to achieve an accuracy of 0.2 cents will result in a better accuracy than if we didn't try to reach this goal.
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www.howtotunepianos.com

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#2305731 - 07/23/14 02:04 AM Re: Theoretical tuning sequence [Re: Mark Cerisano, RPT]
Olek Offline
7000 Post Club Member

Registered: 03/14/08
Posts: 7540
Loc: France
thanks Mark, of course I know that and compare intervals, but I use more consonance than intervals comparation anyway as a primary but supplementary test.
On another (?) subject,
I wonder if beat masking can be used when tuning FBI, then you can obtain intervals with a certain amount of activity but no particularly striking beats.
More precisely, the activity make beat discriminating too difficult.

Best regards.


Edited by Olek (07/23/14 02:05 AM)
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#2306238 - 07/24/14 12:03 AM Re: Theoretical tuning sequence [Re: DoelKees]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1258
Loc: Montreal, Quebec, Canada
Well, FGBD has a certain sound that GBDF doesn't (when GB = FD). Try it.

Also, I sometimes use "phasing" to determine if I have a pure 12th.

Play Ab2F3C5 all together.
Ab2F3 = Ab2C5 for a pure 12th.

Ab2F3 beats.
Ab2C5 beats.

If they are not equal, their beating will beat. Like this:
waa-Waa-WAA-Waa-waa-Waa-WAA-Waa-waa-Waa-WAA-Waa-waa
(Does that make sense?)

When you can only hear one clean beat from Ab2F3C5, that means Ab2F3 = Ab2C5.
(I got 98% on the RPT exam treble portion with that little trick, and my stretch curve followed the string diameters. Can an ETD do that? Does anybody care? Don't answer that last question.)

BTW, I know Verituner can do that.
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