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#2303184 - 07/16/14 10:19 PM Re: Theoretical tuning sequence [Re: DoelKees]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1495
Loc: Montreal, Quebec, Canada
I hear an octave G ringing all over the place. Maybe it's my speaker or the quality of the recording. I don't hear any beat cancellation, or whatever we want to call it, although I have experienced it.
_________________________
Mark Cerisano, RPT
www.howtotunepianos.com

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#2303243 - 07/17/14 03:06 AM Re: Theoretical tuning sequence [Re: DoelKees]
Olek Offline
7000 Post Club Member

Registered: 03/14/08
Posts: 7904
Loc: France
if it add something to the tone it is interesting, if it retracts something, way less in my opinion.

I hear 2 intervals riding approx the same beat . I feel it can be done in additive or substractive modes, and that the result will differ in regard of global consonance.
I ten to prefer a sound that flows without being retained, to one that seem to have some slowing.

It makes me think of the Viennese style unison, where the original conflict around center string provides a thickened tone but the process have absorbed/ruled part of the attack energy so the tone is shortened.

In a goal of providing different tonal outputs between intervals I understand that effect can be included - it sound too present to me, not sounding "naturally" but I am not used to that "extreme"



Edited by Olek (07/17/14 03:10 AM)
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#2303279 - 07/17/14 07:42 AM Re: Theoretical tuning sequence [Re: Bill Bremmer RPT]
Chris Leslie Offline
500 Post Club Member

Registered: 01/01/11
Posts: 761
Loc: Canberra, ACT, Australia
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.


Edited by Chris Leslie (07/17/14 07:42 AM)
_________________________
Chris Leslie
Piano technician
http://www.chrisleslie.com.au

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#2303284 - 07/17/14 07:56 AM Re: Theoretical tuning sequence [Re: Chris Leslie]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3325
Loc: Madison, WI USA
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.
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#2303285 - 07/17/14 08:00 AM Re: Theoretical tuning sequence [Re: Olek]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3325
Loc: Madison, WI USA
Originally Posted By: Olek
if it add something to the tone it is interesting, if it retracts something, way less in my opinion.

I hear 2 intervals riding approx the same beat . I feel it can be done in additive or substractive modes, and that the result will differ in regard of global consonance.
I ten to prefer a sound that flows without being retained, to one that seem to have some slowing.

It makes me think of the Viennese style unison, where the original conflict around center string provides a thickened tone but the process have absorbed/ruled part of the attack energy so the tone is shortened.

In a goal of providing different tonal outputs between intervals I understand that effect can be included - it sound too present to me, not sounding "naturally" but I am not used to that "extreme"



I use this method to tune 99% of the pianos that I tune, including for concert artists. They get on the stage to announce to the public how wonderful the piano sounds, so I will go with their opinion, not yours.
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#2303286 - 07/17/14 08:02 AM Re: Theoretical tuning sequence [Re: Mark Cerisano, RPT]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3325
Loc: Madison, WI USA
Originally Posted By: Mark Cerisano, RPT
I hear an octave G ringing all over the place. Maybe it's my speaker or the quality of the recording. I don't hear any beat cancellation, or whatever we want to call it, although I have experienced it.


You tune your pianos your way and I will tune mine my way. I didn't hear what you hear and you didn't hear what I hear.
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#2303344 - 07/17/14 11:31 AM Re: Theoretical tuning sequence [Re: Bill Bremmer RPT]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4980
Loc: Bradford County, PA
Originally Posted By: Bill Bremmer RPT
Originally Posted By: Mark Cerisano, RPT
I hear an octave G ringing all over the place. Maybe it's my speaker or the quality of the recording. I don't hear any beat cancellation, or whatever we want to call it, although I have experienced it.


You tune your pianos your way and I will tune mine my way. I didn't hear what you hear and you didn't hear what I hear.


Why do I have this image in my mind of a parakeet fighting it's reflection in a mirror?
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#2303479 - 07/17/14 06:14 PM Re: Theoretical tuning sequence [Re: Bill Bremmer RPT]
Chris Leslie Offline
500 Post Club Member

Registered: 01/01/11
Posts: 761
Loc: Canberra, ACT, Australia
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.


Just to confirm I listened again:

Firstly, there is a definite G2 ringing out from somewhere when the G3 is played in some combination.
Secondly, with the G3C4E4 inversion, there is a significant reduction in apparent beating, but the beats are still very audible.
Thirdly, the C3E3G3(C4) inversion, the is hardly any apparent reduction in beating of the CE interval, just a slight reduction in volume but replaced by a muddy quality.

I am just trying to give an honest evaluation.
_________________________
Chris Leslie
Piano technician
http://www.chrisleslie.com.au

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#2303504 - 07/17/14 08:11 PM Re: Theoretical tuning sequence [Re: DoelKees]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3325
Loc: Madison, WI USA
Am I supposed to just give up and tune ET from now on because everything I have done for the last 30 years has all been an illusion? Did Bach really write the Well Tempered Clavier Music for ET as Jeff "proved"? Was ET "firmly in place" by the time of Brahms as the PTG Journal guy claimed? So Brahms just had a girl like Vanna White spin the Wheel of Fortune to determine which key signature he would use for each piece he wrote because one key signature was the same as another?

Equal Beating only makes things sound worse, not better?

Sorry folks, I don't buy any of it. I know what I hear and so do my clients. I haven't tuned a piano in true ET now for 25 years, almost exactly to the date when I heard a set of Brahms performed in the Rameau-Rousseau-Hall composite 18th Century Modified Meantone Temperament from Owen Jorgensen's Handbook of Equal Beating Temperaments.

I can't prove anything with a few examples recorded on a cellphone. I only know what I have heard and experienced in the last 25 years and I am not about to turn back from it. There is a reason why I have so many loyal clients and why performing artists choose to acknowledge what I have done publicly and why they also do the same for my local colleagues who also choose the non-equal temperaments: Musical satisfaction and often great epiphanies that they have never experienced from ET.
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#2303522 - 07/17/14 09:24 PM Re: Theoretical tuning sequence [Re: Bill Bremmer RPT]
prout Offline
500 Post Club Member

Registered: 11/14/13
Posts: 902
Originally Posted By: Bill Bremmer RPT
Am I supposed to just give up and tune ET from now on because everything I have done for the last 30 years has all been an illusion? Did Bach really write the Well Tempered Clavier Music for ET as Jeff "proved"? Was ET "firmly in place" by the time of Brahms as the PTG Journal guy claimed? So Brahms just had a girl like Vanna White spin the Wheel of Fortune to determine which key signature he would use for each piece he wrote because one key signature was the same as another?

Equal Beating only makes things sound worse, not better?

Sorry folks, I don't buy any of it. I know what I hear and so do my clients. I haven't tuned a piano in true ET now for 25 years, almost exactly to the date when I heard a set of Brahms performed in the Rameau-Rousseau-Hall composite 18th Century Modified Meantone Temperament from Owen Jorgensen's Handbook of Equal Beating Temperaments.

I can't prove anything with a few examples recorded on a cellphone. I only know what I have heard and experienced in the last 25 years and I am not about to turn back from it. There is a reason why I have so many loyal clients and why performing artists choose to acknowledge what I have done publicly and why they also do the same for my local colleagues who also choose the non-equal temperaments: Musical satisfaction and often great epiphanies that they have never experienced from ET.


The more I attempt to wring out the physics of piano tuning using sophisticated electronic equipment, the more I realize that the equipment does not hear what we hear. A centimetre shift in the microphone will entirely change the partial amplitudes and phases, which means that your recordings do not capture the quality of the tuning that you hear.

I find that what the tuner hears from the piano bench and what the pianist hears from that same bench is all that matters. Your tunings clearly work and your clientele are happy. Keep up the good work.

Cheers.

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#2303532 - 07/17/14 10:03 PM Re: Theoretical tuning sequence [Re: Bill Bremmer RPT]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1766
Loc: Vancouver, Canada
Originally Posted By: Bill Bremmer RPT
There is a reason why I have so many loyal clients and why performing artists choose to acknowledge what I have done publicly and why they also do the same for my local colleagues who also choose the non-equal temperaments: Musical satisfaction and often great epiphanies that they have never experienced from ET.

You are confusing the issue. It is possible that people like EBVT3 tuning (or some other unequal temperament) but this may have nothing to do with supposed beat (equal or unequal) cancellation/masking.

Please note the word "may" in the previous sentence. I keep an open mind, but I haven't seen any evidence for equal beat cancellation or unequal beat masking in my own experiments on my piano or in any recordings by others.

If this really worked it should be easy to demonstrate by showing a chord with and without equal beats for comparison.

Of course in a chord individual beats are always less prominent. For example play F4A4 and hear it beating. Now put your forearm on that octave: you can't hear F4A4 beating any more.

Kees

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#2303539 - 07/17/14 10:53 PM Re: Theoretical tuning sequence [Re: UnrightTooner]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1766
Loc: Vancouver, Canada
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

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#2303552 - 07/18/14 12:04 AM Re: Theoretical tuning sequence [Re: Bill Bremmer RPT]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1495
Loc: Montreal, Quebec, Canada
Originally Posted By: Bill Bremmer RPT
Am I supposed to just give up and tune ET from now on because everything I have done for the last 30 years has all been an illusion? Did Bach really write the Well Tempered Clavier Music for ET as Jeff "proved"? Was ET "firmly in place" by the time of Brahms as the PTG Journal guy claimed? So Brahms just had a girl like Vanna White spin the Wheel of Fortune to determine which key signature he would use for each piece he wrote because one key signature was the same as another?

Equal Beating only makes things sound worse, not better?

Sorry folks, I don't buy any of it. I know what I hear and so do my clients. I haven't tuned a piano in true ET now for 25 years, almost exactly to the date when I heard a set of Brahms performed in the Rameau-Rousseau-Hall composite 18th Century Modified Meantone Temperament from Owen Jorgensen's Handbook of Equal Beating Temperaments.

I can't prove anything with a few examples recorded on a cellphone. I only know what I have heard and experienced in the last 25 years and I am not about to turn back from it. There is a reason why I have so many loyal clients and why performing artists choose to acknowledge what I have done publicly and why they also do the same for my local colleagues who also choose the non-equal temperaments: Musical satisfaction and often great epiphanies that they have never experienced from ET.


Re-listen to your recording. Try to confirm what other people are saying. Re-record with better equipment so that you can confirm that what you hear in person, is there on the recording. Don't give them any opportunity to say what you know is there, is not there.
_________________________
Mark Cerisano, RPT
www.howtotunepianos.com

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#2303554 - 07/18/14 12:11 AM Re: Theoretical tuning sequence [Re: DoelKees]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1495
Loc: Montreal, Quebec, Canada
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


Kees,

I have discovered another equality test. (No one has every told me about it nor have I read of it anywhere, but that doesn't mean anything) But I thought you might be able to prove it mathematically. I call it the m7b5 equality. Example: FA# = BD#.

I know it works in my sequence, but is there any theoretical proof that it is valid?
_________________________
Mark Cerisano, RPT
www.howtotunepianos.com

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#2303562 - 07/18/14 12:41 AM Re: Theoretical tuning sequence [Re: Mark Cerisano, RPT]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1766
Loc: Vancouver, Canada
Originally Posted By: Mark Cerisano, RPT

Kees,

I have discovered another equality test. (No one has every told me about it nor have I read of it anywhere, but that doesn't mean anything) But I thought you might be able to prove it mathematically. I call it the m7b5 equality. Example: FA# = BD#.

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

FA# = BD#? Typo?

Kees

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#2303580 - 07/18/14 01:41 AM Re: Theoretical tuning sequence [Re: DoelKees]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1495
Loc: Montreal, Quebec, Canada
Whoops. Yes.

FG# = BD#

Thanks.
_________________________
Mark Cerisano, RPT
www.howtotunepianos.com

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#2303592 - 07/18/14 02:38 AM Re: Theoretical tuning sequence [Re: Mark Cerisano, RPT]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1766
Loc: Vancouver, Canada
Originally Posted By: Mark Cerisano, RPT
Whoops. Yes.

FG# = BD#

Thanks.


The M3/M6 test is (for example)
F3A3 = D3#C4.
However the M6 is really a tiny bit faster theoretically.
Your test, when you invert the m3 is
F3A3 = D3B3
and here the M6 is a little bit slower, but not a tiny bit.

So it may be a good bracketing test: the M3 has to beat at a rate in-between those two, but closer to D#3C4.

Kees

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#2303594 - 07/18/14 03:09 AM Re: Theoretical tuning sequence [Re: DoelKees]
Gadzar Offline
1000 Post Club Member

Registered: 12/15/06
Posts: 1910
Loc: Mexico City
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.


Edited by Gadzar (07/18/14 03:11 AM)
_________________________
Rafael Melo
Piano Technician
rafaelmelo@afinacionpianos.com.mx

Serving Mexico City and suburbs.

http://www.afinacionpianos.com.mx

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#2303603 - 07/18/14 04:14 AM Re: Theoretical tuning sequence [Re: DoelKees]
Bernhard Stopper Offline
Full Member

Registered: 09/22/08
Posts: 219
Loc: Germany
Originally Posted By: DoelKees
For example play F4A4 and hear it beating. Now put your forearm on that octave: you can't hear F4A4 beating any more.
Kees


(With "putting the forearm on that octave" Kees means striking all the notes together within that octave)
At least one example, where you confirm that beat masking is possible. We have a good starting point here, more later...
_________________________
Bernhard Stopper
www.piano-stopper.de

Salieri: "Mediocrities everywhere, now and to come: I absolve you all! Amen! Amen! Amen!"
(Amadeus, the movie)

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#2303604 - 07/18/14 04:19 AM Re: Theoretical tuning sequence [Re: Bill Bremmer RPT]
Bernhard Stopper Offline
Full Member

Registered: 09/22/08
Posts: 219
Loc: Germany
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?




Edited by Bernhard Stopper (07/18/14 06:05 AM)
_________________________
Bernhard Stopper
www.piano-stopper.de

Salieri: "Mediocrities everywhere, now and to come: I absolve you all! Amen! Amen! Amen!"
(Amadeus, the movie)

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#2303642 - 07/18/14 08:53 AM Re: Theoretical tuning sequence [Re: DoelKees]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1495
Loc: Montreal, Quebec, Canada
Originally Posted By: DoelKees
Originally Posted By: Mark Cerisano, RPT
Whoops. Yes.

FG# = BD#

Thanks.


The M3/M6 test is (for example)
F3A3 = D3#C4.
However the M6 is really a tiny bit faster theoretically.
Your test, when you invert the m3 is
F3A3 = D3B3
and here the M6 is a little bit slower, but not a tiny bit.

So it may be a good bracketing test: the M3 has to beat at a rate in-between those two, but closer to D#3C4.

Kees


Thank you Kees.

You say the M6 is faster in the dom7 theoretically. Is there a mathematical proof? Thanks.

I use that relational logic to prove the m7b5 relationship but thought there might be a numerical one.

I think your relationship logic assumed a pure 6:3, but I couldn't completely follow it because, while you say my test's m3 inverted is a D3B3, that would refer to B3D4 and that equality is B3D3=F4A4 so I'm completely confused. I'm hoping a few more instructions will clear it up for me. Thanks for your time.

Like I said, I'm looking for a mathematical proof. Here's my relational proof, with assumptions.

Gadzar, I'm always tuning wide 4:2 and narrow 6:3 in the temperament octave.

The m7b5 Proof.

Assumptions:
Dom7 equality. GB=FD and all transpositions within and near the temperament octave, but not over the break.
Narrow 6:3 octaves given by F3G#3>G#3F4 and all transpositions within the temperament octave and nearby, where > means "every so slightly faster; barely noticeable."
The difference in beat speed between the m3 and M6 in a narrow 6:3, is equal to the difference between chromatic M6's in ET.

(These assumptions are valid for the level of accuracy I am going for, which is high compared to some other popular tests, especially since I am using "bracketing" exclusively in my temperament sequence, which seems to produce good precision. My sequence requires that you bisect all the windows equally when setting beat speeds.)

Relational Proof

F3G#3>G#3F4 (narrow 6:3)
A3F#4>G#3F4 (ET relationship)

Therefore,
F3G#3=A3F#4 (assuming beat speed differences are equal)

A3F#4=B3D4 (Dom7 equality)

Therefore,
F3G#3=B3D#4

I'm tuning high accuracy ET temperaments (increasing RBI's and P4's) using these relationships to produce windows into which other beat speeds must bisect (that's the beauty, the bisecting may not be 100% accurate, but it seems to be way more accurate than any other one sided tests like F3A3<F3D4 for the P4, for example. Just how much faster is F3D4 supposed to be? I know some people know the answer intuitively, but how to explain that to a beginner? The windows sequence I've developed gives a solid relationship that students have to reproduce, and even if the formulas and bisecting are not 100% accurate, the process of constantly fitting beat speeds into such small windows develops aural beat speed difference sensitivity, which is the most important skill for tuning aurally using RBI, IMHO. The tolerances on the windows I've developed are 3x more precise than CM3's)

Sorry for the long winded thesis, but that's what this is; my thesis. Your help and criticism is invaluable to helping me produce a method that other technicians and students will find credible.

As always, your objective and scientific criticism is appreciated by me and many others on this forum.

(Addendum. Gadzar, your post was much more concise than mine and proved the relationship nicely, given that the 6:3 was narrow as you say. Nice job.)


Edited by Mark Cerisano, RPT (07/18/14 09:01 AM)
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#2303647 - 07/18/14 09:10 AM Re: Theoretical tuning sequence [Re: DoelKees]
Gadzar Offline
1000 Post Club Member

Registered: 12/15/06
Posts: 1910
Loc: Mexico City
Originally Posted By: Mark Cerisano, RPT
The difference in beat speed between the m3 and M6 in a narrow 6:3, is equal to the difference between chromatic M6's in ET.


Why? It may be greater or lesser, depending on how much the 6:3 octaves are stretched, in this case contracted.

There is no mathematical proof here.
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#2303657 - 07/18/14 09:55 AM Re: Theoretical tuning sequence [Re: DoelKees]
Gadzar Offline
1000 Post Club Member

Registered: 12/15/06
Posts: 1910
Loc: Mexico City
P.S. In the treble we tune narrow 6:3 octaves. In the tenor we tune still narrow 6:3 octaves but less narrow than in the treble. In the bass we tune pure 6:3 octaves. And in the low bass, in some pianos, we even tune wide 6:3 octaves. So your equality will only work on a particular region of the scale where the 6:3 octaves have the required size.
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#2303733 - 07/18/14 02:24 PM Re: Theoretical tuning sequence [Re: Gadzar]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1495
Loc: Montreal, Quebec, Canada
Originally Posted By: Gadzar
Originally Posted By: Mark Cerisano, RPT
The difference in beat speed between the m3 and M6 in a narrow 6:3, is equal to the difference between chromatic M6's in ET.


Why? It may be greater or lesser, depending on how much the 6:3 octaves are stretched, in this case contracted.

There is no mathematical proof here.


The accuracy of the assumption allows me to tune extremely high accuracy, compared to the way I used to tune. (Don't say it!) You need to see the sequence to get it.

Anyway, we don't tune 6:3 octaves in the temperament, unless you're going for some excessive stretch like pure 19ths.

The purpose if this analysis is to explore a sequence that gets good accuracy and precision by placing beat speeds evenly within a window, so that beginners can get decent results and improve their beat speed difference sensitivity.

In the end, it is that sensitivity, and stability, that allow a tuner to accurately make decisions on which way to adjust a note.

The only thing a good sequence can do is get you there faster with less refining.

A bad sequence allows you to get through without showing you the errors, thereby allowing you to move on with a false sense of accomplishment, if you choose not to refine.

Before anybody gets their knickers in a twist, I am NOT saying my sequence is the only one that can do that, or even the best one. I'm just making generalizations on what I think a good sequence should allow you to do.


Anyway, I see what you are saying. Because of the variable size of the 6:3 octave, we can't have a mathematical proof. I'm just wondering how accurate a sequence can be if we assume the m7b5 relationship, when the octave stretch really only confirms FG#=CE? Kees?
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#2303738 - 07/18/14 02:28 PM Re: Theoretical tuning sequence [Re: Gadzar]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1495
Loc: Montreal, Quebec, Canada
Originally Posted By: Gadzar
P.S. In the treble we tune narrow 6:3 octaves. In the tenor we tune still narrow 6:3 octaves but less narrow than in the treble. In the bass we tune pure 6:3 octaves. And in the low bass, in some pianos, we even tune wide 6:3 octaves. So your equality will only work on a particular region of the scale where the 6:3 octaves have the required size.



Agreed. I only use it in the temperament.
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#2303824 - 07/18/14 07:04 PM Re: Theoretical tuning sequence [Re: Mark Cerisano, RPT]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1766
Loc: Vancouver, Canada
Originally Posted By: Mark Cerisano, RPT
Originally Posted By: DoelKees
Originally Posted By: Mark Cerisano, RPT
Whoops. Yes.

FG# = BD#

Thanks.


The M3/M6 test is (for example)
F3A3 = D3#C4.
However the M6 is really a tiny bit faster theoretically.
Your test, when you invert the m3 is
F3A3 = D3B3
and here the M6 is a little bit slower, but not a tiny bit.

So it may be a good bracketing test: the M3 has to beat at a rate in-between those two, but closer to D#3C4.

Kees


Thank you Kees.

You say the M6 is faster in the dom7 theoretically. Is there a mathematical proof? Thanks.

I use that relational logic to prove the m7b5 relationship but thought there might be a numerical one.

I think your relationship logic assumed a pure 6:3, but I couldn't completely follow it because, while you say my test's m3 inverted is a D3B3, that would refer to B3D4 and that equality is B3D3=F4A4 so I'm completely confused. I'm hoping a few more instructions will clear it up for me. Thanks for your time.

Like I said, I'm looking for a mathematical proof. Here's my relational proof, with assumptions.

Gadzar, I'm always tuning wide 4:2 and narrow 6:3 in the temperament octave.

The m7b5 Proof.

Assumptions:
Dom7 equality. GB=FD and all transpositions within and near the temperament octave, but not over the break.
Narrow 6:3 octaves given by F3G#3>G#3F4 and all transpositions within the temperament octave and nearby, where > means "every so slightly faster; barely noticeable."
The difference in beat speed between the m3 and M6 in a narrow 6:3, is equal to the difference between chromatic M6's in ET.

(These assumptions are valid for the level of accuracy I am going for, which is high compared to some other popular tests, especially since I am using "bracketing" exclusively in my temperament sequence, which seems to produce good precision. My sequence requires that you bisect all the windows equally when setting beat speeds.)

Relational Proof

F3G#3>G#3F4 (narrow 6:3)
A3F#4>G#3F4 (ET relationship)

Therefore,
F3G#3=A3F#4 (assuming beat speed differences are equal)

A3F#4=B3D4 (Dom7 equality)

Therefore,
F3G#3=B3D#4

I'm tuning high accuracy ET temperaments (increasing RBI's and P4's) using these relationships to produce windows into which other beat speeds must bisect (that's the beauty, the bisecting may not be 100% accurate, but it seems to be way more accurate than any other one sided tests like F3A3<F3D4 for the P4, for example. Just how much faster is F3D4 supposed to be? I know some people know the answer intuitively, but how to explain that to a beginner? The windows sequence I've developed gives a solid relationship that students have to reproduce, and even if the formulas and bisecting are not 100% accurate, the process of constantly fitting beat speeds into such small windows develops aural beat speed difference sensitivity, which is the most important skill for tuning aurally using RBI, IMHO. The tolerances on the windows I've developed are 3x more precise than CM3's)

Sorry for the long winded thesis, but that's what this is; my thesis. Your help and criticism is invaluable to helping me produce a method that other technicians and students will find credible.

As always, your objective and scientific criticism is appreciated by me and many others on this forum.

(Addendum. Gadzar, your post was much more concise than mine and proved the relationship nicely, given that the 6:3 was narrow as you say. Nice job.)


I was just being simple minded and looked at the beat rates for zero IH and 2^(1/12) tuning. Then it is just a numerical coincidence that M3 and M6 beat about the same in the 3d inversion of dom 7th chord. They are not exactly the same though, M6 is a touch faster.

When there is IH and octave stretch I don't know what happens.

Regarding your proof attempt:

If x>y and z>y it does not follow that x=z unless you assume (as you did in your post) that x-y = z-y, but that is the same as assuming x=z so all you prove is that x=z if x=z.

Generally I like very much the idea of bracketing beatrates between two other beatrates.

Kees

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#2303846 - 07/18/14 09:18 PM Re: Theoretical tuning sequence [Re: DoelKees]
Mark Cerisano, RPT Offline
1000 Post Club Member

Registered: 01/24/10
Posts: 1495
Loc: Montreal, Quebec, Canada
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?
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#2303882 - 07/18/14 11:11 PM Re: Theoretical tuning sequence [Re: Mark Cerisano, RPT]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1766
Loc: Vancouver, Canada
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

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#2303894 - 07/19/14 12:03 AM Re: Theoretical tuning sequence [Re: Mark Cerisano, RPT]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1766
Loc: Vancouver, Canada
Originally Posted By: Mark Cerisano, RPT

I have discovered another equality test. (No one has every told me about it nor have I read of it anywhere, but that doesn't mean anything) But I thought you might be able to prove it mathematically. I call it the m7b5 equality. Example: FA# = BD#.

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


I have written a program for this. It computes a temperament range with IH data from any piano, using tunelab's model. I tune octaves as 4:2 or 6:3 or in-between equal beating. Then I compute F3A3C#4F4A4 to be progressive, and interpolate the notes in between assuming the stretch does not change over a M3.

Below the beat rates computed for 4 IH models.

If we take the usual M3/M6 test on A3 as an example, we should
see that the M3 on A3 beats the same as the M6 on G3. For ih=0 we see the M6 is 0.2 bps is faster, whereas the M6 on F#3 is 0.3 slower. So it would be more accurate to say M3 on A3 is in-between M6's on F#3 and G3.

On the Steinway D the Cerisano M3/M6 test (A3C#4=F#3D#4) is already more accurate than the usual test (A3C#4=G3E4): M3 on A3 is closer to M6 on F#3 than on G3. On the uprights the Cerisano M3/M6 test is the more accurate one.

So it seems the usual M3/M6 test should be replaced by the Cerisano M3/M6 test (or bracketing). I ran the simulation also with 4:2 and 6:3 octaves with the same results.

Did I earn a complementary copy of your book?

Kees
--
octave: equal beating 4:2 and 6:3
ih=0

M3 M6 m3 m6 P4 P5
C#3 5.5 6.3 -7.5 -8.7 0.6 -0.5
D3 5.8 6.7 -7.9 -9.3 0.7 -0.5
D#3 6.2 7.1 -8.4 -9.8 0.7 -0.5
E3 6.5 7.5 -8.9 -10.4 0.8 -0.6
F3 6.9 7.9 -9.4 -11.0 0.8 -0.6
F#3 7.3 8.4 -10.0 -11.7 0.8 -0.6
G3 7.8 8.9 -10.6 -12.4 0.9 -0.7
G#3 8.2 9.4 -11.2 -13.1 0.9 -0.7
A3 8.7 10.0 -11.9 -13.9 1.0 -0.7
A#3 9.3 10.6 -12.6 -14.7 1.1 -0.8
B3 9.8 11.2 -13.3 -15.6 1.1 -0.8
C4 10.4 11.9 -14.1 -16.5 1.2 -0.9
C#4 11.0 -- -15.0 -17.5 1.3 -0.9
D4 11.7 -- -15.9 -- 1.3 -1.0
D#4 12.4 -- -16.8 -- 1.4 --
E4 13.1 -- -17.8 -- 1.5 --
F4 13.9 -- -18.9 -- -- --
F#4 -- -- -20.0 -- -- --

ih = Steinway D

M3 M6 m3 m6 P4 P5
C#3 5.5 6.3 -7.8 -9.1 0.7 -0.5
D3 5.8 6.7 -8.3 -9.7 0.7 -0.5
D#3 6.2 7.1 -8.9 -10.3 0.7 -0.5
E3 6.5 7.5 -9.4 -10.9 0.8 -0.6
F3 6.9 8.0 -10.1 -11.6 0.8 -0.6
F#3 7.3 8.5 -10.7 -12.4 0.9 -0.6
G3 7.8 9.0 -11.4 -13.2 0.9 -0.7
G#3 8.2 9.5 -12.2 -14.1 1.0 -0.7
A3 8.7 10.1 -13.1 -15.0 1.1 -0.7
A#3 9.2 10.7 -13.9 -16.0 1.1 -0.8
B3 9.8 11.3 -14.9 -17.1 1.2 -0.8
C4 10.3 12.0 -16.0 -18.2 1.3 -0.9
C#4 10.9 -- -17.1 -19.5 1.4 -0.9
D4 11.6 -- -18.3 -- 1.5 -1.0
D#4 12.3 -- -19.7 -- 1.6 --
E4 13.0 -- -21.1 -- 1.7 --
F4 13.8 -- -22.7 -- -- --
F#4 -- -- -24.5 -- -- --

ih = HellasHelsinki
M3 M6 m3 m6 P4 P5
C#3 5.4 6.3 -7.9 -10.8 0.7 -0.3
D3 5.7 6.7 -8.4 -11.4 0.8 -0.3
D#3 6.1 7.1 -8.9 -12.0 0.8 -0.3
E3 6.4 7.5 -9.5 -12.7 0.9 -0.3
F3 6.8 8.0 -10.1 -13.4 0.9 -0.4
F#3 7.2 8.4 -10.8 -14.2 1.0 -0.4
G3 7.7 9.0 -11.5 -15.1 1.0 -0.4
G#3 8.1 9.5 -12.2 -16.0 1.1 -0.5
A3 8.6 10.1 -13.1 -17.0 1.2 -0.5
A#3 9.1 10.7 -13.9 -18.0 1.2 -0.6
B3 9.6 11.3 -14.8 -19.1 1.3 -0.6
C4 10.2 12.0 -15.8 -20.4 1.4 -0.6
C#4 10.8 -- -16.9 -21.7 1.5 -0.7
D4 11.4 -- -18.1 -- 1.6 -0.7
D#4 12.1 -- -19.3 -- 1.7 --
E4 12.8 -- -20.6 -- 1.8 --
F4 13.6 -- -22.1 -- -- --
F#4 -- -- -23.7 -- -- --

ih = large Heintzmann upright

M3 M6 m3 m6 P4 P5
C#3 5.4 6.3 -8.2 -10.6 0.7 -0.3
D3 5.7 6.7 -8.7 -11.2 0.8 -0.3
D#3 6.1 7.1 -9.3 -11.9 0.8 -0.4
E3 6.4 7.6 -9.9 -12.6 0.9 -0.4
F3 6.8 8.0 -10.6 -13.5 0.9 -0.4
F#3 7.2 8.5 -11.4 -14.3 1.0 -0.5
G3 7.7 9.0 -12.2 -15.3 1.1 -0.5
G#3 8.1 9.5 -13.0 -16.3 1.1 -0.5
A3 8.6 10.1 -13.9 -17.5 1.2 -0.6
A#3 9.1 10.8 -15.0 -18.7 1.3 -0.6
B3 9.6 11.4 -16.1 -20.0 1.4 -0.6
C4 10.2 12.1 -17.3 -21.5 1.4 -0.7
C#4 10.7 -- -18.6 -23.1 1.6 -0.7
D4 11.4 -- -20.0 -- 1.7 -0.7
D#4 12.1 -- -21.5 -- 1.8 --
E4 12.8 -- -23.2 -- 1.9 --
F4 13.5 -- -25.0 -- -- --
F#4 -- -- -27.1 -- -- --

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#2303916 - 07/19/14 02:20 AM Re: Theoretical tuning sequence [Re: DoelKees]
Hakki Offline
2000 Post Club Member

Registered: 05/26/01
Posts: 2778
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?
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