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#1856606 - 03/05/12 01:39 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
If Dr. White's method does not produce a good tuning, it is because it was not followed very well. And if there was an "acceleration of the inharmonic effect" then stacked M3s wouldn't work either.

And, believe it or not, "tuning to higher partial alignments" will cause problems in setting a temperament across a jump in iH, unless a ghastly wide octave is used.

But I agree that it's not at all related to what you where saying above. Inharmonicity has nothing to do with it! And that is why I still cannot accept what you are saying. The iH in pianos did not cause a problem in tuning until ETDs were used and did not cause a musical shift to ET either.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1856611 - 03/05/12 01:50 PM Re: I've made up my mind about Bach's WTC and ET [Re: Tunewerk]
Ed Foote Offline
1000 Post Club Member

Registered: 05/03/03
Posts: 1186
Loc: Tennessee
[quote=Tunewerk]
>>Equal temperament was known as the lute tuning, before being known as equal temperament, but it was so on the lute for melodic reasons.<<

Greetings,
My understanding is that the lutes and guitars were tuned in equal temperament because of the straight frets' requirements. Melodically, there is much to be gained by sharpening leading tones, etc. but no way to tune a guitar to favor one key without destroying many others.

>>Equal temperament is sort of a broad subject.

This could, perhaps, be the one statement that everybody on the planet can agree with!

>>The bases for implementation on the modern piano are melodic and harmonic.

I would submit that the implementation resulted from musicians ceasing to tune their own keyboards, and tuners becoming a profession. That process was occurring during the ascent of ET. Composition was also leaving its dependence on a tonal center, rendering inequality of the keys a liability rather than an asset.

>>The first ideas of even scale division were for even melodic step. The whole reason why ET was apparently not implemented successfully until recently on the piano was not knowing how to tune the harmonics with enough precision. We didn't understand string vibration in enough detail.<<

I think the system of ET was well figured out in China many centuries before Western civilization existed. And I believe that it was done for reasons of proving the math. No real evidence of it ever being used, that I know of. The intellectuals and theorists would certainly have been earlier proponents of something new than the working tuner in the countryside, so, even if we assume that the science was understood, and ET could be produced,there wasn't sufficient exposure to change the publics tastes very quickly.

>>The harmonic implementation of ET as a preferred tuning on the 12-tone scale of the modern piano is directly related to the effect of inharmonicity in modern instrument design; to equalize the effects of dissonance. <<

The dissonance that is being equalized in ET is there, regardless of the inharmonicity involved, so I would be skeptical of claims that inharmonicity was the driving force behind the widespread adoption of ET. The actual ability to hear beating basically exists in the middle four octaves of the piano, where inharmonicity is at its lowest effect.

>>Pure interval tunings lose their desirability and equal temperament becomes more psycho-acoustically sound.<<

I am lost, here. Pure interval tunings are what? And how do we measure the "soundness" of a temperament?
Regards,

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#1856622 - 03/05/12 02:17 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
rxd Offline
1000 Post Club Member

Registered: 03/11/09
Posts: 1770
Loc: London, England
Wait, we should understand that lutes had moveable frets made of pieces of gut tied around the neck of the instrument well into Bachs' time and later.

This would cause us to rethink many assumptions.


Edited by rxd (03/05/12 03:23 PM)
_________________________
Concert & Recording tuner-tech, London, England.
"in theory, practice and theory are the same thing. In practice, they're not." - Lawrence P. 'Yogi' Berra.

Eschew obfuscation.



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#1856655 - 03/05/12 03:44 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
Withindale Offline
1000 Post Club Member

Registered: 02/09/11
Posts: 1968
Loc: Suffolk, England
Earlier on in this thread, Jeff, you produced Occam's Razor.

The simple answer to the enigma you pose is that Bach wrote his music for instruments like his harpsichords and clavichords which he tuned in his version of the circulating temperaments of that time. No need to ask what the point of composing two books to be played in equal temperament would have been.


Edited by Withindale (03/06/12 05:12 AM)
_________________________
Ian Russell
Schiedmayer & Soehne, 1925 Model 14, 55" upright
Ibach, 1922 49" upright (project piano)

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#1856660 - 03/05/12 03:48 PM Re: I've made up my mind about Bach's WTC and ET [Re: Ed Foote]
Tunewerk Offline
Full Member

Registered: 03/26/11
Posts: 414
Loc: Boston, MA
Originally Posted By: UnrightTooner
If Dr. White's method does not produce a good tuning, it is because it was not followed very well. And if there was an "acceleration of the inharmonic effect" then stacked M3s wouldn't work either.

And, believe it or not, "tuning to higher partial alignments" will cause problems in setting a temperament across a jump in iH, unless a ghastly wide octave is used.

But I agree that it's not at all related to what you where saying above. Inharmonicity has nothing to do with it! And that is why I still cannot accept what you are saying. The iH in pianos did not cause a problem in tuning until ETDs were used and did not cause a musical shift to ET either.


Well I'm sure you are able to implement the White tuning with skill, so you can achieve a good tuning despite.

Wider intervals enable tuning with greater accuracy (spanned M3rds as an example). This is the entire basis for Dr. Sanderson's work with Jim Coleman and the FAC method in the Accutuner algorithm.

In reference also to above, I didn't mean tuning exclusively to higher partials. Yes, this will cause more stretch. I was using that as an example to show another thing that can help dial in accuracy.

It seems that you are missing what I'm saying..

Originally Posted By: Ed Foote
My understanding is that the lutes and guitars were tuned in equal temperament because of the straight frets' requirements. Melodically, there is much to be gained by sharpening leading tones, etc. but no way to tune a guitar to favor one key without destroying many others.


Yes, Ed. This is exactly what I was saying. Melodically (not the psychoacoustic desire for melodic stretch) the frets were arranged to provide somewhat equal spacing. Harmonically, it was not an equal temperament yet.

Originally Posted By: Ed Foote
The dissonance that is being equalized in ET is there, regardless of the inharmonicity involved, so I would be skeptical of claims that inharmonicity was the driving force behind the widespread adoption of ET. The actual ability to hear beating basically exists in the middle four octaves of the piano, where inharmonicity is at its lowest effect.


Ok here I can see I'm being completely misunderstood. Normally, I wouldn't try to correct it, but I see how my point got spun by Jeff in a different direction without context. I'll try to explain the idea again.

I am not saying that an increase in inharmonicity was the singular driving force behind the adoption of ET. I am saying that clearly, it was one.

Here's why:

Modern pianos = Greater tonal volume

Tonal volume is governed by tension in scaling. This is why, when you look at PSCALE charts, length and gauge change to keep tension constant.

Thicker gauge wire is required to gain tension.

Thicker gauge wire has higher inharmonicity by an order greater than the decrease caused by the tension increase tolerated.

Louder pianos = Greater inharmonicity (in general)

When you look at two inharmonic curves governed roughly by an [n^2-1] relation, the greater the inharmonicity, the more acute the curve. This results in less and less tangent area for any given consonant interval.

You could picture it like two spheres, one sitting within the other. When the curvature of one tightens (the radius becomes smaller, i.e. increased iH) the contact footprint shrinks (consonance decreases).

This is a relatively accurate analogy because the iH curve tightens by squares as it reaches higher, so the upper note is less affected in its lower partial regions, as the tighter curve from the lower note interacts with it.

So consonant intervals are less 'sweet' by definition - by mathematics and tested physics - in a higher iH piano. This is when the intervals are tuned pure! (Forgetting about ET for the time being.)

Now, enter ET. Why does it 'solve' this problem? There is less of a sensation of consonance for pure intervals, but by order of magnitude!

The octave still will have good consonance, followed by the octave-fifth and the double octave. Where you will begin to see sour consonances are on interactions like the M3rd!

In the M3rd, there is a 5/4 interaction. This is high enough that the square laws of iH begin to have traction and the curves don't match very well at that point. What does ET do? It widens it! Psychoacoustically, this actually eases the pain a little, as spreading these curves helps with the sense of flatness or sourness.

In any case, ET evenly spreads this iH shell dissonance all over instead of focusing on pure intervals, which would reveal the sourness of high iH pianos in some cases. More like, tuned unified dissonance in the case of ET.
_________________________
www.tunewerk.com

Unity of tone through applied research.

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#1856988 - 03/06/12 04:11 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
Mark R. Offline
2000 Post Club Member

Registered: 07/31/09
Posts: 2052
Loc: Pretoria, South Africa
Tunewerk, I don't get it. (Granted, I'm a beginner.)

Whether the iH of a piano is high or low, our ear will perceive the M3 as pure when the 5th partial of the lower note is beatless with the 4th of the upper note. The exact frequency of these partials doesn't matter. (Or does it?)

I would have thought that F3-A3 will sound pure when it's beatless, regardless of whether the 5:4 partial match takes place at 880, 881, 882 or 883 Hz.

How could a pure, beatless M3 sound "sour", and and top of that, be "improved" by making it wide?
_________________________
Autodidact interested in piano technology.

1922 49" Zimmermann, project piano.
1970 44" Ibach, daily music maker.

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#1857020 - 03/06/12 07:43 AM Re: I've made up my mind about Bach's WTC and ET [Re: Withindale]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
Originally Posted By: Withindale
Earlier on in this thread, Jeff, you produced Occam's Razor.

The simple answer to the enigma you pose is that Bach wrote his music for instruments like his harpsichords and clavichords which he tuned in his version of the circulating temperaments of that time. No need to ask what the point of composing two books to be played in equal temperament would have been.


When using Occam's Razor you need to look at the available evidence. When considering the chromatic order of the pieces and the chromatic passages in the pieces the Title is outweighed in my opinion. Like I said, if it looks like a duck... That is a proper use of Occam's Razor.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1857024 - 03/06/12 07:52 AM Re: I've made up my mind about Bach's WTC and ET [Re: Tunewerk]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
Tunewerk:

Sorry, I do not follow you at all. I do not think you know what you are talking about.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1857030 - 03/06/12 08:14 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
Withindale Offline
1000 Post Club Member

Registered: 02/09/11
Posts: 1968
Loc: Suffolk, England
Originally Posted By: UnrightTooner
Originally Posted By: Withindale
Earlier on in this thread, Jeff, you produced Occam's Razor.

The simple answer to the enigma you pose is that Bach wrote his music for instruments like his harpsichords and clavichords which he tuned in his version of the circulating temperaments of that time. No need to ask what the point of composing two books to be played in equal temperament would have been.


When using Occam's Razor you need to look at the available evidence.


Exactly, there does not seem to be any evidence that Bach tuned his instruments in ET as we know it today. As Kees has said.

Given that Bach composed and played WTC in some form of WT he must have been conscious of key colour and character.

Originally Posted By: UnrightTooner
When considering the chromatic order of the pieces and the chromatic passages in the pieces the Title is outweighed in my opinion. Like I said, if it looks like a duck... That is a proper use of Occam's Razor.


I don't disagree. How Bach tuned harpsichords in the eighteenth century does not necessarily determine the best temperament for playing WTC on a modern piano.

One day I'll get a digital keyboard with switchable temperaments ...

_________________________
Ian Russell
Schiedmayer & Soehne, 1925 Model 14, 55" upright
Ibach, 1922 49" upright (project piano)

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#1857057 - 03/06/12 09:23 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
RonTuner Offline
1000 Post Club Member

Registered: 01/21/04
Posts: 1666
Loc: Chicagoland
Originally Posted By: UnrightTooner
Tunewerk:

Sorry, I do not follow you at all. I do not think you know what you are talking about.


This seems to be your entire take on tonal temperaments - Just so we're clear, just because YOU can't follow, doesn't mean someone doesn't know what they are talking about...

I appreciate that you don't GET the reason, or value of tonal temperaments. It would be reasonable if you would realize that others DO appreciate what tonal temperaments can do for the interpretation of music and just leave it at that.

Ron Koval
_________________________
Piano/instrument technician
www.ronkoval.com
@ronkoval

my piano videos:
http://www.youtube.com/profile_videos?user=drwoodwind


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#1857084 - 03/06/12 10:08 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
Ed Foote Offline
1000 Post Club Member

Registered: 05/03/03
Posts: 1186
Loc: Tennessee
Greetings,
It appears that the original title says it all. His mind is made up.
I suppose the bigger question is why are we all gathered around this mind like a bunch of yokels staring at a closed and rusted bear trap?
Regards,

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#1857091 - 03/06/12 10:17 AM Re: I've made up my mind about Bach's WTC and ET [Re: RonTuner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
Originally Posted By: RonTuner
Originally Posted By: UnrightTooner
Tunewerk:

Sorry, I do not follow you at all. I do not think you know what you are talking about.


This seems to be your entire take on tonal temperaments - Just so we're clear, just because YOU can't follow, doesn't mean someone doesn't know what they are talking about...

.....

Ron Koval


I don't understand what he is talking about in relation to what he says iH does. This has nothing to do with temperaments.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1857093 - 03/06/12 10:18 AM Re: I've made up my mind about Bach's WTC and ET [Re: Ed Foote]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
Originally Posted By: Ed Foote RPT
Greetings,
It appears that the original title says it all. His mind is made up.
I suppose the bigger question is why are we all gathered around this mind like a bunch of yokels staring at a closed and rusted bear trap?
Regards,


I wonder, too.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

Top
#1857132 - 03/06/12 11:45 AM Re: I've made up my mind about Bach's WTC and ET [Re: RonTuner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
Originally Posted By: RonTuner
.....

I appreciate that you don't GET the reason, or value of tonal temperaments. It would be reasonable if you would realize that others DO appreciate what tonal temperaments can do for the interpretation of music and just leave it at that.

Ron Koval


Actually I DO appreciate that others value tonal variation in their temperaments. But I see no reason to "leave things at that" if common sense must be left to fly in the breeze.

In a previous Topic I asked for evidence in pieces that were supposedly written for temperaments with tonal variation. There were very, very few examples given. And now after taking a personal look at WTC, I find chromatic evidence that it was written with ET in mind. And the more I think about it the more it makes sense for it to be designated for well tempered instruments. Equal tempered instruments weren't popular... YET!

Frankly, Ron, I think you better look in the mirror if you want to see someone that does not appreciate other's points of view.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

Top
#1857144 - 03/06/12 12:21 PM Re: I've made up my mind about Bach's WTC and ET [Re: Mark R.]
Tunewerk Offline
Full Member

Registered: 03/26/11
Posts: 414
Loc: Boston, MA
Originally Posted By: Mark
Tunewerk, I don't get it. (Granted, I'm a beginner.)

Whether the iH of a piano is high or low, our ear will perceive the M3 as pure when the 5th partial of the lower note is beatless with the 4th of the upper note. The exact frequency of these partials doesn't matter. (Or does it?)

I would have thought that F3-A3 will sound pure when it's beatless, regardless of whether the 5:4 partial match takes place at 880, 881, 882 or 883 Hz.


Hey Mark, that's a great question.. I didn't think about explaining this further when I initially wrote what I did.

What happens in all instruments is an alignment of a whole series of frequencies, not just the ones we focus on when we tune, or talk about in theory.

In an organ with almost no inharmonicity, the series is almost linear and you can get near perfect alignment (hence the pleasure in meantone tunings). In other instruments, it is curved. On top of that, we see 'phantom' partials, varying bandwidth over decay and paraharmonicity, which is not predicted easily by equation.

What's that great quote? I think rXd, the very experienced technician from London, says it.. 'In theory, there is no difference between theory and practice. In practice, there is.'

Consonance is the sense of all partials in both series aligning for a given interval. For example: Even though the M3rd frequencies that we focus on typically when we tune are in the alignment of the first 5/4 ratio, the M3rd consonance really consists of an alignment 'density' of all partials in a 5/4 ratio of the two series:

5/4, 10/8, 15/12, 20/16, 25/20, etc. - theoretically infinite, but realistically degrading around partial 30

This is the basis, incidentally, for Virgil Smith's whole tone tuning.

What creates the degree of consonance for the ear is the number of aligning frequencies vs. non-aligning (% wise), how the bandwidths behave, etc.

This is what makes the octave, fifth and fourth more powerful and pure resonances before the M3rd. They are always favored more in temperament structure too, for this reason.

The octave has a 75% theoretical alignment ratio ([1/2+1/1]/2), the fifth, 41.7% ([1/3+1/2]/2), the fourth, 29.2% ([1/4+1/3]/2), and the M3rd, 22.5% ([1/5+1/4]/2). This is a simple expression for percentage of aligned partials vs. non-aligned partials, or possible consonance 'density'.

These percentages are theoretical. All inharmonicity reduces these numbers by introducing curves into the linear system. These curves can only have a single point of tangency. This is what I was talking about when referring to the spheres, sitting one within the other.

In practice, there's an area of tangency because partials have bandwidth and when they become close, there is an attraction towards sympathetic vibration. This phenomenon is well known by all tuners in unison tuning (the Virgil Smith phenomenon) and written about extensively by Dr. Brian Capleton, Royal College, UK. I don't remember the name of the physicist who first described this phenomenon, but the work is relatively old and well proven.

In any case, there can only be a limited alignment area of the two curves (with iH) and this is what we move when we stretch an interval. This area decreases when iH increases. Also, because the deviation is exponential, the inscribed radii at upward areas of the curve are always theoretically decreasing.

This brings us to your next question:

Originally Posted By: Mark
How could a pure, beatless M3 sound "sour", and and top of that, be "improved" by making it wide?


Sour is a descriptive word, not scientific, so insert whatever word describes the sensation that you hear. But go to a Pramberger, older Young Chang, or any other new Asian import piano and tune a pure M3rd. (The effect is more obvious on these because of higher inharmonicity.)

Then go to a harpsichord and do the same thing.

The piano will have a whining or sour tone comparatively.

This is caused by the close dissonances of these two curves approaching and then departing from one another, no matter how you tune them. However, tuning the interval a little wider will help with the psychoacoustic perception of this problem. Helmholtz did a lot of research on this.

Tuning the interval slightly wide will help lessen the dissonances much higher in the frequency range by making the top note sound less flat. (The sensation originates from the upper partials of the upper note falling flat of the upper partials of the lower note.)

Tuning the M3rds very wide (as in, 14c with ET) takes these audible partials away from near dissonance and causes the top note to be conclusively sharper (its whole series in the near audible range), thereby eliminating the sense of sourness or flatness completely.

We can see how, through this phenomenon, temperaments are suited to instrument design and co-evolve with them. To know the temperament Bach intended for the WTC, we need to look at the instrument he composed upon, as well as the construction of his compositions.
_________________________
www.tunewerk.com

Unity of tone through applied research.

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#1857184 - 03/06/12 01:24 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
Tunewerk:

What you are now saying, without the analogies, is comprehensible. I think it is an enormous leap to say it had anything to do with the shift to ET.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1857527 - 03/06/12 10:48 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1760
Loc: Vancouver, Canada
Originally Posted By: UnrightTooner
In a previous Topic I asked for evidence in pieces that were supposedly written for temperaments with tonal variation. There were very, very few examples given.

None of which survive scrutiny in my mind.
Originally Posted By: UnrightTooner
And now after taking a personal look at WTC, I find chromatic evidence that it was written with ET in mind.

I find evidence only that Bach's writing does not take temperament into account in any manner whatsoever; if he had any particular temperament in mind for the WTC it must have been sufficiently mild to allow all keys to be usable. This is not specific to WTC though; many of his pieces modulate to distant keys.

I personally like the results of believing he didn't care much about temperament as long as things are reasonably in tune. I think it sounds a bit harsh in Werckmeister 3 as Ton Koopman likes to use, and too out of tune in ET on the harpsichord, but fine in ET on the piano. On the clavichord it is not an issue as you can (and should) play all intervals in tune by varying the key pressure.
If you really don't know how to tune your instrument you can use his curls on top of WTC1 which give reasonably mild WT tunings no matter how you interpret it.

Of course if you can't even tune your own instrument you're not a musician.

Originally Posted By: UnrightTooner

And the more I think about it the more it makes sense for it to be designated for well tempered instruments. Equal tempered instruments weren't popular... YET!

Well OK, you changed your mind rather quickly; in the space of one post!

Kees

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#1857548 - 03/06/12 11:31 PM Re: I've made up my mind about Bach's WTC and ET [Re: DoelKees]
Chris Storch Offline
Full Member

Registered: 08/13/09
Posts: 204
Loc: Massachusetts
Originally Posted By: DoelKees

...I find evidence only that Bach's writing does not take temperament into account in any manner whatsoever; if he had any particular temperament in mind for the WTC it must have been sufficiently mild to allow all keys to be usable. This is not specific to WTC though; many of his pieces modulate to distant keys.

I personally like the results of believing he didn't care much about temperament as long as things are reasonably in tune....


Ahhhh... An even-tempered voice of reason. (Notice I did NOT say equal-tempered or well-tempered) Thanks, Doel. Reading this was refreshing! lol I share a similar sentiment.
_________________________
Chris Storch
Acoustician / Piano Technician

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#1857562 - 03/06/12 11:57 PM Re: I've made up my mind about Bach's WTC and ET [Re: Withindale]
Ocngypz Offline
Full Member

Registered: 11/06/11
Posts: 75

Exactly, there does not seem to be any evidence that Bach tuned his instruments in ET as we know it today. As Kees has said.

Given that Bach composed and played WTC in some form of WT he must have been conscious of key colour and character.

Originally Posted By: UnrightTooner
When considering the chromatic order of the pieces and the chromatic passages in the pieces the Title is outweighed in my opinion. Like I said, if it looks like a duck... That is a proper use of Occam's Razor.


I don't disagree. How Bach tuned harpsichords in the eighteenth century does not necessarily determine the best temperament for playing WTC on a modern piano.

One day I'll get a digital keyboard with switchable temperaments ...

[/quote]

I have a dp with switchable temperments.

ET, Pure M, Pure m, Pythagorean, Meantone, Werkmeister, Kirnberger, Equal Flat.

I took the hymn "O Sacred Head Now Wounded" harmony by JS Bach 1701 and played through each temperment.

It finally sounds right!

After 40 odd years grimacing at the harmony in several measures...if JSB didn't use Werkmeister...I'll eat the ivories.

Also Beethoven's "Moonlight" no longer sounds "muddy".

I really have this forum to thank. When I was shopping for a new dp I happened upon this discussion of temperments which got me thinking about my unhappiness at times with my techs and my prior acoustic gps.

It wasn't my temperment, it was my piano's!

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#1857585 - 03/07/12 01:13 AM Re: I've made up my mind about Bach's WTC and ET [Re: DoelKees]
Tunewerk Offline
Full Member

Registered: 03/26/11
Posts: 414
Loc: Boston, MA
Originally Posted By: Kees
I find evidence only that Bach's writing does not take temperament into account in any manner whatsoever; if he had any particular temperament in mind for the WTC it must have been sufficiently mild to allow all keys to be usable. This is not specific to WTC though; many of his pieces modulate to distant keys.


I agree, Kees. Good to have you here.

Originally Posted By: Kees
...and too out of tune in ET on the harpsichord, but fine in ET on the piano.


This is the effect I was referring to.

I think all the available evidence rather clearly points to the conclusion that Bach utilized some form of a mild circulating well temperament. (In my own opinion, probably experimenting with many, as Kees alluded to).
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#1857617 - 03/07/12 03:42 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
Mark R. Offline
2000 Post Club Member

Registered: 07/31/09
Posts: 2052
Loc: Pretoria, South Africa
Thank you, Tunewerk, for your detailed answer to my query. Unfortunately, I don't have a harpsichord, but my parents have one, and I tune it for them regularly. When next I get a chance, I'll be sure to experiment with whole tone tuning (matching the whole partial envelope) on the harpsichord vs. the piano.

However, to rephrase Jeff's statement:

It might well be that ET suits instruments with higher iH because it "fudges" the smaller tangential alignment between partial curves by making M3s decidedly and conclusively wide. But can one conlude from this that instruments with higher iH were actually a driving force for the introduction of ET, or did it develop anyway, and happened to suit those high-iH instruments rather well?
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1922 49" Zimmermann, project piano.
1970 44" Ibach, daily music maker.

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#1857650 - 03/07/12 05:32 AM Re: I've made up my mind about Bach's WTC and ET [Re: Mark R.]
Tunewerk Offline
Full Member

Registered: 03/26/11
Posts: 414
Loc: Boston, MA
Sure, Mark. Glad to.

Originally Posted By: Mark
But can one conclude from this that instruments with higher iH were actually a driving force for the introduction of ET, or did it develop anyway, and happened to suit those high-iH instruments rather well?


I don't think anywhere in my writing, did I say instruments with higher inharmonicity were the driving force for the development of ET.

Equal temperament is a completely independent concept even from music, reaching back to ancient China, being developed by the mathematicians of the enlightenment era, and further defined for music, to be tuned on harmonic stringed instruments as the understanding of string vibration developed.

I think the last part of what you said is closer to what I was saying: '...or did it develop anyway, and happen to suit those high iH instruments rather well?'

Higher inharmonicity instruments didn't lead to the development of the idea of ET in any way. Certainly they led to its refinement and implementation.. but the idea was already there as part of the repository of universal understanding.

Higher inharmonicity instruments were, I think, one driving force behind the increasing use of equal temperament over unequal (those favoring pure intervals in one way or another).

I think another major driving force was the culture that arose at the time. I included this in my earlier posts as an important part of the big picture.

There's a praise of reason and social conformity/uniformity over the course of the industrial revolution. We became dependent on the machine for our survival and addictions. The individual man lost his value and priority to the assembly line and greater social order.

We appear to be even further down that road now. Unequal temperaments appear to be this esoteric indulgence today when before, they were an important and honored expression and part of the social value. (Just as the individual is deemed irrelevant today to the mass social order.)

So I think this finds its way into our music. Equal temperament is a mathematical 'perfection' of uniformity that reflects our social order. After all, haven't we evolved?

These are the two biggest reasons that I can see for the widespread implementation of ET. It's never just one thing.

I think musical composition then followed these driving forces, seeing them as 'advances'; the same roots of culture causing the conditions, then causing the formulation of its reflection in creativity to take advantage of those conditions.
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#1857666 - 03/07/12 07:34 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
Originally Posted By: DoelKees
.....

Originally Posted By: UnrightTooner

And the more I think about it the more it makes sense for it to be designated for well tempered instruments. Equal tempered instruments weren't popular... YET!

Well OK, you changed your mind rather quickly; in the space of one post!

Kees


No, there is a difference between "designate" and "compose." I can "compose" a piece for one purpose and then "designate" it for some other general purpose.
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#1857668 - 03/07/12 07:38 AM Re: I've made up my mind about Bach's WTC and ET [Re: Mark R.]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
Originally Posted By: Mark R.
.....

However, to rephrase Jeff's statement:

It might well be that ET suits instruments with higher iH because it "fudges" the smaller tangential alignment between partial curves by making M3s decidedly and conclusively wide. But can one conlude from this that instruments with higher iH were actually a driving force for the introduction of ET, or did it develop anyway, and happened to suit those high-iH instruments rather well?


Actually, Mark, I only acknowledged that Tunewerk's latter post was comprehensible. I think M3s are off the chart as far as harmonic dissonance is concerned and do not have multiple partial matches that are significant.
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Who taught the first chicken how to peck?

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#1857848 - 03/07/12 02:31 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
DoelKees Offline
1000 Post Club Member

Registered: 05/01/10
Posts: 1760
Loc: Vancouver, Canada
Originally Posted By: UnrightTooner
I think M3s are off the chart as far as harmonic dissonance is concerned and do not have multiple partial matches that are significant.

Agreed and the reason lies in the hammer. The partials of an ideal struck string decay in amplitude as 1/n^2 (n is the partial number) whereas for a plucked string as 1/n. Hence the out of tune M3s of ET are 6 times more prominent on a harpchichord.

Kees

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#1857912 - 03/07/12 04:19 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
Tunewerk Offline
Full Member

Registered: 03/26/11
Posts: 414
Loc: Boston, MA
Originally Posted By: UnrightTooner
Actually, Mark, I only acknowledged that Tunewerk's latter post was comprehensible. I think M3s are off the chart as far as harmonic dissonance is concerned and do not have multiple partial matches that are significant.


Jeff, I have no problem with you judging my posts as incomprehensible. But if you do, at least get the facts straight!

To so consistently misinterpret my explanations and act as though you are summarizing, you are doing a great disservice to those who are reading and trying to understand.

Originally Posted By: Kees
The partials of an ideal struck string decay in amplitude as 1/n^2 (n is the partial number) whereas for a plucked string as 1/n. Hence the out of tune M3s of ET are 6 times more prominent on a harpchichord.


Interesting, Kees. Further evidence.
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Unity of tone through applied research.

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#1858234 - 03/08/12 06:59 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
Tunewerk:

Yes, I have been keeping my comments brief. I am trying to not stray too far off topic.

Can you start a new topic on this subject? I think we have both misunderstood each other. Not that we will agree, but there is much to be said. smile
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Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1858294 - 03/08/12 09:55 AM Re: I've made up my mind about Bach's WTC and ET [Re: DoelKees]
Jake Johnson Offline
Full Member

Registered: 10/20/08
Posts: 84
Originally Posted By: DoelKees
Originally Posted By: UnrightTooner
I think M3s are off the chart as far as harmonic dissonance is concerned and do not have multiple partial matches that are significant.

Agreed and the reason lies in the hammer. The partials of an ideal struck string decay in amplitude as 1/n^2 (n is the partial number) whereas for a plucked string as 1/n. Hence the out of tune M3s of ET are 6 times more prominent on a harpchichord.

Kees


I'm not sure that I understand the equation 1/N^2, for I don't understand the meaning of the "^" symbol. One site ( http://en.wikipedia.org/wiki/List_of_mathematical_symbols ) says that "u ∧ v means the wedge product of vectors u and v. This generalizes the cross product to higher dimensions."

Sorry, But I just don't know what a wedge product, or here, a vector, are. Is there a source that you could point me to? Thanks.


Edited by Jake Johnson (03/08/12 09:56 AM)

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#1858299 - 03/08/12 10:01 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
Phil D Offline
500 Post Club Member

Registered: 01/15/10
Posts: 551
Loc: London, England
The ^ symbol is a substitute for superscript. It tells you that the following numbers/symbols should be written up above, as an index. It's used on the internet when only plain text can be written. It usually means 'to the power of'
So the equation 1/N^2 means "one over N to the power of two" - "one over N squared"
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Phil Dickson
The Cycling Piano Tuner

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#1858300 - 03/08/12 10:04 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4949
Loc: Bradford County, PA
^ means "to the power of..." It is what is used when typing on a keyboard.

So 3^2 is 3 squared, or 3 x 3.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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