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#1854833 - 03/02/12 10:16 AM Re: I've made up my mind about Bach's WTC and ET [Re: Chris Storch]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
Originally Posted By: Chris Storch
Originally Posted By: RonTuner
My experience seems to indicate that:
1. The vast majority don't care, just give them clean unisons and usable octaves
2. A small minority likes the added palette of tonal tunings.
3. A small minority can't stand anything but near ET.


I'd second this wholeheartedly, Ron. And I'd also maybe add that some of the folks in categories 2 and 3 may have something to say about the octaves in category 1. Clean double octave? Triple? Quadruple? Or clean octave+fifths? It's this last part that makes the piano sound in tune with itself, and on some instruments, brings out the sonority of the thing as a whole.

Chris S.

P.S. I thought I read somewhere that one must, of course, turn the curli-cue UPSIDE DOWN in order to understand how Bach tuned.


Pure twelfths also demand a precise ET, in my experience.
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Who taught the first chicken how to peck?

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#1854886 - 03/02/12 12:10 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: 1707
Loc: Vancouver, Canada
Originally Posted By: UnrightTooner
So is there still a debate or is there a consensus?

Neither. The positions have become entrenched and scolars from different camps avoid interaction.

I've read a lot of the literature on the subject but am not an expert at the level of going to Germany to decipher old manuscripts in cold crumbling church archives in the hope of finding a hint, or learning to write old German with a feather to determine how if the curlies are easier to write upside down.

From what I've read I conclude that there is no direct evidence either way regarding how Bach tuned. The strongest indirect evidence is the title of the WTK as there was a word for ET: Gleichzwebend.

Kees

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#1854896 - 03/02/12 12:43 PM Re: I've made up my mind about Bach's WTC and ET [Re: DoelKees]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
Originally Posted By: DoelKees
.....

The strongest indirect evidence is the title of the WTK as there was a word for ET: Gleichzwebend.

Kees


Hmmm, if Werckmeister defined Wohltemperierte and then Bach used this term, hmmm...
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Jeff Deutschle
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Who taught the first chicken how to peck?

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#1854931 - 03/02/12 01:50 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
Originally Posted By: UnrightTooner
Originally Posted By: DoelKees
.....

The strongest indirect evidence is the title of the WTK as there was a word for ET: Gleichzwebend.

Kees


Hmmm, if Werckmeister defined Wohltemperierte and then Bach used this term, hmmm...


Nope, I can't unconvince myself. If it looks like a duck, walks like a duck and quacks like a duck, its a duck regardless of what it is called. I have to think that Bach's version of Well Temperament had evolved to be distinguishable from ET in name only.

I agree with rXd in that Bach was a "devious old prankster."
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Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1854970 - 03/02/12 02:57 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
meadpiano Offline
Full Member

Registered: 11/30/10
Posts: 131
Loc: East TN
I recently aquired Owen Jorgensen's book "Tuning the Historical Temperaments by Ear" (the 1977 book and the 1991). Of course he has a good bit to say about this subject. In the foreword of the 1977 book Peter Yates writes, "to recognize that many of the keyboard variations works by Mozart and early variations works by Beethoven seem to have composed or improvised on an instrument tuned to meantone, whereas none of their sonatas accept that tonal limitaion but were composed for a well-tempered instrument; to appreciate the esthetic gain of playing of playing the works by Chopin and the younger Liszt in a well temperament; and will be able to outline with historic accuracy the gradual change of tuning from an unequal to equal temperament, followed by the compensating shifts from diatonicism to chromaticism to atonality. Similar consideration can be given to the correct tuning of orchestral instruments.
Realizations of this historical information will undoubtedly cause more than a little inconvenience, but the gain in real music and musicianship will be worth the trouble-especially in recorded music, which until now has maintained a nearly solid barrier of incorrect performances."

Chew on that! I have never heard anyone mention that Mozart and Beetoven may have used more than one temperament. Thought that might be interesting to some of you.

-Daniel
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Mead Piano Works
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#1855000 - 03/02/12 03:40 PM 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: 1653
Loc: Chicagoland
Originally Posted By: UnrightTooner
]

Nope, I can't unconvince myself. If it looks like a duck, walks like a duck and quacks like a duck, its a duck regardless of what it is called. I have to think that Bach's version of Well Temperament had evolved to be distinguishable from ET in name only.


Well, there is the whole semantic thing that muddles up the concept. Consider that what we know as the Broadwood's Best temperament was considered ET of the day and you realize that there may have been a bit more... variety to the equal temperaments then that we would accept today!

Ron Koval
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#1855023 - 03/02/12 04:12 PM 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: 1134
Loc: Tennessee
Greetings,
It is entirely plausible that tuners of the late 1700's would have been tuning meantone, or modified meantone, or equal, or whatever else was the easiest way to placate their clientele. It has always been thus. I doubt that the average tuner of the time was anywhere near as well versed in the theory of what they were doing, nor in contact with other tuners. Makes me think that the speed of change would be far slower than today, and it ain't exactly been turning over the last 150 years.
Bach's WTC played in ET and WT are profoundly different musical experiences for some people, identical to others, and to a few individuals, a clear indication that ET was desired for these 24 pieces. There is no accounting for taste. HOwever, I have found that many people have epiphanies when first hearing alternatives to ET, and many fine musicians have lost all interest in it.
Mozart's near total avoidance of the 4 most remote keys for much of his compositional life has to have reason, and the possibility of virtually all keyboards tuned in what was then known as "keyboard tuning" is as sensible a reason as I can imagine. He certainly wasn't afraid of F# etc, why never a composition in it?
It is hard to hear Beethoven's piano music played side by side with a WT and ET and believe he wanted it to sound like ET. I cannot.
Regards,

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#1855041 - 03/02/12 04:28 PM Re: I've made up my mind about Bach's WTC and ET [Re: Chris Leslie]
kpembrook Offline
1000 Post Club Member

Registered: 04/06/10
Posts: 1308
Loc: Michigan
Originally Posted By: Chris Leslie
Quote:
Equal temperament emerged after the development of logarithmic tables -- which weren't around in Bach's time.


Keith, logarithms were developed in the early 1600s, and equal temperament was well known long before Bach's time. Lutenists were, and still are along with guitarists, constrained to equal temperament because of the parallel frets although pitch could be subtly varied in performance.



I'm not a historian of mathematical history so I may be in error on the specifics here. I do recall either attending a lecture or reading a research article that correlated the emergence of the use of equal temperament with mathematical developments. In my memory that was something to do with logarithms.

Thanks for your comments.
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#1855113 - 03/02/12 06:12 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
BDB Online   content
Yikes! 10000 Post Club Member

Registered: 06/07/03
Posts: 21433
Loc: Oakland
Temperaments exist independent of the ability to describe them mathematically.
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#1855189 - 03/02/12 07:42 PM Re: I've made up my mind about Bach's WTC and ET [Re: BDB]
Tunewerk Offline
Full Member

Registered: 03/26/11
Posts: 405
Loc: Boston, MA
I'm not sure what that means. All temperaments can be described mathematically. First theoretically, and then correcting for inharmonicity. The final steps need to be refined by ear, but that is just due to the peculiarities of each instrument.

Maybe you mean the goal is not mathematical?

Maybe you are pointing to how temperaments were developed before the understanding of logarithmic functions, and so being practised artistically, are independent of mathematics?

Mathematics refines the blind and somewhat irregular implementation of all temperament, but of course the end goal is not to satisfy a mathematical equation, but tune an instrument.

I see mathematics here as the servant to better understanding reality, as it usually is.
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#1855428 - 03/03/12 09:33 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
It doesn't take the maths of logarithms to describe an Equal Temperament. Nor is it required to work out how to tune one pretty well - tune all 5ths 1/12 comma narrow is a good starting point, and it doesn't take much of a leap to work out this means for the other intervals, at least roughly.
Logarithms are required for it to be precisely described, but not for it to be tuned.
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#1855512 - 03/03/12 11:56 AM Re: I've made up my mind about Bach's WTC and ET [Re: Tunewerk]
BDB Online   content
Yikes! 10000 Post Club Member

Registered: 06/07/03
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Originally Posted By: Tunewerk
I'm not sure what that means. All temperaments can be described mathematically...


Mathematics was developed to describe temperaments that existed previously by the Pythagoreans and others. The mathematics was not necessary for the temperaments to exist.
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#1855612 - 03/03/12 03:40 PM Re: I've made up my mind about Bach's WTC and ET [Re: BDB]
David-G Offline
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Registered: 01/17/06
Posts: 1242
Loc: London
The earth orbits the sun independently of the mathematical equation which describes its orbit. That does not mean that the mathematics is not helpful in describing and understanding the motion.

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#1855616 - 03/03/12 03:52 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
David-G Offline
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Registered: 01/17/06
Posts: 1242
Loc: London
Just a thought ... given that the mathematics of pitches and intervals is known, it is evident that an ET can exist with semitones related by the twelfth root of 2. However, if the mathematics had not yet been understood, it seems to me that the concept of ET becomes much less obvious; and even if someone thought of it, it might take a lot of research to discover what the semitonal relationship for ET might be.

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#1855972 - 03/04/12 07:14 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
rxd Offline
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Registered: 03/11/09
Posts: 1703
Loc: London, England
Take an average 12 year old with 3-4 years of playing an instrument. Give him some tuning equipment and let him loose with a keyboard to tune.

An aware youngster will have heard that keyboards are tuned by 4ths and 5ths.
So off he goes, C-G-D-E..... Oh, here's my favorite interval, a Maj 3rd. ...why does it sound so awful? What's wrong? It seems it isn't even possible to tune a simple major scale without compromise. He checks a few things and realises it can't be done to have all intervals pure, something has to be done. He tunes a pure major third and juggles the 4ths and5ths to fit. He doesn't know it, but he has come up with something near meantone.
Then he tries how far he can sharpen his major third and juggle the 4ths & 5ths to fit.

At this point, he probably goes to the library to try to find out what's going on.

All this without realising there is any mathematical principle involved.

Something approaching ET can be acheived with no mathematical involvement whatsoever. This could have happened at any time in history where diatonic and chromatic scales were used.

A child could do it. (and did).


Edited by rxd (03/04/12 03:58 PM)
_________________________
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"in theory, practice and theory are the same thing. In practice, they're not." - Lawrence P. 'Yogi' Berra.

Eschew obfuscation.



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#1856427 - 03/05/12 07:28 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
As soon as it is understood that there is a "comma" (pythagorean) when trying to complete the circle of 4ths and 5ths to create an octave, the solution presents itself: Spread the comma out evenly, just like you would when drawing the waist string on your pajamas. It doesn't take any math at all.

Or you could use math, if the relationship of pitches were investigated by shortening the length of a musical string, and you wanted to design an instrument. You would soon realize that you needed to find the 12th root of 2, which can be found with trial and error to what ever accuracy is needed. I wonder why Meantone and then the Well Temperaments ever existed at all, let alone why people are retro about it. No accounting for taste, I guess.

As Bach was truly ahead of his time in composition, I think he was also ahead of his time in temperament. What a throw-back (throw-up?) that moron Mozart was, especially if he stuck with Meantone. But then I really don't care for Mozart. A little bit of him goes a long way with me. Again, no accounting for taste, I guess.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1856455 - 03/05/12 09:02 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
Tunewerk Offline
Full Member

Registered: 03/26/11
Posts: 405
Loc: Boston, MA
Boy, you certainly do know how to incite controversy, eh Jeff? wink

This subject is way too deep for an online thread. Both Stuart Isacoff's book and Ross Duffin's were insufficient to explain it.

It makes sense that simpler music structures centered on meantone. There are so many elements to look at.

First, our evolving sense of musicality. In earlier ages, people were more grounded into nature and the earth. We are today, increasingly removed from nature in our ever heightening self-importance and reliance on technology.. self-reinforcing worlds of egotistical architecture and artificial means.

This means that our sensibilities have changed. We think and live digitally. Why not have a digital instrument? The equal tempered scale reminds us and reassures us of the ideals of the technological age and its perfection.

Equal temperament has co-evolved with the piano, whose power and inharmonicity has increased - the bellowing, egotistical voice for our industrial dominance. Equal temperament sounds better on a high inharmonicity instrument because of the effect of already inherent non-alignment of the partial series in consonant intervals. Carefully balanced non-consonance (rather, equality in dissonance) sounds better than to be confronted with the ear-bending sound of so very near consonance, but in fact dissonance.

So in exchange for power, we have given up consonance. Sounds like our society, doesn't it? Music usually reflects the harmony or unrest of society.

In Mozart and Bach's time, instruments were lower in inharmonicity. Consonant intervals aligned with a sweetness. To hear that sweetness today, you'd have to listen to the organ, guitar, harpsichord. Not the piano.

Hence the desire for meantone and well temperaments: they allowed different combinations of modulation when that kind of consonance was available. Why do you think meantone temperaments are still used on organs today? For instruments with the partial alignment possibility of complete resonance and consonance, equal temperament is consistent and utter dissonance.

From the music, the instrument, and the cultural enlightenment, comes the tuning.

Nevermind the social psychology in recent past generations being more rooted in the individual - and power actually resting in him. Today, sure everyone is an individual - just choose a box from the vending machine and a color to vote for, that has been preselected for you! Express yourself all you want (as long as you fit carefully into your box, and choose from the options given to you).

Our society values uniformity more as the industrial age of increased prosperity requires the 'individual' to function more as a cog to keep the all-important machine running.. hence the value for an all-encompassing solution to tuning!

What for should someone desire styles of expression such as meantone and well temperament? Such silly things of shallow contrivance! (In an industrial society when people have lost their depth of soul.)

Equal temperament is not the best solution! It is not evolved! I could go into so many specifics for this! This is simply what aligns with our instrument design, the current and predominant keyboard pattern for fixed instruments, and what our culture thinks is valuable right now.

About Bach and the WTC: I think it is more important to ask the right questions. Why would Bach care to write these pieces for equal temperament? Would equal temperament have inspired him on the instruments of the day? Is it shown that equal temperament was being precisely implemented in his time (which it requires for any semblance of consonance)?

I am certain that educated musicians of the day (no one could argue any more than Bach) had at least a qualitative understanding of equal temperament. A quantitative understanding was completely derived and fresh on the books for the European world, written by mathematician Marin Mersenne, who died almost 40 years before Bach was born! (Although not the instructions to precisely tune it.)

No one can know what Bach was doing without going back in time and speaking to Bach himself. Although, my opinion is the idea wouldn't even have interested him; he was far more inventive and complex. The instruments then were more subtle and consonant, and experimentation was with well temperament - evidence shows that quasi equal was being implemented, but not precise equal [Jorgensen]. Even if Bach had tuned precise equal temperament, as we define it today, I doubt he would have been interested.

The idea of equal temperament (or simply equal spacing between notes) is extremely old, going back to the fretted lute and early recorded evidence of equal tone division in China to at least 400 AD.

My opinion is that temperament experimentation in the well tempered era grew around the already pre-existing understanding of equal temperament. I think the subtle and expressive variation of well temperament was the goal in itself.
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#1856485 - 03/05/12 09:54 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
Tunewerk:

Pianos had inharmonicity for a long time before the phenomenon was even noticed. Until specific frequencies were measured, it didn't matter a bit, yet ET became prominent. Why? Because it makes it possible to do more than without it. I wouldn't say ET "evolved", it was always there. I would say that music instruments evolved in the direction of ET.
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Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1856497 - 03/05/12 10:25 AM Using historic temperaments [Re: UnrightTooner]
Wally Scherer Offline
Junior Member

Registered: 10/10/05
Posts: 14
Loc: South Florida
I have a lot of historic temperaments in my Verituner 100, which I occasionally mention as being available. But I have NEVER been asked to tune with one. And I have only tried it once, without telling the customer. I wasn't pleased with the results; I don't think she noticed any difference.

But my question is this: Has anyone made a concerted effort in a music school environment to have practice pianos available with different temperaments so the students can compare?

Wally
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#1856518 - 03/05/12 11:18 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
Tunewerk Offline
Full Member

Registered: 03/26/11
Posts: 405
Loc: Boston, MA
Originally Posted By: UnrightTooner
Pianos had inharmonicity for a long time before the phenomenon was even noticed. Until specific frequencies were measured, it didn't matter a bit, yet ET became prominent. Why? Because it makes it possible to do more than without it. I wouldn't say ET "evolved", it was always there. I would say that music instruments evolved in the direction of ET.


Yes, they had inharmonicity, but much less. This is very critical information.

ET was implemented specifically because of the cultural co-evolution that led to instruments with higher inharmonicity. Culture then led to more chaotic, atonal music, further enabled by ET, so we have this view of it allowing more. But allowing more what? (The music that evolved with it and the culture!)

Well temperaments allow everything! (If your goal is very ordered harmonic modulation from key to key.) Meantone temperaments allow everything! (If you are on an organ and your goal is Gregorian-like spiritual consonance in a restricted group of keys.) But our goals are different now aren't they?
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#1856527 - 03/05/12 11:33 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: 4919
Loc: Bradford County, PA
Originally Posted By: Tunewerk
Originally Posted By: UnrightTooner
Pianos had inharmonicity for a long time before the phenomenon was even noticed. Until specific frequencies were measured, it didn't matter a bit, yet ET became prominent. Why? Because it makes it possible to do more than without it. I wouldn't say ET "evolved", it was always there. I would say that music instruments evolved in the direction of ET.


Yes, they had inharmonicity, but much less. This is very critical information.

ET was implemented specifically because of the cultural co-evolution that led to instruments with higher inharmonicity. Culture then led to more chaotic, atonal music, further enabled by ET, so we have this view of it allowing more. But allowing more what? (The music that evolved with it and the culture!)

Well temperaments allow everything! (If your goal is very ordered harmonic modulation from key to key.) Meantone temperaments allow everything! (If you are on an organ and your goal is Gregorian-like spiritual consonance in a restricted group of keys.) But our goals are different now aren't they?


I know I am posting in a more gruff manner than I'd like. Bad dreams..... again. After reading this, I wonder if I have woken up. Inharmonicity caused ET to become popular???
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1856531 - 03/05/12 11:45 AM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
Tunewerk Offline
Full Member

Registered: 03/26/11
Posts: 405
Loc: Boston, MA
Yes. wink

That's part of the equation.

The other parts I tried to write carefully above, to the best of my perception.
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#1856541 - 03/05/12 12:00 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: 4919
Loc: Bradford County, PA
Then I suppose the lutes, even before Bach's time, had significant inharmonicity and that was why they were made in ET intonation. Hoodathunkit!
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1856546 - 03/05/12 12:04 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: 405
Loc: Boston, MA
No, that's not at all what I'm inferring.
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#1856552 - 03/05/12 12:13 PM 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: 4919
Loc: Bradford County, PA
Originally Posted By: Tunewerk
No, that's not at all what I'm inferring.


Of course not. That is why I cannot accept what you are saying.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1856573 - 03/05/12 12:41 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: 1707
Loc: Vancouver, Canada
Originally Posted By: UnrightTooner
What a throw-back (throw-up?) that moron Mozart was, especially if he stuck with Meantone. But then I really don't care for Mozart. A little bit of him goes a long way with me.

Piano technicians are not musicians and should just turn the pins and not try to think too much about it. If I take my car in for an oil change I do not expect the mechanic to have an opinion on manual versus automatic transmission. {tc}*

Kees
(*) Could not find tongue-in-cheek emoticon

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#1856576 - 03/05/12 12:45 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: 405
Loc: Boston, MA
Well, maybe you need to read what I wrote more carefully.

Two entirely different relations.

Equal temperament was known as the lute tuning, before being known as equal temperament, but it was so on the lute for melodic reasons. Equal temperament is sort of a broad subject.

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

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.

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. Pure interval tunings lose their desirability and equal temperament becomes more psycho-acoustically sound.
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#1856578 - 03/05/12 12:46 PM Re: I've made up my mind about Bach's WTC and ET [Re: UnrightTooner]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
Yes, I couldn't possibly have any musical sense: I do not care for Mozart! {tc}
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Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1856584 - 03/05/12 12:51 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: 4919
Loc: Bradford County, PA
Tunewerk:

Something that is not generally understood is that because iH changes logarithmically on a set of frequencies that are also logarithmic, the effects are largely self-correcting. Otherwise we would have to tune very differently than how we do. Dr. White's method works even though iH is not taken into account.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1856590 - 03/05/12 01:04 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: 405
Loc: Boston, MA
Well, that's not at all related to what I was saying above.

In relation to what you mentioned, inharmonicity accelerates the logarithmic curvature of fundamentals in tuning, it doesn't cancel it out. We aren't talking about an exponential function on log paper, this is an exponential function compounding an exponential function.

This is why William Braid White's tuning doesn't inherently produce a very good one. Dr. Sanderson's aural method and others after, considered double octave intervals and stacked M3rds to more accurately gauge the acceleration of the inharmonic effect; similar principle as tuning to higher partial alignments for greater accuracy.
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www.tunewerk.com

Unity of tone through applied research.

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