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On the other hand the C# major prelude from WTC2 was written in C and transposed to C# to fit in. And the Ab major fugue was originally in F major. And the B minor Ouverture and suite was originally in C minor. To me that proves key signature "sound" was of almost no significance to Bach.

Kees

Prout,

Did I misunderstand? So, you were not saying that in 12TET octaves (beat rates, as I understood) are progressive, you were saying that also the value of the octave is part of the geometric progression. Sorry, too obvious to miss.

Regards, a.c.,
.

Greetings,
Proof is a very strong word, so I need more info. It is obvious that Bach moved pieces he had written around to various key signatures, so there had to be a reason. Was the fingering easier for a given piece in certain keys? I haven't had a keyboardist yet explain why that would be a causative reason. Perhaps Bach was taking the easy way out and just assigning pieces to keys he needed to make up the whole 24. That doesn't explain why pieces would be placed in other keys elsewhere than in the "collections".

Or, could it be that Bach decided that the piece sounded better in another key because of the emotional impact he could create with a variety of tempering? I don't think that would be beyond his sensibilities, and I don't know that he was so perfect that he wouldn't change his mind. I know that how a pierce is played has more to do with its impact than the temperament,, so do we know that Bach played the pieces in the same manner, regardless of key? The change of key could have been done to favor the way he played the score.

Simply taking the fact that he tried various keys before settling on one for a piece doesn't prove, to me, that he was insensitive to various qualities of key signatures. It seems more plausible that the various harmonic qualities caused the transpositions rather than proving Bach didn't hear any differences.

regards,

Language is such an issue - words are so easily misunderstood.

The concept of 'progressive' when referenced in tuning is one such word. I think I mis-used the word.

I was thinking about the actual 'ordinal' size or rate, if you will, of each and every interval in a theoretical ET, not in cents, but in Hz. All intervals are progressive.

On the other hand, in cents, all intervals are the same size and therefore not progressive.

On the the third hand, the beat rates of a given interval are progressive with the single exception of octaves, which are not progressive.

Surely, we can't be getting back to Bach invented ET and wrote the Well Tempered Clavier Music to prove how great it was, that ET and WT are synonymous and that ET was universally accepted ever since? That is what a lot of people seem to believe and it is written in far more places than any other explanation but you'll never get me to buy it.

If you ask me why Bach reassigned key signatures for various pieces, it was because he found a better match after initially trying some of them in a particular key. WT does not preclude transposition from one key to another. It only dictates that the effects of the new key will be somewhat different depending upon the strength of the WT involved.

To me, the most difficult proposition there ever would be to accept would be that key signature is now and always was meaningless, just chosen randomly and for no particular reason. I tune the way I do for a reason and nobody's rhetoric or theory that virtually all music from the 18th Century to the present was conceived with all key signatures having the same character is going to change my mind.

Not wanting to sound like ganging up, but those facts could be interpreted as Bach changing keys to deliberately exploit the change in sound. A new key changes the sound for the better or worst and not simply a change in pitch or fingering.

Hi Bill, again you say "..the same character", though the fact that all keys sound nicely "in tune" does not mean that one key or another would produce - musically - the same result. And what counts far more than that is the interpretation of the artist, once the ground we offer is fair, meaning in tune. Your description sounds like a simplification, who are you addressing it to, these days?

Regards, a.c.
.

Yes, ."..the third hand.." is what I was referring to, and you replied that all 12TET intervals are equally (100 cents) spaced. Ok, here is what I wrote:

"Saying that ET is called so because all keys sound "equal" is spreading a very ignorant idea. The first ET introduced a geometric progression for the frequency values. Unfortunately, wanting to preserve a zero-beating 2:1 octave, that model could not offer a geometric beat-rate progression for all intervals. Nevertheless, in recent times piano tuners are able to achieve that, which means that every key has a specific and unique "tension". That is what I do every day, assigning different beat rates to all intervals, progressive beat rates in spick-and-span order. This returns the widest "color" palette, while preserving the chords hierarchy faithfully."

And I explained what happens with beats and tensions, depending on how the piano is played:

"Also pianists, like other musicians, can determine and share their intentions, and they do not need an arbitrary Well-cage to enhance emotions, at the opposite, they can go far beyond that - if the scale is in tune - and re-invent harmonic and melodic forms (you do not work with sounds and beats so your ignorance is justified). Concert pianists can exploit dynamics, with the way they (approach and) hit every individual key, the "quantity" and "quality" of energy they put and the timing between one note and the others. By doing that, they change the way beats appear, how beats fuse together and perform an emotional event, in this way and without misplaced out-of-tune interferences pianists control and determine the "tension" of the chords and the melodies they play."

Anything to add on beat-rates, and on tensions or intonation?

What would you say came first, tempering, good intonation or music?

Kind regards, a.c.
.

One final comment:

If 'modern Equal Temperament' tuning is, in fact, ideal, why do the great professional string quartets not use it, but, instead, play in exquisite beatless chords when the music requires it, and in excruciatingly dissonant massively beating chords when the music also requires it?

Hi Prout,

How did you conclude that "...great professional string quartets... ...play in exquisite beatless chords when the music requires it, and in excruciatingly dissonant massively beating chords when the music also requires it"?

Did you read that somewhere? Do you have some papers that you can share?

I use my ears.

alfredo, are you suggesting that this be played using expanded octaves?

[img]http://forum.pianoworld.com/ubbthreads.php/galleries/2643031/quartet.html[/img]


Here is one of many sources for quartet intonation: http://ir.uiowa.edu/cgi/viewcontent.cgi?article=6409&context=etd

For instance, the paper below is about "Fundamental frequency adjustment in barbershop singing":

http://www.speech.kth.se/prod/publications/files/qpsr/1980/1980_21_1_028-042.pdf

Interesting, I was asking because I too use my ear, but never could I come to that conclusion. I believe that if I noticed "beatless chords" or "excruciatingly dissonant massively beating chords", I would conclude that the quartet is not in tune.
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Prout,

In order to produce "harmony" and "emotions" we do not need to go from "beat-less chords" to "excruciatingly dissonant massively beating chords".

You see, good intonation and harmoniousness is not a question of quantity of beats, it is a question of quality, if you know what I mean, quality of beats. This requires control on beats and the tension relative to simple and complex chords, and this makes good intonation (and tuning) a very difficult area.

You ask about "expanded octaves" but most likely if you heard the actual expanded octave I (and I guess many others) tune, you would say that it is a beat-less octave.

Regards, a.c.
.

Yes there is no conclusive information either way so all opinions are subjective.
I believe the French Ouverture was transposed from C to B minor because it had to fit in the Clavier Uebung cycle which already had a piece in C minor.
And similar with the WTC pieces. Also at the time great emphasis was placed in teaching on transposing at sight (to match ambitus of singer/instrument, or match a prelude to the first hymn sung in service) which also makes it hard for me to believe in the "key characteristic" theories.
Finally I play all the WTC pieces with 4 or more sharps also a semitone down. In a WT it sounds slightly different but to not better or worse. I hear a slight change of timbre, much less than the change by using a different harpsichord.

All my personal interpretations of course...

Kees

Hi Prout,

Re-read your statement above. Do you remember when I wrote .."I understand that you associate temperament to music, not to "intonation"?

Your comment above seems to confirm my statement. You seem to use your ear in order to check/confirm a notion that you relate to music, you say for what "music requires". My ear, I would say, seems to focus on good intonation. This means that for you it is a notion that dictates/admits faintly beating and very tense intervals, for me it is "harmoniousness", meaning that intervals can be faintly beating or very tense, but I need to hear them (and chords in general) as part of an "in tune" whole.

Not saying one way is correct and the other is not, just saying that perhaps we use two different filters?

Regards, a.c.
.

The original concept of 'in tune' or 'good intonation', to the best of my knowledge, is based on integral relationships between two frequencies (and their associated harmonics). A perfect octave is 'in tune' when the two fundamental frequencies are in a ratio of 2:1 and all of the harmonics are exact integer multiples of the fundamental frequencies. This does not mean that this perfect octave 'sounds harmonious'. It does sound extraordinarily clear however, if a bit 'sterile' - perhaps not as 'harmonious' as it could be. I like to think of this sound as similar to a quartet singing in a small, dead room - very sterile, as opposed to hearing them sing in a cathedral, where the subtle errors in their tuning (and Doppler shifts of their heads, body relationships) are reflected from the walls and kept alive for a long time, adding to the sonority of the sound.

Tuning a fixed pitch keyboard is a compromise in 'good intonation'. Tuning a piano is even more of a compomise. IMHO, the piano needs to be able to produce clear, sterile sounds using no sustain pedal when the music demands it, and more 'harmonious', ethereal, resonant, alive sounds using the sustain pedal and the entire inharmonic and tempered vibrations that emanate from a good piano, 'well' tuned.

Both ETs and UTs can do this, to some extent, on a piano since the inharmonicity prevents even a single struck string from being 'in tune' with itself. If we define 'harmonious' and 'in tune' as meaning 'nice sounding' as opposed to 'perfectly tuned', then I am in agreement with you.

Regards,

prout

Hi Prout,

You wrote: "The original concept of 'in tune' or 'good intonation', to the best of my knowledge, is based on integral relationships between two frequencies (and their associated harmonics)."...

Yes, I agree on the original concept. The fact is that I have never evaluated 'in tune' and 'good intonation' basing on a concept. I use my musical ear, in general, and beats relations too when I tune a piano.

..."..A perfect octave is 'in tune' when the two fundamental frequencies are in a ratio of 2:1 and all of the harmonics are exact integer multiples of the fundamental frequencies."..

Yes, I am aware of that "original" notion and I had to dismiss that precise notion at the beginning of my research. In fact, I could not tune 2:1 beat-less octaves because that (to my ear) would produce narrower (unpleasant) 5ths and 12ths. Today I can say with both iH or non-iH tones.

..."..This does not mean that this perfect octave 'sounds harmonious'. It does sound extraordinarily clear however, if a bit 'sterile' - perhaps not as 'harmonious' as it could be. I like to think of this sound as similar to a quartet singing in a small, dead room - very sterile, as opposed to hearing them sing in a cathedral, where the subtle errors in their tuning (and Doppler shifts of their heads, body relationships) are reflected from the walls and kept alive for a long time, adding to the sonority of the sound."...

Nice description. A beat-less octave (at 2:1 match), to me, sounded OK. Only later I discovered the octave threshold, a ground where the beat starts appearing, right the beginning of a beating octave. Then I was enabled to work around that area and distinguish a dead-on 2:1 octave from a breathing octave.

..."Tuning a fixed pitch keyboard is a compromise in 'good intonation'."..

Today you state that basing on the best of "your knowledge" (read "..integral relationships between two frequencies.."). I needed to reject the word "compromise", I could not stand it, got reed of the state-of-the-art notions and decided to start a kind of "fresh" path altogether.

..."Tuning a piano is even more of a compromise."..

Yes, it still is a compromise, but only if we consider piano settling, time available for tuning, false strings, wrong scaling and things like that. If we consider the reference scale it does not need to be a compromise anymore.

..."..IMHO, the piano needs to be able to produce clear, sterile sounds using no sustain pedal when the music demands it, and more 'harmonious', ethereal, resonant, alive sounds using the sustain pedal and the entire inharmonic and tempered vibrations that emanate from a good piano, 'well' tuned."...

I am not sure I like the word "sterile" and it would not be music, for me, to "demand" the sound, but my musical ear. I like the rest of your sentence.

..."..Both ETs and UTs can do this, to some extent, on a piano since the inharmonicity prevents even a single struck string from being 'in tune' with itself."..

What you say, a string "..being 'in tune' with itself.." is what I have worked on. The "original" notion is that partials 2 and 3 cannot be combined, numbers - today - prove that that notion is wrong.

..."..If we define 'harmonious' and 'in tune' as meaning 'nice sounding' as opposed to 'perfectly tuned', then I am in agreement with you."

"Harmonious", "in tune" and "nice sounding" does not need to oppose "perfectly tuned". I would reword your sentence and say ..'harmonious' and 'in tune' as meaning 'nice sounding' by ear, as opposed to "non-perfectly tuned" on purpose.

Regards, a.c.
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