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#1360796 - 01/29/10 10:35 AM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence exist? [Re: Gadzar]
Olek Offline
6000 Post Club Member

Registered: 03/14/08
Posts: 6333
Loc: France
Originally Posted By: Gadzar
BDB, I've heard your file.

The piano sounds good. I can not distinguish a temperament by listening to a unique isolated piece without comparing it to another temperament. But I've found no disonances in the harmonies, I guess this means ET.

I am with Kamin, I don't know what he exaclty means by "tight" but I didn't like the sound of some unisons.

For the stretch I believe the bass matches perfectly the tenor and treble.


tight : close , not enough air, or light, or whatever very immaterial concept you can come with (while I am sure there is a acoustical explanation which is pretty simple, probably ...)

2 parts in tone : strenght/timing of the stabilisation,(called "attack") and spread of the partial halo.

We tune from those both directions, listening with ears and fingers, checking for the energy spread and its use, and checking as well for the coupling of partials.

With only those 2 concepts you can tune a whole range of unisons.

One of the nicest thing that add energy is the "charging" of upper part of the system : tuning pin and the little wire segment under it. (patented concept !) done with the test blow at the speed of light, it manage to put the system in the best position for an open tone.




Edited by Kamin (01/29/10 10:42 AM)
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#1360882 - 01/29/10 12:42 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence exist? [Re: Gadzar]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3036
Loc: Madison, WI USA
Originally Posted By: Gadzar
Bill,

Now that it is detuned, I mean tuned in ET, I wonder if I can tune it in reverse well and make some tests.

And after showing the reverse well, retune it in EBVT III and record again the same 3 pieces, getting a way to compare the results. But, as I am a bad player, I'm afraid that would be not worth.



Yes, Rafael, do it! Transpose each of the values for the EBVT III up one half step, example, C= the value for C#, C#=D, D=D#, etc. and use the Verituner or whichever ETD.

Ask people to guess which is reverse well and which is ET. I bet that more than half of the guesses will be wrong. It will show that the really bad sound is actually ET and that even a backwards version of a WT actually sounds better than ET does.
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#1360909 - 01/29/10 01:11 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Bill Bremmer RPT]
BDB Online   content
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Registered: 06/07/03
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Originally Posted By: Bill Bremmer RPT
Ask people to guess which is reverse well and which is ET. I bet that more than half of the guesses will be wrong. It will show that the really bad sound is actually ET and that even a backwards version of a WT actually sounds better than ET does.

You should do this, Mr. Bremmer, rather than asking someone else to do it, particularly someone without your experience.
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#1360915 - 01/29/10 01:18 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: BDB]
Silverwood Pianos Offline
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The trouble with anyone doing this is the temperaments could be sabotoaged to make one sound better than the other.
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#1360921 - 01/29/10 01:21 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Silverwood Pianos]
BDB Online   content
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Yes, and very few people would know the difference. Temperaments are not the first thing people listen to. After all, most instruments use different temperaments when they are played normally. Stringed instruments use different temperaments from wind instruments, and neither of them are the same as pianos.
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#1360924 - 01/29/10 01:28 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: BDB]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4789
Loc: Bradford County, PA
Wouldn't it be interesting if people preferred a well temperament transposed up a semitone, or better yet 6 semitones, (I choose not to use the made-up term for this) to a traditional well temperament?

But this is gong down a path other than the subject of this Topic. A new Topic would be appropriate.
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#1360933 - 01/29/10 01:39 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: BDB]
Silverwood Pianos Offline
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Registered: 03/10/08
Posts: 4182
Loc: Vancouver B. C. Canada
Originally Posted By: BDB
Yes, and very few people would know the difference. Temperaments are not the first thing people listen to. After all, most instruments use different temperaments when they are played normally. Stringed instruments use different temperaments from wind instruments, and neither of them are the same as pianos.


Very astute BDB. I was hoping for this to come out here.
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#1360934 - 01/29/10 01:41 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: UnrightTooner]
Silverwood Pianos Offline
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Registered: 03/10/08
Posts: 4182
Loc: Vancouver B. C. Canada
Originally Posted By: UnrightTooner
Wouldn't it be interesting if people preferred a well temperament transposed up a semitone, or better yet 6 semitones, (I choose not to use the made-up term for this) to a traditional well temperament?

But this is gong down a path other than the subject of this Topic. A new Topic would be appropriate.


Now this would surprise a lot of members here, and this is the type of thing that should be attempted...
New topic Jeff??
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#1360940 - 01/29/10 01:50 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Silverwood Pianos]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4789
Loc: Bradford County, PA
Not me. I am not all that interested. But I do wonder sometimes how the Bb trumpet, as the lead voice in North American music education, might effect what people expect to hear. The Bb chord (in the power range) is in just intonation while the C chord has a major third that is 2 cents wider than ET.
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#1360944 - 01/29/10 01:53 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: UnrightTooner]
Gadzar Offline
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Registered: 12/15/06
Posts: 1432
Loc: Mexico City
Well, here it is: Reverse Well Temperament.

This is the Anton Bemetzrieder Reverse Well Temperament of 1808.

It is a strong colored Well Temperament identical to the Vallotti-Young transposed down one semitone.

In this Reverse Well temperament six fifths are pure and the other six fifths are equally tempered, twice as tempered than in Equal Temperament.

The aural procedure to tune this temperament is to tune pure all fifths between white keys and to tune tempered all fifths that involve one or more black keys.

Pure fifths = C-G, D-A, E-B, F-C, G-D, A-E
Tempered fifths = C#-G#, D#-A#, F#-C#, G#-D#, A#-F, B-F#

When testing this temperament it may be possible to find that a theoretical pure fifth is in fact a little tempered to the wide side. Also a theoretical tempered fifth may sound almost pure. Why? Because iH of the strings tends to bring wider than theoretical intervals. Though Major thirds in this temperament behave as theoretically expected. That is M3 of the simpler keys sound harsh and M3s of the remote keys sound more harmonious.

In the video I made a mistake when saying that thirds sound horrible. What I mean is that some of them sound horrible. Some of them in fact sound better than in ET.

The color of the Keys is also inverted. Simpler keys sound busy and remote keys sound harmonious. So in this temperament pieces in the remote keys will sound better than in ET but music written in the simpler keys will sound harsh. So what says UnrightTooner of people liking more Reverse Well would be no surprise. It will depend on how strong is the coloration of the keys. And what keys are most used by this person.

Reverse Well
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#1360946 - 01/29/10 01:58 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Gadzar]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4789
Loc: Bradford County, PA
Gadzar:

What exactly did Anton Bemetzrieder call his temperament?
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#1360982 - 01/29/10 02:47 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: BDB]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3036
Loc: Madison, WI USA
Originally Posted By: BDB
[quote=Bill Bremmer RPT]

You should do this, Mr. Bremmer, rather than asking someone else to do it, particularly someone without your experience.


Well, he already did it and did a fine job! I only wish some music had been played (although the You Tube video stalled before the end, so I don't know what may have been there). I have heard many pianos which have sounded more or less like this although most are not so precisely done. This had the pure 4ths and 5ths perfectly pure and the tempered 4ths & 5ths all tempered twice as much as in ET and all the same. Since reverse well is virtually never deliberately done, they are usually not quite as regularly structured.

That is why I recommended tuning the EBVT III electronically but transposing all the values up 1/2 step (down 1/2 step would work too). The EBVT III is an irregular Victorian style WT. "Irregular" only means that the tempering of the 4ths and 5ths varies from one to another. A reverse EBVT III but without the carefully constructed octaves would be a better representation of what I usually come across. Even that would be more precisely constructed than the usual results of erroneous tempering found by 4ths & 5ths tuners for whom the piano did not "tell" the tuner when the tempering was correct (on topic).
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#1360991 - 01/29/10 02:57 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: UnrightTooner]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3036
Loc: Madison, WI USA
Originally Posted By: UnrightTooner
Gadzar:

What exactly did Anton Bemetzrieder call his temperament?



There is a long section on this in Owen Jorgensen's book, Tuning. He Bemetzreider did not have a name for what he did, only several attempts at what by his description of what he was trying to accomplish would be called ET today. He also berated the methods of other tuners of his time saying that they were too wrapped up in theory and did not follow what was obvious to a musically trained ear.
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Madison WI USA
www.billbremmer.com

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#1360996 - 01/29/10 03:00 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: UnrightTooner]
Gadzar Offline
1000 Post Club Member

Registered: 12/15/06
Posts: 1432
Loc: Mexico City
He beilieved he was tuning Equal Temperament.

It is known as a Pytagorean Temperament used as a substitute from Equal Temperament.

Pytagorean because he tunes pure fifths. Substitute from Equal Temperament because he tempers 6 fifths at 1/6 of a comma aiming to make it "disappear".

He was aware that in just intonation there is a difference betwen flats and sharps i.e. Eb and D#. He called this a "differential coma".

In his own words:

"the differential comas disappear under the hammer of the tuner, and both chromatic and diatonic half tones become equal, the one loosing what the other gaines".

He wrote:

"Tune the fifths for the flats and sharps not quite so full as your ear can bear them, you will make differential comas disappear, without any further tempering. C and F flats will coincide with B and E, and B and E sharps will coincide with F and C".

He used an F fork to start with.

If you do the mathematics of these tuning instructions you'll come to the Vallotti-Young Well Temperament, transposed one half tone down.



Edited by Gadzar (01/29/10 03:29 PM)
_________________________
Rafael Melo
Piano Technician
rafaelmelo@afinacionpianos.com.mx

Serving Mexico City and suburbs.

http://www.afinacionpianos.com.mx

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#1360997 - 01/29/10 03:01 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Silverwood Pianos]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3036
Loc: Madison, WI USA
Originally Posted By: Silverwood Pianos

The trouble with anyone doing this is the temperaments could be sabotoaged to make one sound better than the other.


If you already took Gadzar's rendering of ET as acceptable, there would be no reason to sabotage it. Just put the two recordings, one ET, one reverse well side by side in random order and ask people to identify which is which.

Nobody is trying to fool anybody. I already know the entire world has been fooled already.
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#1361002 - 01/29/10 03:09 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Gadzar]
Gadzar Offline
1000 Post Club Member

Registered: 12/15/06
Posts: 1432
Loc: Mexico City
Here is Bach's Prelude N° 1, in C Major, from the Well Tempered Clavier, with the piano still tuned in Reverse Well.

The prelude loses all its musical meaning. The harmonies create tension where they must release it and visceversa.

Prelude N°1 in C Major. Bach. Reverse Well.



Edited by Gadzar (01/29/10 03:36 PM)
_________________________
Rafael Melo
Piano Technician
rafaelmelo@afinacionpianos.com.mx

Serving Mexico City and suburbs.

http://www.afinacionpianos.com.mx

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#1361011 - 01/29/10 03:18 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Gadzar]
Inlanding Offline
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Registered: 08/05/09
Posts: 1631
Loc: Colorado
Hi Rafael, Bill, and all...

As a novice wanna-be tuner/tech, I have been following this thread with great interest.

Ultimately, is it the musicality of the instrument's tuning as a whole the concern, or exactly which temperament is pressing against the bridge?

Here is a recording of my attempt at trying to acheive an equal temperament. My main checks are m3, M3, M6, M10ths, 17ths. I tune the fourths slightly wider than the fifths are narrow (too old school?). As I do the checks up the scale it is clear that some of the intervals 6ths/10ths/17ths do not increase equally. Does this uncategorically mean it is not equal, or can this be expected due to the scale of the piano, inharmonicity, etc?

I suspect it is not even close to being equal, even though playing in the key of C Major is equally as musical as playing in any other key. What should I be listening for with this? True, some minor/major keys ring more than others, etc. Yes, consistency is key, I know...it is a work in progress. wink

Bach's Prelude in C Major BWV 846

Glen
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#1361032 - 01/29/10 03:44 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Inlanding]
Gadzar Offline
1000 Post Club Member

Registered: 12/15/06
Posts: 1432
Loc: Mexico City
Originally Posted By: Inlanding
Ultimately, is it the musicality of the instrument's tuning as a whole the concern, or exactly which temperament is pressing against the bridge?


Musicality of course! But it is attained through temperament.

Nice tuning. Good unisons. What kind of piano are you playing? How did you do the recording?
_________________________
Rafael Melo
Piano Technician
rafaelmelo@afinacionpianos.com.mx

Serving Mexico City and suburbs.

http://www.afinacionpianos.com.mx

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#1361036 - 01/29/10 03:51 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Gadzar]
Gadzar Offline
1000 Post Club Member

Registered: 12/15/06
Posts: 1432
Loc: Mexico City
Here is: Behind the Waterfall & Desert Rain Medley. By David Lanz. Played in Reverse Well Temperament.

As it is written in the key of A flat Major, it may sound better in Reverse Well than in Equal Temperament. I personally believe that it lacks character, it is too mellow, too much harmonious.

Behind the Waterfall & Desert Rain. David Lanz. Reverse Well Temperament.






Edited by Gadzar (01/29/10 04:05 PM)
_________________________
Rafael Melo
Piano Technician
rafaelmelo@afinacionpianos.com.mx

Serving Mexico City and suburbs.

http://www.afinacionpianos.com.mx

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#1361078 - 01/29/10 04:47 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Gadzar]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3036
Loc: Madison, WI USA
Originally Posted By: Gadzar
Here is Bach's Prelude N° 1, in C Major, from the Well Tempered Clavier, with the piano still tuned in Reverse Well.

The prelude loses all its musical meaning. The harmonies create tension where they must release it and visceversa.

Prelude N°1 in C Major. Bach. Reverse Well.



Beautiful example, Rafael! That is approximately the way most people are used to hearing it. The harmony is so confused, you really don't know why you're modulating, you just are and for most people, that is all ET means to them. This was ET to Bemetzrieder, why not to everybody else for the entire 20th Century?
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#1361080 - 01/29/10 04:49 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Gadzar]
Olek Offline
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Registered: 03/14/08
Posts: 6333
Loc: France
Excellent tone, Inlanding, I like it !
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#1361090 - 01/29/10 04:57 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Gadzar]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3036
Loc: Madison, WI USA
Originally Posted By: Gadzar
Here is: Behind the Waterfall & Desert Rain Medley. By David Lanz. Played in Reverse Well Temperament.

As it is written in the key of A flat Major, it may sound better in Reverse Well than in Equal Temperament. I personally believe that it lacks character, it is too mellow, too much harmonious.

Behind the Waterfall & Desert Rain. David Lanz. Reverse Well Temperament.

There are a lot of people who play a lot of music among mostly the black keys, Debussy, Chopin, etc. and Jazz who have long accepted reverse well tunings by the tuners who habitually tune that way.

As Piqué once suggested, there are people who do NOT hire me because they know I am known for the way I tune the piano and I fully accept that. There are some who proudly profess their loyalty to a certain tuner in town who of course, says he only tunes in ET but what he really does is a fairly strong reverse well. His clients talk about the "sweetness" of his reverse well tunings and are firmly convinced that they are ET. Most of them are the Jazz, late Romantic and Impressionist era players who have become accustomed to the sound. There is even a concert by George Winston that was recorded here in Madison in reverse well.



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Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#1361101 - 01/29/10 05:14 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence exist? [Re: Gadzar]
Mark Cerisano, RPT Offline
500 Post Club Member

Registered: 01/24/10
Posts: 783
Loc: Montreal, Quebec, Canada
Hi,
I use both methods interchangeably when the situation calls for it. M3s definately offer a higher level of precision than 4ths/5ths and that's why they are favoured by many aural tuners. Having said that, tuning using only 4ths/5ths uses M3s and M6s when using check notes, so it is hard to find a technique that is purely 4ths/5ths.
I imagine some technicians preference to using 4ths/5ths may be due to the challenge of being able to hear the coincidental partial (CP) beating clearly. I give courses to students and technicians learning to tune and repair pianos or improve their skills and this is the approach I use to help them hear thirds (and their CPs) better

1. Ghost the coincidental partial. (play interval with no sound. Strike C.P. loudly and listen) Only works if CP is close to freq. of harmonics of each interval note.

2. Turn and move head slowly, all around. This changes the filtering characteristics of the ear canal and filters different frequencies depending on angle. Also, frequencies are more alive at different spots in the room. This technique actually can filter out completely the CP you are trying to hear or actenuate it depending on head angle and position. Just think of Rockwell's painting of The Tuner.

3. Focus the ear like a radio. Gently play the CP a number of times and listen carefully to its frequency. Think of attenuating your ear to that frequency. Then play the interval mf and see if it helps. Use the techniques in #2 to help as well.

Having said that, tuning using only 4ths/5ths does in fact let the piano tell you if, after tuning the whole temperament, you "temper" all the intervals to give the same quality of rolling 5ths and noisy 4ths. It should only fit for that piano if the sizes are those that fit with that piano. But good precision is harder to get than with 3rds in my opinion. Some people say there is no such thing as ET anyway; there is always a variation, perfectly even 3rds are impossible to attain. But we can get very close. Others say perfect ET is over-rated anyway; many professional concert pianists are not able to tell if the thirds are not increasing evenly throughout the temperament. It can become quite the philosophical question.

Mark Cerisano, RPT, Mech.Eng.
Mr. Tuner Piano Service and Techncian's School
mark@mrtuner.com
http://mrtuner.com
514-771-8666
1-866-MR-TUNER(678-8637)
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#1361112 - 01/29/10 05:29 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Olek]
Inlanding Offline
1000 Post Club Member

Registered: 08/05/09
Posts: 1631
Loc: Colorado
Originally Posted By: Gadzar
Originally Posted By: Inlanding
Ultimately, is it the musicality of the instrument's tuning as a whole the concern, or exactly which temperament is pressing against the bridge?


Musicality of course! But it is attained through temperament.

Nice tuning. Good unisons. What kind of piano are you playing? How did you do the recording?


Thanks, Rafael. Thanks to Bill, Issac, and Kent, I am slowly, but surely getting the hang of it, working on consistency, with proper unisons the key. The piano is a remarkably fine condition 1917 Steinway O, the NY Steinway hammers are about 20 years old, unplayed until the last year, the treble strings are original, only the wound strings have been replaced (6 months ago).

As for recording, it is a borrowed Olympus digital recorder. Placement is key - I think I finally found a good placement for it. I am looking at which recorder to purchase for my own.


Originally Posted By: Kamin
Excellent tone, Inlanding, I like it !


Thanks, Issac. Improvements are being made, time and practice are the key elements. Thanks for all your help!



Enjoy the weekend, gentlemen.

Glen
_________________________


March piano audio
https://app.box.com/s/evl3yyp1kj52ve8l069u


A Bit of YouTube

PTG Associate Member

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#1361128 - 01/29/10 05:47 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: UnrightTooner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4789
Loc: Bradford County, PA
Mark:

Welcome Aboard!

All:

I made a large error in this earlier post:

Originally Posted By: UnrightTooner
Not me. I am not all that interested. But I do wonder sometimes how the Bb trumpet, as the lead voice in North American music education, might effect what people expect to hear. The Bb chord (in the power range) is in just intonation while the C chord has a major third that is 2 cents wider than ET.


Actually the major third in the (concert) C major chord of a Bb trumpet, in the power range, would be 16 cents wider than ET.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1361327 - 01/29/10 11:21 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: UnrightTooner]
Gadzar Offline
1000 Post Club Member

Registered: 12/15/06
Posts: 1432
Loc: Mexico City
Today, while tuning the Bemetzrieder Temperament I wonder how difficult it was to the tuners of 19th Century to tune a piano.

They didn't know anything about iH. So I see them trying to tune a pure octave, a pure fifth and finding that the resulting fourth was not so pure! Why? because of iH, we now know that but at the time they didn't.

For them tuning was an art. For us it is only knowledge and technique.


Edited by Gadzar (01/29/10 11:22 PM)
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#1361333 - 01/29/10 11:40 PM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Gadzar]
BDB Online   content
Yikes! 10000 Post Club Member

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There is nothing in the way that I tune that could not have been done by a 19th century tuner.
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#1361369 - 01/30/10 01:01 AM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: BDB]
Gadzar Offline
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Registered: 12/15/06
Posts: 1432
Loc: Mexico City
So you make 19th century tunings? (just kidding)

At that time they didn't know how to tune true ET! The best they could do was Quasi Equal Temperaments.

It was until 20th Century that Braid White wrote his method of tuning ET.

In 19th century tuners did not tuned fast beating intervals, i.e. thirds, sixths, tenths, 17ths. Nor they heared at them. At least not as we now do it estimating beat rates, they only judged their "color".

How do I know what they did? Because of Owen H. Jorgensen's big red book:


Tuning
The Perfection of 18th Century Temperament
The Lost Art of 19th Century Temperament
And
The Science of Equal Temperament




Edited by Gadzar (01/30/10 01:10 AM)
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Rafael Melo
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rafaelmelo@afinacionpianos.com.mx

Serving Mexico City and suburbs.

http://www.afinacionpianos.com.mx

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#1361374 - 01/30/10 01:11 AM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Gadzar]
BDB Online   content
Yikes! 10000 Post Club Member

Registered: 06/07/03
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Nonsense! Nobody knows for certain what was done, but they could have tuned equal temperament at any time.
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#1361493 - 01/30/10 08:42 AM Re: Does a "Let the piano tell you" ET 5ths/4ths sequence ex [Re: Gadzar]
Silverwood Pianos Offline
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Registered: 03/10/08
Posts: 4182
Loc: Vancouver B. C. Canada
Originally Posted By: Gadzar

It was until 20th Century that Braid White wrote his method of tuning ET.


In the year 1911 Joseph C. Miller published the frequencies of all the notes in ET as well as the beating frequencies of all the testing intervals. In Miller’s tables, one can observe the truths of the nearly equal-beating pairs of intervals that are used by the finest aural tuners.

Braid-White republished the same tables for his book in 1917.

Owen Jorgensen /Oct.2009
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