"Theoretical/Mathematical Definition of Equal Well Temperament:
A temperament that makes every key usable and has identical semitones
throughout."
Given that this is a forum for discussion of piano tuning topics, I
need to point out that there is a well-established definition of equal
temperament on the piano:
Equal temperament is that temperament in which the beat rates of all
like intervals progress smoothly across the chromatic scale.
There is no ultimate authority when it comes to piano tuning theory.
And piano tuning theory is not fully discussed in the wider world of
music theory, probably because the field of piano tuning is small, and
the problem of applying the theoretical mathematical temperaments to
the inharmonicity-laden piano is a very challenging problem that is
not widely understood.
Even so, piano tuning theory exists among the piano tuning
practitioners and is well-developed and should be recognized and
respected for what it is, one of the great human endeavors.
True, piano tuners do not always agree about what constitutes a good
piano tuning. However, piano tuning organizations around the world
have had some success establishing standards for piano tuning, no
small feat!
There is a great deal of material whose source is piano tuning
practitioners, and this material is worthy of your study. You will not
ever understand piano tuning theory just by studying general music
tuning theory. Give us the credit due our endeavor.
We have repeatedly suggested appropriate piano tuning theory sources.
Do we need to list them yet again?
Original Message------
Peter,
It is theoretically possible to have identical semitones throughout an equal temperament. The definition that I have proposed for Equal Well Temperament is the theoretical definition. For example, every semitone, theoretically, has a size of 100 cents in 12 tones to the pure octave equal temperament.
I like your modification because it accounts for the variations that occur in practice.
Theoretical definition of Equal Well Temperament:
A temperament that makes every key usable and has identical semitones throughout.
Practical definition of Equal Well Temperament:
A temperament that makes every key usable and has virtually identical semitones throughout.
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Roshan Kakiya
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Original Message:
Sent: 06-11-2019 07:49
From: Peter Grey
Subject: Well-Temperament
I think "well temperament" needs to have a standard definition.
I have proposed a simple, clear standard definition and I have also shown how it can be modified to account for equal and unequal temperaments:
"Well Temperament: A temperament that makes every key usable.
Equal Well Temperament: A temperament that makes every key usable and has identical semitones throughout.
Unequal Well Temperament: A temperament that makes every key usable and does not have identical semitones throughout."
Roshan,
Personally, I actually like your proposed definitions, however I would suggest insertion of one qualifying expression in EWT as follows:
Equal Well Temperament: A temperament that makes every key usable and has virtually identical semitones throughout.
Reason: It is actually impossible to achieve total equality in these intervals from a mathematical standpoint. However we can come close enough so that what slight variations do exist in practice become musically irrelevant (in context).
Pwg
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Peter Grey
Stratham NH
603-686-2395
pianodoctor57@gmail.com
Original Message:
Sent: 06-11-2019 04:37
From: Roshan Kakiya
Subject: Well-Temperament
12 tones to the pure octave equal temperament:
Semitone = 1200 × log2(2)1/12 = 1200 × 1/12 × log2(2) = 1200 × 1/12 × 1 = 1200 × 1/12 = 100 cents.
19 tones to the pure twelfth equal temperament:
Semitone = 1200 × log2(3)1/19 = 1200 × 1/19 × log2(3) = 100.10 cents.
7 tones to the pure fifth equal temperament:
Semitone = 1200 × log2(3/2)1/7 = 1200 × 1/7 × log2(3/2) = 100.28 cents.
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Roshan Kakiya
Original Message:
Sent: 06-10-2019 22:16
From: Paul Klaus
Subject: Well-Temperament
ET1/2STEP = (8800 + INHARMONICITY) cents/ 88 ?
Original Message:
Sent: 06-10-2019 18:05
From: Roshan Kakiya
Subject: Well-Temperament
Definition of equal temperament:
"An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio."
Source: https://en.m.wikipedia.org/wiki/Equal_temperament
In this case, equal temperament has not been defined in terms of deviations from pure intervals.
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Roshan Kakiya
Original Message:
Sent: 06-10-2019 17:54
From: Steven Rosenthal
Subject: Well-Temperament
Aren't both Well and Equal temperaments defined by deviations from pure intervals?
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Steven Rosenthal
Honolulu HI
808-521-7129
Original Message:
Sent: 06-10-2019 16:39
From: David Pinnegar
Subject: Well-Temperament
The definition is standard among all who understand the term -
https://en.wikipedia.org/wiki/Well_temperament
"Well temperament means a mathematical-acoustic and musical-practical organisation of the tone system within the twelve steps of an octave, with the goal of impeccable performance in all tonalities, based on the natural-harmonic tone system [i.e., extended just intonation], while striving to keep the diatonic intervals as pure as possible. This temperament acts, while tied to given pitch ratios, as a thriftily tempered smoothing and extension of the meantone, as unequally beating half tones and as equally beating [i.e., equal] temperament."
Equal temperament is not based on the natural-harmonic tone system.
Best wishes
David P
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David Pinnegar, B.Sc., A.R.C.S.
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+44 1342 850594
Original Message------
I think "well temperament" needs to have a standard definition.
I have proposed a clear standard definition and I have also shown how it can be modified to account for equal and unequal temperaments:
https://my.ptg.org/communities/community-home/digestviewer/viewthread?MessageKey=da22d3bf-9ef3-4b7d-acbf-fdaa773c281c&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf
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Roshan Kakiya
Original Message:
Sent: 06-10-2019 14:16
From: David Pinnegar
Subject: Well-Temperament
With respect in my opinion and experience of alternatives, equal temperament or an approximation thereto is not a well temperament as it does not express the music well ;-)
My experience is that a good unequal temperament or "well temperament" can substitute for equal temperament being able to work for music not composed for it, but that equal temperament is inadequate to work for music composed with liklihood of chromatic exploration and exploitation.
Were your TV or computer screen be confined to black and white you would not consider it a "well" device at all whereas colour screens can serve well even for black and white.
Equal temperament is a genre unto itself.
There is actually an important reason for using a well temperament rather than a pseudo equal temperament. Whereas equal temperament gives the piano a glistening conflict of harmonics with scale notes, so a shimmering characteristic in which nothing is still, your pseudo equal calms that down. When we have a well designed well temperament giving variations between keys then depending on the key we can choose what is still and what is glistening. Sunshine and shade rather than constant grey skies.
Best wishes
David P
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David Pinnegar, B.Sc., A.R.C.S.
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+44 1342 850594
Original Message------
Is a piano that has been tuned to an approximation of equal temperament a "well-tempered piano"?
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Roshan Kakiya
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