Isn't it the case that 88 keys is a modern standard? Harpsichords had fewer keys.
I think the real question is what composers, especially those pushing the boundaries of the piano such as Beethoven, Lizst, and Rachmaninoff were demanding. Is there evidence of how the modern piano industry came up with 88 keys? There are also physical limits; C8 and A0 rarely sound like a real pitch and are difficult to tune. It must have been harder to tune those end notes 100 years ago so no one wanted the notes beyond them.
My theory is that it's convenient for the piano technician: when you have to replace 88 parts, they often send a bag of 90 in case you screw it up somehow.
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Scott Cole
Talent OR
541-601-9033
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Original Message:
Sent: 06-13-2018 00:13
From: Geoff Sykes
Subject: Why do most modern pianos contain 88 keys?
Because too much is always better than not enough.
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Geoff Sykes, RPT
Los Angeles CA
Original Message:
Sent: 06-12-2018 16:19
From: Roshan Kakiya
Subject: Why do most modern pianos contain 88 keys?
I have based all my thoughts and ideas on information that is hundreds of years old and still relevant today.
Music theory - The Circle of Fifths:
Source: https://www.classicfm.com/discover-music/music-theory/what-is-the-circle-of-fifths/
Tuning - 12-tone equal temperament:
"In 1584, Zhu Zaiyu was the first in the world to systematically calculate the equal temperament of the music scale. His book, New Rule of Equal Temperament, explains a system using 12 equal intervals that is identical with that used around the world today."
Source: http://www.chinaembassy.se/eng/zrgx/wh/t100768.htm
Tuning - Well temperament:
I have combined all this information to "discover" how to theoretically correct the Pythagorean comma (12 "perfect" fifths (the ratio of each fifth is 3/2) is not equal to 7 octaves (the ratio of each octave is 2/1)) and how to physically correct the circle of fifths. We can already see compromises will be needed to achieve this.
Correcting the Pythagorean Comma Theoretically
The perfect fifth has a value of 701.955 cents (3 d.p.). The octave has a value of 1200 cents.
12 perfect fifths = 701.955 × 12 = 8423.46.
7 octaves = 1200 × 7 = 8400.
Pythagorean comma = 12 perfect fifths - 7 octaves = 8423.46 - 8400 = 23.46 cents.
The circle of fifths is broken because of the Pythagorean comma.
This comma can be "tempered out" by reducing the value of all 12 perfect fifths by 1.955 cents.
701.955 - 1.955 = 700 cents.
12 fifths tempered by a twelfth of the Pythagorean comma = 700 × 12 = 8400.
This compromise is necessary to close the circle of fifths.
12 fifths tempered by a twelfth of the Pythagorean comma = 7 octaves.
This can also be achieved by well temperament which contains unequal fifths (some fifths are tempered and some fifths are pure).
The values, in cents, of all 12 fifths (pure and tempered) can be added together to get 8400.
Therefore, the sum of all 12 fifths = the sum of all 7 octaves if well temperament is used.
The Pythagorean comma has been theoretically corrected now. The circle of fifths must be physically corrected now.
Correcting the Circle of Fifths Physically
Keyboards contain octaves that are divided into 12 semitones. Each octave contains 8 white keys and 5 black keys. The mathematics above indicates that, as long as 12-tone equal temperament and well temperament are used, 12 fifths = 7 octaves.
This is the chain of 12 fifths:
C-G-D-A-E-B-F#-C#-G#-D#-A#-F-C.
This is the chain of 7 octaves:
C1-C2-C3-C4-C5-C6-C7-C8.
The number of keys between C1 and C8 (including C1 and C8) must be added to calculate the number of keys that are needed to ensure 12 fifths = 7 octaves. There are 85 keys.
85 keys are needed to physically correct the circle of fifths.
Well temperament is just as valid as equal temperament because it also theoretically corrects the circle of fifths.
Due to the nature of 12-tone equal temperament, every semitone will have a value of 100 cents so it is possible to have an infinite number of keys and still maintain 12-tone equal temperament. The circle of fifths does not need to be physically corrected in this case.
However, well temperament causes semitones to have different sizes. It makes sense to physically correct the circle of fifths in order to effectively use well temperament.
How can the extra 3 keys of an 88-key piano be tuned if well temperament is used? There are many different well temperaments which could make it difficult to track the different sizes of the remaining 3 semitones (85th, 86th and 87th).
How can a fraction of a fifth be tuned if well temperament is used to tune a piano that contains a chain of fifths and a fraction of a fifth?
An 85-key piano is "ideal" for well temperament because a chain of 12 complete fifths can be tuned and there will not be a fraction of a fifth remaining that needs to be tuned.
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Roshan Kakiya
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