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Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743514
06/10/18 07:38 PM
06/10/18 07:38 PM

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Roshan Kakiya
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Re: Why do most modern pianos contain 88 keys?
[Re: Kenny Cheng]
#2743549
06/10/18 11:24 PM
06/10/18 11:24 PM

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Probably not much differently. Now, if they had pencils with erasers, or ballpoint pens, that would have made a real difference!
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Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743577
06/11/18 04:49 AM
06/11/18 04:49 AM

Joined: Mar 2013
Posts: 914 Germany
patH
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C Major is a popular key. A is the note used to tune.
The only problem with 88 keys is that in Germany, piano lovers should avoid naming their clubs with an 88 in it. In Germany, since 1933, 88 has another connotation than pianos; one that nonextremists should try to avoid.
My grand piano is a Yamaha.



Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743593
06/11/18 08:27 AM
06/11/18 08:27 AM

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Fareham
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I think a piano with 85 keys is mathematically ideal because 12 perfect fifths = 7 octaves. Well, it's not. Pythagorus discovered this around 2500 years ago. The circle of 12 perfect fifths is a frequency ratio of 129.75 (approx) while 7 octaves is a frequency ratio of 128. JS Bach and a few others in the late 17th / early 18th century sought to correct this by 'fudging' the intervals. Bach showed it worked in all 24 keys by writing "The Welltempered Klavier". Being so popular he did another 24, and hence the '48'.
The English may not like music much, but they love the sound it makes ... Beecham



Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743604
06/11/18 09:44 AM
06/11/18 09:44 AM

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Roshan Kakiya
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Well, it's not. Pythagorus discovered this around 2500 years ago. The circle of 12 perfect fifths is a frequency ratio of 129.75 (approx) while 7 octaves is a frequency ratio of 128. That is the famous Pythagorean comma which has a value of 23.46 cents. JS Bach and a few others in the late 17th / early 18th century sought to correct this by 'fudging' the intervals. Bach showed it worked in all 24 keys by writing "The Welltempered Klavier". Being so popular he did another 24, and hence the '48'.
This comma was corrected a full century before J. S. Bach was born (1685  1584 = 101). Zhu Zaiyu invented 12tone equal temperament during the late 16th century in 1584: 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://pl.chinaembassy.org/pol/wh/wh/t129181.htm.Here are more sources that confirm this: https://www.teoria.com/en/articles/temperaments/05equaltemperament.phphttps://books.google.co.uk/books?id...%20temperament%20zhu%20zaiyu&f=false12 perfect fifths = 7 octaves = 85 keys. This is mathematically ideal and it can be achieved by using either 12tone equal temperament or well temperament because they both eliminate the Pythagorean comma. Therefore, everything I have said in the following quote is correct: I also don't get why specifically 12 perfect fifths are ideal. 12 perfect fifths (1.5 ^ 12) later, you arrive 129.746 x base frequency, which deviates from the "ideal" octave of 128 x base frequency. 12tone equal temperament and welltemperament solve that problem. 12tone equal temperament: 12 perfect fifths = (2 ^ (7/12)) ^ 12 = 128. 7 octaves = 2 ^ 7 = 128. Therefore, 12 perfect fifths = 7 octaves. x = 12 (the number of semitones within an octave) × 7 (7 octaves = 12 perfect fifths) + 1 = 85 keys. This proves that a piano with 85 keys is mathematically ideal, although it is no longer practically ideal because pieces have probably been composed that utilise the extra 3 keys of a piano with 88 keys. The 85key keyboard is only mathematically ideal. It is no longer practically ideal.
The 88key keyboard has been standard for many years now. Therefore, it is practically ideal. It seems to be just right because it has been effective for many years.



Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743623
06/11/18 12:19 PM
06/11/18 12:19 PM

Joined: Jun 2016
Posts: 306 Maryland, USA
Davdoc
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Well, it's not. Pythagorus discovered this around 2500 years ago. The circle of 12 perfect fifths is a frequency ratio of 129.75 (approx) while 7 octaves is a frequency ratio of 128. That is the famous Pythagorean comma which has a value of 23.46 cents. JS Bach and a few others in the late 17th / early 18th century sought to correct this by 'fudging' the intervals. Bach showed it worked in all 24 keys by writing "The Welltempered Klavier". Being so popular he did another 24, and hence the '48'.
This comma was corrected a full century before J. S. Bach was born (1685  1584 = 101). Zhu Zaiyu invented 12tone equal temperament during the late 16th century in 1584: 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://pl.chinaembassy.org/pol/wh/wh/t129181.htm.Here are more sources that confirm this: https://www.teoria.com/en/articles/temperaments/05equaltemperament.phphttps://books.google.co.uk/books?id...%20temperament%20zhu%20zaiyu&f=false12 perfect fifths = 7 octaves = 85 keys. This is mathematically ideal and it can be achieved by using either 12tone equal temperament or well temperament because they both eliminate the Pythagorean comma. Therefore, everything I have said in the following quote is correct: I also don't get why specifically 12 perfect fifths are ideal. 12 perfect fifths (1.5 ^ 12) later, you arrive 129.746 x base frequency, which deviates from the "ideal" octave of 128 x base frequency. 12tone equal temperament and welltemperament solve that problem. 12tone equal temperament: 12 perfect fifths = (2 ^ (7/12)) ^ 12 = 128. 7 octaves = 2 ^ 7 = 128. Therefore, 12 perfect fifths = 7 octaves. x = 12 (the number of semitones within an octave) × 7 (7 octaves = 12 perfect fifths) + 1 = 85 keys. This proves that a piano with 85 keys is mathematically ideal, although it is no longer practically ideal because pieces have probably been composed that utilise the extra 3 keys of a piano with 88 keys. The 85key keyboard is only mathematically ideal. It is no longer practically ideal.
The 88key keyboard has been standard for many years now. Therefore, it is practically ideal. It seems to be just right because it has been effective for many years.Being a lay person myself, I am reasonably aware of the existence of temperament, or tempered pitch of these notes. A "perfect" (3:2) perfect fifth, times 12 fifths, will not lead to a "perfect" (2:1) octave times 7 octaves. Temperaments exist to solve the problem by musically acceptable compromises. So your formula still, to me, does not add anything. The current most popular design, as others mentioned above, came from a gradual process of demand, practicality, then de facto standard.
1969 Hamburg Steinway B, rebuilt by PianoCraft in 2017 2013 New York Steinway A Kawai MP11
Previously: 2005 Yamaha GB1, 1992 Yamaha C5



Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743633
06/11/18 12:55 PM
06/11/18 12:55 PM

Joined: May 2017
Posts: 463 Rural UK
Fareham
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I also don't get why specifically 12 perfect fifths are ideal. 12 perfect fifths (1.5 ^ 12) later, you arrive 129.746 x base frequency, which deviates from the "ideal" octave of 128 x base frequency. One can also argue whether fifths are more "harmonious" than octaves. 12 perfect fifths = (2 ^ (7/12)) ^ 12 = 128. This proves that a piano with 85 keys is mathematically ideal, although it is no longer practically ideal because pieces have probably been composed that utilise the extra 3 keys of a piano with 88 keys.On my planet a perfect fifth is a ratio of 1.5 (3/2), not 2^(7/12) which is 1.49831. Incidentally your link makes the reference "Starting in the mid 18th century, equal temperament became the universal standard tuning system for all the instruments used today", so maybe, just maybe JS Bach was in the vanguard when he posited that you don't need to retune your harpsichord when a piece is in a different key. There is an excellent Howard Goodall series which goes into all this in a bit more depth, and I suspect you really, really need to watch it. As for the numbers of keys on a piano, you need to consider the history of the harpsichord, virginals, spinet and other keyed instruments for which so much music was written before the piano. I find your arguments extremely hard to understand, in light of what I know of this history. Incidentally in 60+ years of playing the piano, I have come across 1 piece by Debussy that uses those top three keys and "Jet d'eau", a seriously awful piece by Sidney Smith , a thankfully long forgotten Victorain composer, which also does.
The English may not like music much, but they love the sound it makes ... Beecham



Re: Why do most modern pianos contain 88 keys?
[Re: Fareham]
#2743644
06/11/18 01:37 PM
06/11/18 01:37 PM

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Roshan Kakiya
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On my planet a perfect fifth is a ratio of 1.5 (3/2), not 2^(7/12) which is 1.49831. Thank you for correcting me. I think the term "perfect fifth" should be reserved for a fifth whose frequency ratio is 3:2. Any other fifth can be called a "fifth". I think the mathematics just indicates how 1.5 ^ 12 ≠ 2 ^ 7 can be solved both theoretically and practically. The circle of fifths can be closed, theoretically, by tempering every perfect fifth so that the value of each fifth is 700 cents. Additionally, the circle of fifths can be closed, practically, by ensuring a keyboard has at least 85 keys. It is with this combination of 12tone equal temperament (or well temperament) and a keyboard with 85 keys that the circle of fifths can be truly closed both theoretically and practically. The mathematics just shows us that keyboards should have at least 85 keys to make this possible. This is a musical consideration. I am merely using mathematics to show how this can be achieved. More keys can be added to achieve other objectives but there should be at least 85 keys to ensure the circle of fifths remains both theoretically and practically closed.



Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743677
06/11/18 04:48 PM
06/11/18 04:48 PM

Joined: Jun 2016
Posts: 306 Maryland, USA
Davdoc
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On my planet a perfect fifth is a ratio of 1.5 (3/2), not 2^(7/12) which is 1.49831. Thank you for correcting me. I think the term "perfect fifth" should be reserved for a fifth whose frequency ratio is 3:2. Any other fifth can be called a "fifth". I think the mathematics just indicates how 1.5 ^ 12 ≠ 2 ^ 7 can be solved both theoretically and practically. The circle of fifths can be closed, theoretically, by tempering every perfect fifth so that the value of each fifth is 700 cents. Additionally, the circle of fifths can be closed, practically, by ensuring a keyboard has at least 85 keys. It is with this combination of 12tone equal temperament (or well temperament) and a keyboard with 85 keys that the circle of fifths can be truly closed both theoretically and practically. The mathematics just shows us that keyboards should have at least 85 keys to make this possible. This is a musical consideration. I am merely using mathematics to show how this can be achieved. More keys can be added to achieve other objectives but there should be at least 85 keys to ensure the circle of fifths remains both theoretically and practically closed. We shouldn't limit ourselves to octaves or fifths. We should explore the harmonic beauty of 2nd (2 semitones), diminished 3rd (3), diminished 5th (6), ninth (14), thirteenth (21), all factors of 84, and design pianos accordingly. Your formula is essentially just X = Y + 1. Make Y a nonprime natural number, you'll have endless possibilities. Your formula also could not capture these two endeavors: Sauter microtone pianoSinnakken allwhitekey piano
1969 Hamburg Steinway B, rebuilt by PianoCraft in 2017 2013 New York Steinway A Kawai MP11
Previously: 2005 Yamaha GB1, 1992 Yamaha C5



Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743689
06/11/18 05:50 PM
06/11/18 05:50 PM

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Roshan Kakiya
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The following formula is a simple formula that has a simple purpose. It does not need to be more complicated than it already is (not complicated): I have created a formula that seems to correctly identify the number of keys that fit within a given quantity of a specified interval:
x = yz + 1.
x = The number of keys. y = The number of semitones within a specified interval. z = The quantity of the specified interval. The following formula is an artificial adaptation of the formula above. It should be disregarded and ignored because it has no purporse or meaning: Revised formula for modern pianos:
x = yz + 1 + 3 = yz + 4. The following information and formulas should be considered carefully. This is because everything that has been included in the quotes below works flawlessly. You will understand this if you also do the maths: 12tone equal temperament:
12 fifths = (2 ^ (7/12)) ^ 12 = 128.
7 octaves = 2 ^ 7 = 128.
Therefore, 12 fifths = 7 octaves.
x = 12 (the number of semitones within an octave) × 7 (7 octaves = 12 fifths) + 1 = 85 keys. I think the mathematics just indicates how 1.5 ^ 12 ≠ 2 ^ 7 can be solved both theoretically and practically. The circle of fifths can be closed, theoretically, by tempering every perfect fifth so that the value of each fifth is 700 cents. Additionally, the circle of fifths can be closed, practically, by ensuring a keyboard has at least 85 keys.
It is with this combination of 12tone equal temperament (or well temperament) and a keyboard with 85 keys that the circle of fifths can be truly closed both theoretically and practically. The mathematics just shows us that keyboards should have at least 85 keys to make this possible. This is a musical consideration. I am merely using mathematics to show how this can be achieved.
More keys can be added to achieve other objectives but there should be at least 85 keys to ensure the circle of fifths remains both theoretically and practically closed.



Re: Why do most modern pianos contain 88 keys?
[Re: Mark_C]
#2743737
06/11/18 08:12 PM
06/11/18 08:12 PM

Joined: Nov 2010
Posts: 5,636 Hobart, Australia
ando
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Why are you so focused on formulas, and what fits them or doesn't?
I think it might be interesting if you said something about that, rather than just keeping on giving more and more formulas.
(And I'm even someone who loves formulas, and anything about them.) These formulas are his simple ones  you should see his threads on the technician's forum and Pianostreet. I think some people see the world in terms of numbers and Roshan is one of those people. They believe truth is found in numbers. To the rest of us, it's utterly baffling and irrelevant.



Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743740
06/11/18 08:18 PM
06/11/18 08:18 PM

Joined: Jun 2003
Posts: 27,107 Oakland
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The mathematics is descriptive, not prescriptive!
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Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743742
06/11/18 08:21 PM
06/11/18 08:21 PM

Joined: May 2006
Posts: 4,755 Georgia, USA
terminaldegree
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Mathematics can be used to definitively prove that it is not necessary to have more than 85 keys. False.




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