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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

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Re: Why do most modern pianos contain 88 keys?
[Re: Fareham]
#2743644 06/11/1812:37 PM06/11/1812:37 PM

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 12-tone 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]
#2743645 06/11/1812:38 PM06/11/1812:38 PM

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 12-tone 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 non-prime natural number, you'll have endless possibilities.

Your formula also could not capture these two endeavors:

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):

Originally Posted by Roshan Kakiya

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:

Originally Posted by Roshan Kakiya

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:

Originally Posted by Roshan Kakiya

12-tone 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.

Originally Posted by Roshan Kakiya

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 12-tone 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]
#2743731 06/11/1806:51 PM06/11/1806:51 PM

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]
#2743738 06/11/1807:15 PM06/11/1807:15 PM

Mathematics is the glue that holds everything I have posted on this thread together. Mathematics combines music theory (circle of fifths), tuning (12-tone equal temperament and well temperament) and keyboard design (12 fifths (each fifth has a value of 700 cents) = 7 octaves = 85 keys). Mathematics can be used to definitively prove that it is not necessary to have more than 85 keys.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743740 06/11/1807:18 PM06/11/1807:18 PM

Mathematics is the glue that holds everything I have posted on this thread together. Mathematics combines music theory (circle of fifths), tuning (12-tone equal temperament and well temperament) and keyboard design (12 fifths (each fifth has a value of 700 cents) = 7 octaves = 85 keys). Mathematics can be used to definitively prove that it is not necessary to have more than 85 keys.

Absolutely not. You have some elementary equation and think, erroneously, that means only 85 keys are needed. Using the same "reasoning" one could argue, again incorrectly, that only 6 octaves are necessary.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743748 06/11/1808:13 PM06/11/1808:13 PM

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):

... trimmed for clarity and simplicity ...

Your formula complicates things in an irrelevant and unnecessary way.

To me, the numbers 88 (or 92, 97, 102, 85) are simple enough. If I'd like to understand these numbers a little more, structurally, then 7+1/4 octaves (and etc.) are good enough.

Your formula fails to support the current standard (88 keys), and fails to predict/extrapolate the advanced/innovative (102 keys). It highly reminds me of the ether theory.

The formula's favorite, 85-key piano, while having great historic merits cannot play popular pieces such as Prokofiev's 3rd concerto, or Rachmaninoff's 3rd concerto.

Last edited by Davdoc; 06/11/1808:17 PM.

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: pianoloverus]
#2743750 06/11/1808:27 PM06/11/1808:27 PM

Absolutely not. You have some elementary equation and think, erroneously, that means only 85 keys are needed. Using the same "reasoning" one could argue, again incorrectly, that only 6 octaves are necessary.

12-TET and well temperament eliminate the Pythagoerean comma. The circle of fifths has been theoretically closed.

A fifth contains 7 semitones. An octave contains 12 semitones.

....Mathematics can be used to definitively prove that it is not necessary to have more than 85 keys.

If you truly believe this, then no matter how advanced you may be in math, you don't understand math.

Understanding math is more than knowing numbers and formulas; it's knowing what the numbers and formulas mean, including the limitations of what they say.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743769 06/11/1810:43 PM06/11/1810:43 PM

Absolutely not. You have some elementary equation and think, erroneously, that means only 85 keys are needed. Using the same "reasoning" one could argue, again incorrectly, that only 6 octaves are necessary.

12-TET and well temperament eliminate the Pythagoerean comma. The circle of fifths has been theoretically closed.

A fifth contains 7 semitones. An octave contains 12 semitones.

I suspect that a lot of the 'amateur' pianists here have studied maths to at least degree level (I'm one of them), and are perfectly capable of elementary algebra and manipulating numbers at GCSE (16 year old) level in their sleep.

It's good to see someone who is full of enthusiasm 'discovering' some of the maths of music as we all did some decades ago. I'm afraid that however much you propose the same thing over and over again, that won't make it any more true than it was in the first instance. Simply put, the 88 keys 'just happened' a bit like Topsy, and became an industry standard.

If you want to see a better explanation of the history of the piano look at this

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: Ken Iisaka]
#2743811 06/12/1804:16 AM06/12/1804:16 AM

Understanding math is more than knowing numbers and formulas; it's knowing what the numbers and formulas mean, including the limitations of what they say.

That is completely right.

Originally Posted by Davdoc

But your 85-key piano still cannot play Tchaikovsky's First Concerto.

I previously mentioned there are likely to be pieces that utilise the 3 extra keys of a piano that contains 88 keys. I have also said keyboards should have at least 85 keys in combination with either 12-tone equal temperament or well temperament to theoretically and physically close the circle of fifths. A keyboard with more than 85 keys is not necessary. The key word is necessary. I have also said more keys can be added to achieve other objectives.

I know what the numbers and formulas mean. They are not random, except for the revised formula for modern pianos which was random (x = yz + 4).

I am aware of the limitations of the 85-key keyboard. That is why I also previously said the 85-key keyboard is mathematically ideal and the 88-key keyboard is practically ideal because the 88-key keyboard has been standard for many years.

What is more mathematically ideal than 12 tempered fifths (each fifth has a value of 700 cents) = 7 octaves = 85 keys?

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743820 06/12/1806:19 AM06/12/1806:19 AM

Understanding math is more than knowing numbers and formulas; it's knowing what the numbers and formulas mean, including the limitations of what they say.

That is completely right.

Originally Posted by Davdoc

But your 85-key piano still cannot play Tchaikovsky's First Concerto.

I previously mentioned there are likely to be pieces that utilise the 3 extra keys of a piano that contains 88 keys. I have also said keyboards should have at least 85 keys in combination with either 12-tone equal temperament or well temperament to theoretically and physically close the circle of fifths. A keyboard with more than 85 keys is not necessary. The key word is necessary. I have also said more keys can be added to achieve other objectives.

I know what the numbers and formulas mean. They are not random, except for the revised formula for modern pianos which was random (x = yz + 4).

I am aware of the limitations of the 85-key keyboard. That is why I also previously said the 85-key keyboard is mathematically ideal and the 88-key keyboard is practically ideal because the 88-key keyboard has been standard for many years.

What is more mathematically ideal than 12 tempered fifths (each fifth has a value of 700 cents) = 7 octaves = 85 keys?

You are contradicting yourself. On one hand you said keyboards with more than 85 keys are not necessary; on the other hand you listed limitation as such.

A larger than 85-key piano is necessary to play a hugely popular piece such as Tchaikovsky's first concerto, premiered 143 years ago in 1875.

And why is 12 tempered fifths mathematically ideal? Why is tempering data (needed for musical reason) ideal for a mathematics?

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]
#2743823 06/12/1806:49 AM06/12/1806:49 AM

Anybody else with such a perfect birthday calc...?...

= = = =

BTW Who is not perfect in VBA calculations, see: & ampersand is the "string addition", to put a ciphre or a letter or some more behind some other. Just try it out.

WORD, Alt-F11 to start VBA, insert the following procedure, and start it by F5 ... The birthday will be seen at the point of your last put-in letters in the open WORD document.

You are contradicting yourself. On one hand you said keyboards with more than 85 keys are not necessary; on the other hand you listed limitation as such.

A larger than 85-key piano is necessary to play a hugely popular piece such as Tchaikovsky's first concerto, premiered 143 years ago in 1875.

I should have clarified that it is not mathematically necessary to have a keyboard that has more than 85 keys because the circle of fifths can be both theoretically and physically closed with a keyboard that has 85 keys. However, it eventually became practically necessary to have a keyboard that has 88 keys because the 88-key keyboard was standardised and pieces that require the extra 3 keys of a keyboard that has 88 keys have been composed. This is the practical limitation of an 85-key keyboard. Therefore, I am not contradicting myself.

Originally Posted by Davdoc

And why is 12 tempered fifths mathematically ideal? Why is tempering data (needed for musical reason) ideal for a mathematics?

The circle or fifths contains 12 fifths. The Pythagorean comma (1.5 ^ 12 is not equal to 2 ^ 7) has been eliminated by 12-TET and well temperament to fix the broken circle of fifths so that 12 fifths = 7 octaves. 7 octaves contain 85 keys.

Everything magically falls into place:

Music theory (circle of 12 fifths) + Tuning (12-TET and well temperament) + Keyboard design (85 keys) = A theoretically and physically complete circle of fifths.

Can you think of another way to theoretically and physically complete the circle of fifths?

A keyboard that has 85 keys is still quite practical. However, the 88-key keyboard has been standard for many years and some composers have utlilised its extra 3 keys so the 88-key keyboard has probably become practically ideal because of this.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743830 06/12/1807:21 AM06/12/1807:21 AM

If you look into the history of the piano you will quickly realize that the instrument didn’t always have 88 keys. In fact, for most of the piano’s history, it had far fewer than 88 keys. It wasn’t until the late 1800s that 88 keys became the standard on pianos. For most of the 1800s the standard for pianos was 85 keys or less. This is why the vast majority of Classical repertoire on the piano only requires between 61-85 keys.

Quote

For the vast majority of pianists 85 keys will not present a serious limitation.

The piano was probably getting developed so it is easy to understand why it had fewer keys than 85 or 88 keys in the past.

Why did the 85-key keyboard become standard? The mathematics seems to explain this quite well but it may or may not be the only way to justify the use of 85 keys.

Why were 3 extra keys added? I cannot find any information that answers this question properly. I will be grateful if somebody can find any research into the transition from the 85-key piano to the 88-key piano. It will clarify many things.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743831 06/12/1807:26 AM06/12/1807:26 AM

I think Robert Estrin @ Living Pianos does a pretty good job of explaining it here from a historical perspective. 88 key pianos began to appear in the 1870 and then 88 was the standard by the 1880's.

By the middle of the 19th century, pianos typically had 85 keys. By the end of the century, pianos began to emerge with the now standard 88 keys. It wasn’t really until the late 1880s when 88 keys became standard on pianos.

Quote

Much like the sostenuto or middle pedal, 88 keys are a late development in the the evolution of the piano and not necessarily something you absolutely need unless you’re playing a great deal of relatively modern music.

By the middle of the 19th century, pianos typically had 85 keys. By the end of the century, pianos began to emerge with the now standard 88 keys. It wasn’t really until the late 1880s when 88 keys became standard on pianos.

Quote

Much like the sostenuto or middle pedal, 88 keys are a late development in the the evolution of the piano and not necessarily something you absolutely need unless you’re playing a great deal of relatively modern music.

However, it still does not explain the rationale behind the addition of the extra 3 keys.

I don't know that a specific, scientific rationale is available and it may be lost to history. I don't know if we can ever definitely say who introduced the 88-key piano and why.

I think it's very interesting that the application of mathematics to music frequencies, the circle of fifths, etc., explains what is going; how intervals work, why we perceive certain intervals and harmonious and others as dissonant, on and on. However, music didn't evolve that way. I doubt that Bach, Beethoven, Mozart, etc., were applying mathematical formulas as they wrote there masterpieces. They simply played what sounded good. They were passionate individuals pushing themselves further and further as composers and artists.

History shows that the piano evolved from the harpsichord over time with one simple innovation after another by passionate individuals seeking to find solutions to the limitations of the instrument. During this important period in music history, a collaboration between many composers, artists, and instrument builders took place which drove the cycle of bring design and engineering to the market place.

At the same time, different manufacturers were bringing designs to the market to differentiate themselves from their competitors to gain market position. Some of these designs failed or did not endure the test of time, while others became industry standards. The 88-key piano may have been just such a move. I have seen Steinway given credit for the 1st 88-key piano, and then I've seen articles that indicate it may have been another manufacturer. But it was probably done as a design to add something to the piano that the others didn't have. At the time, Steinway was a dominant force in the industry. Once they (or another dominant force in the industry) adopted a design or innovation, everyone else followed suit. I believe it's just that simple. Someone added the notes and they sounded good. Consumers began to purchased the 88-key pianos over the 85-key pianos, and everyone got on board.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743845 06/12/1808:30 AM06/12/1808:30 AM

In the late 1880s, piano manufacturer Steinway created the 88-key piano. Other manufacturers followed suit, and Steinway’s model has been the standard ever since.

Steinway & Sons manufactured the first piano with 88 keys in 1869.

It was said that after Steinway & Sons introduced 88-key pianos, other piano makers followed suit in a competitive move, and this configuration stabilised since.

By the middle of the 19th century, pianos typically had 85 keys. By the end of the century, pianos began to emerge with the now standard 88 keys. It wasn’t really until the late 1880s when 88 keys became standard on pianos.

How did Steinway & Sons decide to manufacture pianos with 88 keys rather than 85 keys?

What was the rationale behind this design change?

Originally Posted by Roshan Kakiya

Did Steinway & Sons do this to differentiate its pianos from pianos that were produced by other manufacturers in order to gain a competitive edge?

Did other manufacturers also produce pianos with 88 keys to compete with Steinway & Sons?

Did the abundance of pianos with 88 keys cause the eventual standardisation of the 88-key piano?

Originally Posted by GC13

Originally Posted by Roshan Kakiya

Quote

By the middle of the 19th century, pianos typically had 85 keys. By the end of the century, pianos began to emerge with the now standard 88 keys. It wasn’t really until the late 1880s when 88 keys became standard on pianos.

Quote

Much like the sostenuto or middle pedal, 88 keys are a late development in the the evolution of the piano and not necessarily something you absolutely need unless you’re playing a great deal of relatively modern music.

However, it still does not explain the rationale behind the addition of the extra 3 keys.

I don't know that a specific, scientific rationale is available and it may be lost to history. I don't know if we can ever definitely say who introduced the 88-key piano and why.

I think it's very interesting that the application of mathematics to music frequencies, the circle of fifths, etc., explains what is going; how intervals work, why we perceive certain intervals and harmonious and others as dissonant, on and on. However, music didn't evolve that way. I doubt that Bach, Beethoven, Mozart, etc., were applying mathematical formulas as they wrote there masterpieces. They simply played what sounded good. They were passionate individuals pushing themselves further and further as composers and artists.

History shows that the piano evolved from the harpsichord over time with one simple innovation after another by passionate individuals seeking to find solutions to the limitations of the instrument. During this important period in music history, a collaboration between many composers, artists, and instrument builders took place which drove the cycle of bring design and engineering to the market place.

At the same time, different manufacturers were bringing designs to the market to differentiate themselves from their competitors to gain market position. Some of these designs failed or did not endure the test of time, while others became industry standards. The 88-key piano may have been just such a move. I have seen Steinway given credit for the 1st 88-key piano, and then I've seen articles that indicate it may have been another manufacturer. But it was probably done as a design to add something to the piano that the others didn't have. At the time, Steinway was a dominant force in the industry. Once they (or another dominant force in the industry) adopted a design or innovation, everyone else followed suit. I believe it's just that simple. Someone added the notes and they sounded good. Consumers began to purchased the 88-key pianos over the 85-key pianos, and everyone got on board.

These are probably the only plausible explanations for the standardisation of the 88-key piano.

However, there are other mysteries. Bösendorfer has produced 92-key and 97-key pianos.

The 92-key piano contains 13 complete fifths (x = 7 × 13 + 1 = 92 keys) and the 97-key piano contains 8 complete octaves (x = 12 × 8 + 1 = 97 keys).

Are these instances mathematical coincidences?

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743850 06/12/1808:48 AM06/12/1808:48 AM

The number of fifths has nothing to do with the number of keys on a piano. The range simply kept getting extended when the construction techniques and/or customer desires made it worthwhile to do so. These extra keys often have limited musical value. We now have 102 key pianos too, and soon 108.

What do snowflakes and Chickerings have in common? There are no two exactly alike!