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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.

Example 1:

The specified interval is the octave which contains 12 semitones. There are 7 octaves.

Therefore, y = 12 and z = 7:

x = 12 × 7 + 1 = 85 keys.

Example 2:

The specified interval is the perfect fifth which contains 7 semitones. There are 12 perfect fifths.

Therefore, y = 7 and z = 12:

x = 7 × 12 + 1 = 85 keys.

We can say, at this point, that only 85 keys are needed to close the circle of fifths.

Why do most modern pianos contain 88 keys if only 85 keys are needed to close the circle of fifths?

Revised formula for modern pianos:

x = yz + 1 + 3 = yz + 4.

Verification:

An octave contains 12 semitones and there are 7 octaves.

x = 12 × 7 + 4 = 88 keys.

Example 3:

The Bösendorfer Model 290 Imperial contains 8 octaves and 97 keys. We can use the original formula to verify this.

x = 12 × 8 + 1 = 97 keys.

Why does this piano not contain 3 more keys? We can use the revised formula to calculate the number of keys it should have based on the design of most modern pianos:

x = 12 × 8 + 4 = 100 keys.

Why does the design of the Bösendorfer Model 290 Imperial not match the design of most modern pianos?

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Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743373 06/10/1808:27 AM06/10/1808:27 AM

Because your "formula" was designed to match 88 key piano. There's no reason why a piano with a different number of keys should be described by your formula. The number of keys on a piano at any point in the piano's history has nothing to do with your formula.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743375 06/10/1808:30 AM06/10/1808:30 AM

I think the design of the Bösendorfer Model 290 Imperial is much better than the design of most modern pianos because it contains 8 complete octaves (97 keys) rather than 7 octaves plus a fraction of an octave (88 keys).

88 keys also contain 12 perfect fifths plus a fraction of a perfect fifth so the use of 88 keys is not ideal.

I think pianos with 92 keys are good as well because they contain 13 complete perfect fifths:

x = 7 × 13 + 1 = 92 keys.

Conclusion:

I think a piano with 85 keys is mathematically ideal because 12 perfect fifths = 7 octaves. However, it is not likely to be practically ideal because pianos with 88 keys have been invented which means there are likely to be pieces that utilise the extra 3 keys that are available.

A piano with 97 keys is more practical because every piece that has been composed for a piano with 88 keys can be played on it and it also contains 8 complete octaves. However, it may be difficult to distinguish between the highest and the lowest notes since their sound may be too low or too high for the ears.

Overall, I think a piano with 92 keys is ideal. This is because it contains 13 complete perfect fifths, the ears will probably be able to distinguish between its highest and lowest notes and every piece that has been composed for a piano with 88 keys can be played on it.

The range of the keyboard seems to be getting larger as time passes by. Therefore, a piano with 97 keys could become ideal in the future if many pieces that utilise its extra keys have been composed. I do not think a piano with more than 97 keys is likely to be practical because it may be very difficult, or maybe nearly impossible, for the ears to distinguish between the sound of its highest and lowest notes.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743377 06/10/1808:39 AM06/10/1808:39 AM

The number of keys on the piano at various stages in its development is not related to the mathematical thinking you expressed. What you consider to be "mathematically ideal" is not relevant.

Last edited by pianoloverus; 06/10/1808:40 AM.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743378 06/10/1808:40 AM06/10/1808:40 AM

I equally do not see the point of your formula. You made your formula to fit into current piano designs.

For example, in (Western) music system, the most commonly used intervals are 12 semitones within one octave. This is not necessarily the case with other systems. One can argue that a different piano can be designed to have 22 keys within one octave.

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.

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]
#2743379 06/10/1808:41 AM06/10/1808:41 AM

The number of keys on a piano at any point in the piano's history has nothing to do with your formula.

Pianos in the past contained 85 keys. I cannot think of any other way to justify the use of 85 keys. Please do share another reason, other than 12 perfect fifths = 7 octaves = 85 keys, that justifies the use of 85 keys if you can do that.

Re: Why do most modern pianos contain 88 keys?
[Re: Davdoc]
#2743381 06/10/1808:49 AM06/10/1808:49 AM

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-tone equal temperament and well-temperament solve that problem.

12-tone 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.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743385 06/10/1809:17 AM06/10/1809:17 AM

I'm not sure where that formula came from or how important it is in the scheme of all things pianos. But, quite frankly, I just like the evenly rounded number of 88 keys beginning with A0 and ending with C88. Although, I would venture to say that most pianists/players rarely or never use A0 or C88. It's like the saying it is better to have it and not need it than to need it and not have it.

As far as any mathematical formula proving to be ideal, it's like saying algebra is something you use throughout your life every single day. Hence, theory and reality are often two different things, proven or not.

Interesting discussion...

Rick

Piano enthusiast and amateur musician: "Treat others the way you would like to be treated". Yamaha C7. YouTube Channel

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743386 06/10/1809:18 AM06/10/1809:18 AM

This is all numerology. Most pianos have 88 keys because if you have any more in the treble, it is difficult to fit the mechanism in, and if you have any more in the bass, it is difficult to make the tones distinct enough to distinguish. There are pianos that have more than 88 keys, but they are very large pianos (> 7 feet) with extra notes in the bass. Making a small piano with more than 88 keys is difficult and pointless, and most pianos are small. Since most pianos have 88 keys, that is what composers write for, There is little point to write music that most people would not be able to play.

If you want to do some mathematics that will show this, look at the difference in frequency of the lowest notes of the piano, and the geometry of the highest notes.

Semipro Tech

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743388 06/10/1809:20 AM06/10/1809:20 AM

As others have pointed out math has nothing to do with the range of a piano. In Beethoven's time he wanted more notes and some of his pieces show the range limitations of the pieces of his time. The invention of the iron frame allowed the range to go higher. You don't find composers asking for higher notes anymore. The addition of lower notes in the Bosendorfer Imperial is more to move the more commonly used low notes farther into the soundboard than to extend the low range of the instrument. The piano is a musical instrument and its development was driven by musical considerations not mathematical. I've yet to hear a piece that consists of perfect fifths from the bottom to the top.

The number of keys on a piano at any point in the piano's history has nothing to do with your formula.

Pianos in the past contained 85 keys. I cannot think of any other way to justify the use of 85 keys. Please do share another reason, other than 12 perfect fifths = 7 octaves = 85 keys, that justifies the use of 85 keys if you can do that.

That ONE particular compass MIGHT have been used to have an integer number of octaves but it could have also been for purely musical(how high they could make notes that sounded musical at the time) or aesthetic(not ending on a black key) or design construction(as BDB explained) reasons.

As virtually every responder has indicated, trying to use a formula to describe a totally subjective mathematical ideal for the number of keys on a piano is not relevant.

Last edited by pianoloverus; 06/10/1810:04 AM.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743397 06/10/1810:05 AM06/10/1810:05 AM

I think the only thing I have made clear is that a piano with 88 keys is an ideal compromise.

That's right.

88 is a nice round number: not too big (like 100) and not too small (like 76); in other words, just nice.

Also, it's divisible by 2, 4, 6 (OK, not 6 ), 8 and 11. Therefore it's not a prime number.

Last but not least, anyone with long enough arms can play my latest opus, which requires the pianist to play both the top C and the bottom A simultaneously and repeatedly without strain. Whereas the Bösendorfer Imperial would be a bit of a stretch for a pianist to play the notes at both extremities of the keyboard at the same time without an arm extension........

"I don't play accurately - anyone can play accurately - but I play with wonderful expression. As far as the piano is concerned, sentiment is my forte. I keep science for Life."

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743417 06/10/1811:17 AM06/10/1811:17 AM

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.

This general formula really is useful. It is not tied to any specific keyboard instrument. It is just an easy way to calculate the number of keys a keyboard contains, without counting, if the number of octaves (or any other interval) it contains is known.

For example, a keyboard which contains 5 octaves has 61 keys:

x = 12 (number of semitones within an octave) × 5 (number octaves this specific keyboard contains) + 1 = 61 keys.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743421 06/10/1811:45 AM06/10/1811:45 AM

I think the design of the Bösendorfer Model 290 Imperial is much better than the design of most modern pianos because it contains 8 complete octaves (97 keys) rather than 7 octaves plus a fraction of an octave (88 keys).

88 keys also contain 12 perfect fifths plus a fraction of a perfect fifth so the use of 88 keys is not ideal.

I think pianos with 92 keys are good as well because they contain 13 complete perfect fifths:

x = 7 × 13 + 1 = 92 keys.

Conclusion:

I think a piano with 85 keys is mathematically ideal because 12 perfect fifths = 7 octaves. However, it is not likely to be practically ideal because pianos with 88 keys have been invented which means there are likely to be pieces that utilise the extra 3 keys that are available.

A piano with 85 keys is usable, but it is not ideal because 88 key pianos have been commonly in existence, and the standard, for over 120 years, so just about anything written back through the impressionist period might use the top 3 notes. The mathematical relevance you cite is arbitrary and not supported by any composer's methodology I remember reading.

The design of the Imperial (and other > 88 key instruments, by association) is not "much better", it's just different, and was brought about because some composers in the early 20th century wanted to experiment with low sub-bass notes, and collaborated with the builder. If you've actually seen and played an Imperial, the fundamental pitch clarity of the sub bass notes is practically inaudible, however the clarity of the lowest notes down to the more traditional A0 on this piano is always magnificent, because A0 isn't all the way out at the edge of the bass bridge, near the rim. Since pianists are used to seeing the traditional 88-key setup for about 120 years, some of us (not me, but some of my colleagues) find the visual presence of the extra sub-bass keys to be disorienting, when playing and especially leaping to the lowest pitches on the piano. Finally, I believe the sheer size of more-than 88-key concert pianos, regardless of brand (when you factor in the cheek blocks and necessary thickness of the rim) are troublesome because they do not fit through the width of most double doorways, according to many building codes, which put them at a disadvantage when managing a stage area without costly customizations. Just like in marketing, it's unwise to use one parameter, in a vacuum, as justification for an instrument's absolute superiority.

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.

I think that modern pianos have 88 keys because modern pianos have 88 keys. . . . A lot about musical instrument (and other) design can be attributed to tradition; there may be no other explanation.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743514 06/10/1806:38 PM06/10/1806:38 PM

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 non-extremists should try to avoid.

Everything is possible, and nothing is sure.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743593 06/11/1807:27 AM06/11/1807:27 AM

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 Well-tempered 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 06/11/1808:44 AM06/11/1808:44 AM

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.

Originally Posted by Fareham

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 Well-tempered 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 12-tone equal temperament during the late 16th century in 1584:

Quote

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.

12 perfect fifths = 7 octaves = 85 keys. This is mathematically ideal and it can be achieved by using either 12-tone equal temperament or well temperament because they both eliminate the Pythagorean comma. Therefore, everything I have said in the following quote is correct:

Originally Posted by Roshan Kakiya

Originally Posted by Davdoc

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.

12-tone equal temperament and well-temperament solve that problem.

12-tone 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 85-key keyboard is only mathematically ideal. It is no longer practically ideal.

The 88-key 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 06/11/1811:19 AM06/11/1811:19 AM

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.

Originally Posted by Fareham

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 Well-tempered 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 12-tone equal temperament during the late 16th century in 1584:

Quote

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.

12 perfect fifths = 7 octaves = 85 keys. This is mathematically ideal and it can be achieved by using either 12-tone equal temperament or well temperament because they both eliminate the Pythagorean comma. Therefore, everything I have said in the following quote is correct:

Originally Posted by Roshan Kakiya

Originally Posted by Davdoc

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.

12-tone equal temperament and well-temperament solve that problem.

12-tone 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 85-key keyboard is only mathematically ideal. It is no longer practically ideal.

The 88-key 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]
#2743631 06/11/1811:46 AM06/11/1811:46 AM

I am convinced. Anyone else who is convinced may contact me for quotes on removing 3 notes from your 88 note pianos. Quantity discounts available so ask your friends!

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 06/11/1812:37 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!

Re: Why do most modern pianos contain 88 keys?
[Re: guyl]
#2743859 06/12/1809:28 AM06/12/1809:28 AM

The number of fifths has nothing to do with the number of keys on a piano.

Are you saying every instance I have mentioned on this thread is a happy mathematical accident? I doubt this is entirely true. It is surprising how music theory and tuning are both linked via the circle of fifths and how 85-key pianos magically make it possible to completely close the circle of fifths (theoretically and physically).

Originally Posted by guyl

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.

That is exactly why I mentioned this earlier:

Originally Posted by Roshan Kakiya

The range of the keyboard seems to be getting larger as time passes by.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743865 06/12/1809:47 AM06/12/1809:47 AM

The evolution of all music has been centered around what sounds good to the human ear, including the range of the piano. The math explains the phenomenal capabilities of human hearing and the brain to perceive and interpret variations frequencies as distinct and related pitches. But human hearing does have it range of limitations. I think the 88-keys on the piano pretty much push the limits on both ends. Pitches below A0 and above C88 become generally harder to hear and less "musical" as perceived by the human ear. I don't think it's a "happy mathematical accident". It's the design of our Creator, as a matter of fact.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743871 06/12/1810:25 AM06/12/1810:25 AM

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.

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.

You are contradicting yourself in every step.

First of all, why must a keyboard instrument have one, and only one, circle of fifths? You are discrediting instruments used by Bach, Mozart, and Beethoven, and also the modern standard of 88-key pianos, or 92-, 97, 102-key pianos, or organs. Other non-pianist instrumentalists are also crying that their instruments are not ideal since they don't have 12 x fifths.

Second, your formula placed 12 x fifths as the highest priority ("ideal"), which unfortunately by the perfect 3:2 ratio will deviate from 7 x octaves.

Third, you then argued 12 x 7 semitones = 7 x 12 semitones. This is a circular argument.

Fourth: also for the above (third point) to be valid, some tempering (temperament) has to happen. Your fifth, by all the current temperament methods being used, is no longer as "perfect" and the merit for the second point above, of insisting on 12 x fifths as the highest priority, is further lost.

Fifth, by using equal temperament, one of the many methods to justify the fourth point above, each interval has the same and fixed ratio that a 84-interval (85-key) piano does not have more merit than, say, a 85-interval, or 83-interval, hypothetical piano.

Last edited by Davdoc; 06/12/1810:30 AM.

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]
#2743899 06/12/1812:42 PM06/12/1812:42 PM

I don't know why anyone would care if a piano's compass contained a closed circle of fifths. For one thing, the mathematical expression in question would only work if there were no inharmonicity in the piano, or if the inharmonicity followed some regular pattern, the former of which is impossible, and the latter of which is unlikely, and would require a different math expression. Second, even if a closed circle of fifths existed in some piano, that piano would sound pleasing only if it is well tuned, and nothing magical happens if note 85 is or isn't present or if the compass exceeds 85 notes. In equal temperature, the mathematical expression in question is an identity, and mentioning its existence, in no way that I can discern, has anything to say about any desirable design characteristic of any piano.

Re: Why do most modern pianos contain 88 keys?
[Re: Davdoc]
#2743900 06/12/1812:57 PM06/12/1812:57 PM

First of all, why must a keyboard instrument have one, and only one, circle of fifths? You are discrediting instruments used by Bach, Mozart, and Beethoven, and also the modern standard of 88-key pianos, or 92-, 97, 102-key pianos, or organs. Other non-pianist instrumentalists are also crying that their instruments are not ideal since they don't have 12 x fifths.

Second, your formula placed 12 x fifths as the highest priority ("ideal"), which unfortunately by the perfect 3:2 ratio will deviate from 7 x octaves.

Third, you then argued 12 x 7 semitones = 7 x 12 semitones. This is a circular argument.

Fourth: also for the above (third point) to be valid, some tempering (temperament) has to happen. Your fifth, by all the current temperament methods being used, is no longer as "perfect" and the merit for the second point above, of insisting on 12 x fifths as the highest priority, is further lost.

Fifth, by using equal temperament, one of the many methods to justify the fourth point above, each interval has the same and fixed ratio that a 84-interval (85-key) piano does not have more merit than, say, a 85-interval, or 83-interval, hypothetical piano.

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:

Quote

The Pythagorean Circle was the grandaddy of the Circle of Fifths. Different revisions and improvements were made by Nikolay Diletsky in the 1670s, and Johann David Heinichen in 1728, until finally we reached the version we have today.

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.

The well temperaments used throughout the 17 and 18 hundreds also allow one to modulate amongst different keys. However, the octave is not divided into equal steps.

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.

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 value, 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 a be a fraction of a fifth that needs to be tuned.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743903 06/12/1801:15 PM06/12/1801:15 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.

By the same "logic", it is not necessary to have more than 61 keys. You can have a circle of fourths with 61 keys.

Mathematics are great, but they don't say anything about the necessity of a certain number of keys.

Everything is possible, and nothing is sure.

Re: Why do most modern pianos contain 88 keys?
[Re: patH]
#2743946 06/12/1804:00 PM06/12/1804:00 PM

By the same "logic", it is not necessary to have more than 61 keys. You can have a circle of fourths with 61 keys.

Mathematics are great, but they don't say anything about the necessity of a certain number of keys.

That is interesting. However, a piano with a keyboard that has 85 keys contains both the circle of fourths and the circle of fifths. Therefore, both circles are fully accounted for by a piano with a keyboard that has 85 keys.

It can be argued that a piano with a keyboard that has 88 keys or more fully accounts for both the circle of fifths and the circle of fourths. However, the extra 3 keys it contains are not necessary to theoretically and physically account for both the circle of fifths and the circle of fourths. This is because the circle of fifths and fourths can already be fully accounted for by a piano that has a keyboard with 85 keys.

I am only pointing out what is necessary and what is not necessary to fully account for both the circle of fifths and the circle of fourths.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743950 06/12/1804:20 PM06/12/1804:20 PM

You will only be able to go around the circle of fifths and fourths in one direction with a 61-key piano: anticlockwise. The circle of fourths will have been fixed but the circle of fifths would still be physically broken.

An 85-key piano enables you to go around this circle in both directions: clockwise and anticlockwise.

No more than 85 keys are needed to go around the circle of fifths and fourths in both directions.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743951 06/12/1804:25 PM06/12/1804:25 PM

By the same "logic", it is not necessary to have more than 61 keys. You can have a circle of fourths with 61 keys.

Mathematics are great, but they don't say anything about the necessity of a certain number of keys.

That is interesting. However, a piano with a keyboard that has 85 keys contains both the circle of fourths and the circle of fifths. Therefore, both circles are fully accounted for by a piano with a keyboard that has 85 keys.

It can be argued that a piano with a keyboard that has 88 keys or more fully accounts for both the circle of fifths and the circle of fourths. However, the extra 3 keys it contains are not necessary to theoretically and physically account for both the circle of fifths and the circle of fourths. This is because the circle of fifths and fourths can already be fully accounted for by a piano that has a keyboard with 85 keys.

I am only pointing out what is necessary and what is not necessary to fully account for both the circle of fifths and the circle of fourths.

Other explanations for 88 keys make as much sense mathematically and musically; like the fact that C major is the first key people learn when starting to play the piano; and that therefore the piano ending with a C makes sense, for scales. Plus, the chamber tone used for tuning is an A, which means that starting the piano with an A also makes sense.

You can find all types of patterns in any object; but you will never find anything that points to a necessity one way or another. In fact, to the thesis that not more than 85 keys are necessary, a philistine (or non-music-loving neighbor of a piano player) could reply that the whole piano is not necessary. But on this forum, this would be a minority opinion.

Everything is possible, and nothing is sure.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743953 06/12/1804:26 PM06/12/1804:26 PM

RK, I think you are the only person on Planet Earth who can provide humankind with the equation to solve all equations, namely, the Universal Equation, which eluded Einstein, who was working on it before his untimely demise. As well as the late Stephen Hawking (of A Brief History of Time fame) who tried in vain to combine string theory with The Theory of Everything.

Your Universal Equation will explain, in strict mathematical terms, how Homo sapiens evolved from primeval slime via Tyrannosaurus rex, and how the grand piano evolved from primeval wood to have exactly 88 keys, no more and certainly no less. Apart from upstarts like Bösendorfer and Stuart of course, but they will of course be shunned in your Universal Equation, as mere footnotes in the timeline.

So, how about it? Fame & fortune beckons......... (not to mention the Nobel Prize).

"I don't play accurately - anyone can play accurately - but I play with wonderful expression. As far as the piano is concerned, sentiment is my forte. I keep science for Life."

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2743956 06/12/1804:40 PM06/12/1804:40 PM

... and of course 42 is 9 x 6, according to Douglas Adams - and who are we to argue ? (Incidenrtally he's probably right)

Perhaps we should have a new thread proposing that the keyboard should have exactly equal numbers of white and black keys. The white keys are the whole-tone scale starting with A and the black keys starting with A#.

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]
#2744046 06/13/1802:41 AM06/13/1802:41 AM

There is only one way to destroy nearly all my thoughts and ideas on this thread because nearly everything I have posted is based on the circle of fifths.

The term "85 keys" has no specific meaning on its own. The mathematics I have included above gives it a specific meaning.

You will also be able to comprehend, by doing the maths, that, as long as 12-TET and well temperament are used to theoretically correct the Pythagorean comma, a chain of 12 consecutive fifths = a chain of 7 consecutive octaves = 85 keys based on the fact that every octave contains 8 white keys and 5 black keys. This means that it is necessary to have 85 keys to physically close the circle of fifths.

We can remove the circle of fifths. This will cause the term "85 keys" to have no specific meaning.

Almost all my thoughts and ideas on this thread live with the circle of fifths and die without it.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2744079 06/13/1806:44 AM06/13/1806:44 AM

I'm sorry that you feel so strongly about the circle of fifths and its applicability to keyboard size. As you rightly point out, the circle precedes the keyboard size by around 2000 years.

Its importance is geared to the triads used within a piece. Generally speaking the tonic defers to its dominant, subdominant, submediant and mediant in that order of importance. You can use the 'circle of fifths' to find out the triads for each of these, whatever the home key signature. If you're using perfect fifths for tuning, then those are the triads you want to get as much into tune as possible - hence Bach's concerns / quandry about having to 'sacrifice' accuracy of sound to achieve flexibility. As we almost never hear true fifths any more, and have got used to even / well-tempered tunings, it's very much less of an issue to us as it would have been 300+ years ago.

If you look at organ manuals, most of them are 61 notes, with 25, 30 and 32 notes for the pedal. Even then, composers manage to write organ music that calls for notes beyond that range (Bach requires a bottom B natural that no-one has), and I suspect that the circle of fifths just doesn't get a look-in here (I note that the Atlantic City Convention centre organ has 7 manuals, all C--C with four at 61 notes, 1 at 73 (G--G) and two at 85).

So, as just about everyone has said, it's just coincidence that a nominal circle of fifths which requires 85 notes happens also to be the number on keys on older pianos.

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]
#2744107 06/13/1809:14 AM06/13/1809:14 AM

All of us understand the importance of the circle of fifths and the circle of fourths in music. It's a bedrock of music, and once one understands it, it opens up worlds of insight and possibilities. It's used as the foundation in most teaching methods to enable the student to learn and assimilate musical understanding long before the concept is introduced to the learner.

Does the design of the standard 6-string guitar complete the circle of fifth, or the trombone, or the violin, or the steel guitar, or the mandolin, or any instrument for that matter? I don't know b/c I haven't done the analysis. If they don't complete the circle does that mean their design are inferior and that makes them less viable and all into question their place in the musical universe? Hardly!

Your point that 85 notes on a piano completes the circle of fifths is well noted and an interesting piece of trivia. I just don't see it's significance, in fact I see it at trivial. It doesn't make me want to trade in my 1981 Steinway with 88 keys for an 1885 with only 85-keys, or petition manufacturers to go back to the 85-key design. I use all 88 keys on my piano all the time, especially when I'm playing by ear or improvising. They all sound musical to me, and I can fit them in. Without those extra keys, I'd have to cut my Bb, B, and C arpeggios and octave short!

I guess I'm lost on what your point is in this tread, and what you are trying to convince us of regarding the circle of fifths and the piano keyboard, and the 3 extra notes on the end.

Re: Why do most modern pianos contain 88 keys?
[Re: GC13]
#2744110 06/13/1809:29 AM06/13/1809:29 AM

I guess I'm lost on what your point is in this tread, and what you are trying to convince us of regarding the circle of fifths and the piano keyboard, and the 3 extra notes on the end.

Roshan has a savant-like relationship with numbers which takes precedence over other, more practical, concerns. To most of us, the entire circle of fifths doesn't need to be spanned by the keyboard because you would never play it that way. You can jump up or down to fulfil a dominant-tonic resolution. There's little point in arguing this any further - he will not change his mind about the importance of the numbers, and we will not change our minds about it being unimportant.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2744222 06/13/1806:08 PM06/13/1806:08 PM

I have already shown how an 85-key piano physically completes the circle of fifths and fourths. This is actually a theoretical reason for basing the design of a keyboard on this circle rather than a practical reason for doing this.

There is also a practical reason though. I can also show how an 85-key piano can be completely tuned with only fifths and fourths whose values have been derived from a simple circle of fifths, containing zeros and fractions, of a well temperament.

I will be using Thomas Young's Second Temperament throughout the rest of this post. Here is the circle of fifths for this tuning upon which the rest of this post will be based:

The zeros indicate the fifths should be pure and the fractions indicate the amount by which the fifths should be tempered.

A pure fifth is a fifth whose ratio is 3/2. A logarithm can be used to calculate its value in cents:

Pure fifth = 1200 x log2(3/2) = 701.96 cents.

−1/6 indicates the pure fifth should be reduced by a sixth of the Pythagorean comma. Pythagorean comma = 23.46 cents (I have shown how this can be calculated on a previous reply).

1/6 of the Pythagorean comma = 23.46 / 6 = 3.91 cents.

Tempered fifth = 701.96 - 3.91 = 698.05 cents.

We can confirm that a chain of 12 fifths = a chain of 7 octaves:

7 octaves = 1200 x 7 = 8400.

12 fifths = 701.96 x 6 + 698.05 x 6 = 8400.

12 fifths = 7 octaves. The Pythagorean comma has been eliminated.

Fourths will also need to be used so their values must also be calculated.

A pure fourth is a fourth whose ratio is 4/3. A logarithm can be used to calculate its value in cents:

Pure fourth = 1200 x log2(4/3) = 498.04 cents.

The inversion of a pure fifth is a pure fourth. The value of the pure fourth can also be calculated by subtracting the value of the pure fifth from the value of the octave:

Pure fourth = 1200 - 701.96 = 498.04 cents.

The value of the tempered fourth can be calculated by subtracting the value of the tempered fifth from the value of the octave:

Tempered fourth = 1200 - 698.05 = 501.95 cents.

Summary:

Pure fifth = 701.96 cents (offset from 12-TET = +1.96 cents).

The following procedure can be tested by using an electronic tuning device to tune an 85-key piano:

1. Determine the first and last notes of the piano. They will be the same because 7 octaves are physically equal to 85 keys. The notes could be C1 and C8 or A0 and A7. I will be using C1 and C8 for this procedure. This means every octave should start and end with C for this procedure.

2. Tune 1 octave from C[unison] to C[octave].

3. Tune a chain of alternating tempered fifths and tempered fourths within this octave: C[unison]-G (fifth), G-D (fourth), D-A (fifth), A-E (fourth), E-B (fifth) and B-Gb (fourth).

4. Tune a chain of alternating pure fourths and pure fifths within this octave: Gb-Db (fourth), Db-Ab (fifth), Ab-Eb (fourth), Eb-Bb (fifth) and Bb-F (fourth). F-C[octave] (fifth) should already be pure.

Every interval within this octave should have been tuned now. The chain of fifths and fourths is still unbroken:

C[unison]-G-D-A-E-B-Gb-Db-Ab-Eb-Bb-F-C[octave].

5. C[octave] becomes C[unison] in the next octave. Repeat steps 2 - 5 until all 7 octaves have been tuned.

Conclusion:

Every interval from C1 to C8 should have been tuned now. Only fifths and fourths have been used throughout this entire procedure. I have essentially derived all this information from this simple circle of fifths:

All the information above suggests that any 85-key piano can be tuned with any well temperament by using only fifths and fourths. I think this tuning method can be practical if inharmonicity is controlled.

The mathematics on this entire thread suggests that the 85-key piano physically corrects the circle of fifths and fourths and the circle of fifths and fourths can be used to completely tune the 85-key piano, as long as inharmonicity is controlled.

Everything I have posted on this post should be thoroughly checked and tested.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2744231 06/13/1807:34 PM06/13/1807:34 PM

My question is: do tuners use a circle of fifths to tune every single note? I thought that this was only used to set the temperament a couple of octaves near in the middle, and then the other octaves were tuned by unisons up and down from that middle area, looking for the best compromise considering the inharmonicities of that particular piano. That's how I tune my own piano.

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

Re: Why do most modern pianos contain 88 keys?
[Re: guyl]
#2744251 06/13/1808:25 PM06/13/1808:25 PM

12-tone equal temperament causes every semitone to have a size of 100 cents which means a keyboard can be as large or as small as one wants it to be. 12-TET is standard.

The 88-key keyboard is also standard and has been standard for many years.

Therefore, an 88-key piano tuned with 12-tone equal temperament is the most practical solution for the repertoire.

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2744461 06/14/1803:48 PM06/14/1803:48 PM

You can ask a moderator to close the thread. But they don't usually do that unless things get nasty and personal attacks and insults begin to fly. They don't usually close a thread just because the OP, or others, feel like its time. That is one of the risks of starting a particular thread. You don't have a lot of control over what happens next; the moderators are the only ones that do.

Good luck!

Rick

Piano enthusiast and amateur musician: "Treat others the way you would like to be treated". Yamaha C7. YouTube Channel

Re: Why do most modern pianos contain 88 keys?
[Re: Roshan Kakiya]
#2744650 06/15/1808:26 AM06/15/1808:26 AM