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Re: Why do most modern pianos contain 88 keys? [Re: guyl] #2743859
06/12/18 09:28 AM
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Originally Posted by guyl
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.

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Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2743865
06/12/18 09:47 AM
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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
06/12/18 10:25 AM
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Originally Posted by Roshan Kakiya


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; 06/12/18 10:30 AM.

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Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2743899
06/12/18 12:42 PM
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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
06/12/18 12:57 PM
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Originally Posted by Davdoc
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.


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.


Source: https://www.classicfm.com/discover-music/music-theory/what-is-the-circle-of-fifths/


Tuning - 12-tone equal temperament:

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.


Source: http://www.chinaembassy.se/eng/zrgx/wh/t100768.htm

Tuning - Well temperament:

Quote
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.


Source: http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html


I have combined all this information to "discover" how to theoretically correct the Pythagorean comma (12 "perfect" fifths (the ratio of each fifth is 3/2) is not equal to 7 octaves (the ratio of each octave is 2/1)) and how to physically correct the circle of fifths. We can already see compromises will be needed to achieve this.


Correcting the Pythagorean Comma Theoretically


The perfect fifth has a value of 701.955 cents (3 d.p.). The octave has a value of 1200 cents.

12 perfect fifths = 701.955 × 12 = 8423.46.

7 octaves = 1200 × 7 = 8400.

Pythagorean comma = 12 perfect fifths - 7 octaves = 8423.46 - 8400 = 23.46 cents.

The circle of fifths is broken because of the Pythagorean comma.

This comma can be "tempered out" by reducing the value of all 12 perfect fifths by 1.955 cents.

701.955 - 1.955 = 700 cents.

12 fifths tempered by a twelfth of the Pythagorean comma = 700 × 12 = 8400.

This compromise is necessary to close the circle of fifths.

12 fifths tempered by a twelfth of the Pythagorean comma = 7 octaves.

This can also be achieved by well temperament which contains unequal fifths (some fifths are tempered and some fifths are pure).

The 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
06/12/18 01:15 PM
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You were not addressing any of my critiques of your logic. I will stop here.


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Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2743943
06/12/18 03:43 PM
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Originally Posted by Roshan Kakiya
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
06/12/18 04:00 PM
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Originally Posted by patH
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
06/12/18 04:20 PM
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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
06/12/18 04:25 PM
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Originally Posted by Roshan Kakiya
Originally Posted by patH
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. wink


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Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2743953
06/12/18 04:26 PM
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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......... thumb (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
06/12/18 04:40 PM
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Everything (or almost everything) I have posted on this thread is based on the circle of fifths and fourths.

There is only one question I have left, related to this thread, that I believe is my most important question:

How musically significant is the circle of fifths and fourths?

Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2743961
06/12/18 05:47 PM
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Has even one poster so far thought the OP's comments are relevant or logical? Not as far as I can see.

Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2744005
06/12/18 09:17 PM
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Also, let's not forget that the answer to the ultimate question: the meaning of life, the universe, everything is 42, not 85 or 88


What do snowflakes and Chickerings have in common? There are no two exactly alike!
Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2744044
06/13/18 02:07 AM
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... 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
06/13/18 02:41 AM
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Many tunings began with the note C. They did not begin with the note A. The chain of fifths I have mentioned above starts with the note C.

You have to proceed around the circle of fifths and fourths, as shown by these videos.

Why is the design of the 88-key keyboard not based on the circle of fifths and fourths if many tunings heavily rely on this circle?

The 85-key piano conveniently completes the chain/circle of 12 fifths.

Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2744049
06/13/18 02:52 AM
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All seven octaves can be tuned:

C1-C2-C3-C4-C5-C6-C7-C8.

Finally, each fifth or fourth can be tuned:

C-G-D-A-E-B-F#-C#-G#-D#-A#-F-C.

All 12 fifths and all 7 octaves are contained within an 85-key piano.

The extra 3 keys of an 88-key piano are not necessary.

Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2744053
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They are if you want to play those notes. If you do not want to play any notes, then all 88 keys are not necessary.


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Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2744070
06/13/18 05:49 AM
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Originally Posted by Roshan Kakiya


Many tunings began with the note C. They did not begin with the note A. The chain of fifths I have mentioned above starts with the note C.


In equal temperament, there are many tuners and ETDs that tune an F-F temperament or an A-A temperament...


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Re: Why do most modern pianos contain 88 keys? [Re: Roshan Kakiya] #2744072
06/13/18 06:01 AM
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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.

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