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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 06/12/1810:47 AM06/12/1810:47 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 06/12/1811:25 AM06/12/1811:25 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; 06/12/1811:30 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 06/12/1801:42 PM06/12/1801:42 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.

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Re: Why do most modern pianos contain 88 keys?
[Re: Davdoc]
#2743900 06/12/1801:57 PM06/12/1801:57 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 06/12/1802:15 PM06/12/1802:15 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 06/12/1805:00 PM06/12/1805:00 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 06/12/1805:20 PM06/12/1805:20 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 06/12/1805:25 PM06/12/1805:25 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 06/12/1805:26 PM06/12/1805:26 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 06/12/1805:40 PM06/12/1805:40 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 06/13/1803:41 AM06/13/1803:41 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 06/13/1807:44 AM06/13/1807:44 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 06/13/1810:14 AM06/13/1810:14 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 06/13/1810:29 AM06/13/1810:29 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 06/13/1807:08 PM06/13/1807:08 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 06/13/1808:34 PM06/13/1808:34 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 06/13/1809:25 PM06/13/1809:25 PM