Piano World Home Page Forums Our Most Popular Forums Piano Forum Why do most modern pianos contain 88 keys?

It might be anecdotally and mathematically interesting, but that's all.

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.

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.

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.

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:

https://www.google.com/search?biw=1229&bih=607&tbm=isch&sa=1&ei=oX4hW8SiNOaX0gKuqbCYCQ&q=young%27s+second+temperament+circle+of+fifths&oq=young%27s+second+temperament+circle+of+fifths&gs_l=img.3...6638.7100.0.7306.2.2.0.0.0.0.202.390.0j1j1.2.0....0...1c.1.64.img..0.0.0....0.kvHLTFtOC-Q#imgrc=D-RR301yu-eprM:


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).

Tempered fifth = 698.05 cents (offset from 12-TET = −1.95 cents).

Pure fourth = 498.04 cents (offset from 12-TET = −1.96 cents).

Tempered fourth = 501.95 cents (offset from 12-TET = +1.95 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:

https://www.google.com/search?biw=1229&bih=607&tbm=isch&sa=1&ei=oX4hW8SiNOaX0gKuqbCYCQ&q=young%27s+second+temperament+circle+of+fifths&oq=young%27s+second+temperament+circle+of+fifths&gs_l=img.3...6638.7100.0.7306.2.2.0.0.0.0.202.390.0j1j1.2.0....0...1c.1.64.img..0.0.0....0.kvHLTFtOC-Q#imgrc=D-RR301yu-eprM:

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.

Who cares?

The OP cares!

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.

While there are tuners who have been known to need a fifth before they can tune, some of us get by sober.

What's more important to me is...why is the simplest key C...and not A?

Pwg

This thread needs to be closed now. It's become ridiculous.

I can't believe it's approaching 100 responses! Certainly 85 would have been sufficiently appropriate.

I have reviewed the information on this thread.

These are my final thoughts:

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.

Well, it appears that you knew the answer to your own question all along...

Sometimes we travel around the world, twice, to get to our destination, which was actually just across the street.

Rick

Question: Why do most modern pianos contain 88 keys?

Answer: Standardisation.

Sounds perfectly sensible to me...

Rick

Halleluja!