A NOTE that is both SHARP and FLAT...

The DUDE

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Anybody else ever get a papercut while reading a letter?
 

Stonesy

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I got a haircut while reading the paper.
 

specflec

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They are all sharps and all flats depending on the current company they keep.
 

Nicky

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give me a big fat C## any day.
Ebb (E double flat...) sounds much better.
:shock:
There actually is a difference between the two, depending upon which temperament you use.

Why I became interested in tuning and temperament
What is the relationship between consecutive notes in the scale?
Why are the white keys and black keys arranged the way they are?
What is the difference between 'tuning' and 'tempering'?
What do we mean when we say we are going to tune an instrument?
Most string, brass and wind players know how to tune their instruments, but very few can explain in detail what they are doing.
Explaining what it means to be 'in tune' is quite complex.
We are used to measuring systems where the intervals between larger units are logical and easy to visualize. EX:
4 C + 3(1/3) C = 5 C
7 LB + ½ LB = 8 LB - ½ LB
1' + 7 " = 2' - 5"
Unfortunately, our 12-tone chromatic scale doesn't divide up so neatly.
The first step to understanding this problem is to look at the harmonic series.
A pure tone sounds only at the fundamental frequency or pitch.
Musical tones are complex in that they not only sound at the fundamental pitch, but also at higher frequencies sometimes called overtones. The first 5 harmonics of C are:
INTERVAL RATIO FACTOR
Octave = 1:2 = 2.0X
Fifth = 2:3 = 1.5X
Fourth = 3:4 = 1.33X
Maj 3rd = 4:5 = 1.25X
Min 3rd = 5:6 = 1.2X

It is the presence of these harmonics that give each musical instrument its peculiar sound. If we hear a trumpet, a clarinet and a violin play the same note, we identify the source of the note by the blending of the harmonics. The fundamental gives us the pitch, the harmonics identify the source. The same is true of the various voices of the pipe organ. The pipe builder causes certain harmonics to sound with the fundamental by the shape of the pipe, the material from which it is made and by using a particular scale or ratio (diameter to length). Mutations and mixtures sound at the exact harmonics of the fundamental, adding to the natural harmonics of the fundamental pitches.
These harmonics are what we listen for when we tune an instrument.
For example, if we wish to tune the 5th C - G, the second harmonic of C and the first harmonic of G is G in the next higher octave. If our interval is not tuned pure, you can hear a beat produced by the out-of-tune harmonics. If it is pure, you will not hear a beat.
Let's look at how some of these intervals work together:
A pure 5th + a pure 4th = 1 octave.
If C' = 100, then 100 X 1 ½ = 150 (G') X 1 1/3 = 200 (C").
Trying to divide the octave in half or thirds doesn't work so well.
For example, 3 maj 3rds do not = an octave.
1.25 X 1.25 X 1.25 = 1.953125
To make division of the scale easier to understand, we use a logarithmic system that divides the scale into 1200 cents per octave.
Looking again at the first 5 harmonics:
INTERVAL RATIO CENTS
Octave = 1:2 = 1200
Fifth = 2:3 = 702
Fourth = 3:4 = 498
Maj 3rd = 4:5 = 386
Min 3rd = 5:6 = 316
A pure 5th (702) + a pure 4th (498) = 1 octave (1200).
3 major 3rds (386 X 3) = 1158 cents, 42 cents short of an octave, nearly half of a semitone flat!
Most of us have been told since very early in our musical training that notes with enharmonic names are the same pitch. In theory this is true only in equal temperament. For example:
Pure maj 3rd C - E = 386
Pure maj 3rd E - G# = 386
Pure dim 4th Ab - C = 428
Octave =1200
By changing the G# to an Ab we make the last interval a dim 4th, which completes the octave. The key G#/Ab either must be tuned and used as one or the other, or tempered to serve as both.
A circle of 12 pure 5ths = 8424 cents (12 X 702); 7 octaves = 8400 cents (7 X 1200). This 24 cent discrepancy is known as the ditonic comma.
The elimination or accommodation of the comma is called tempering. Tempering is an adjustment of the intervals between notes away from pure. It is a compromise to make the intervals fit.
It is common to use the term 'tune' for both tuning and tempering.
'Tuning' is static and involves adjusting the pitch of one instrument to another, as when a group of instrumentalists 'tune up'. It is also the correct term to use when a string player 'tunes' the individual strings of his instrument to each other.
When we 'tune', we adjust to unisons or pure intervals. These intervals can be expressed as the ratio of integers.
'Tempering' is the correct term to use when we adjust pitches that are not pure, but set to some other value. This would apply to fretted string instruments and keyboard instruments.
In tempering, we distribute the comma over the 12 semitone intervals in the octave.
These ntervals can only be expressed with irrational numbers.
The remainder of this discussion assumes we are dealing with a keyboard instrument, specifically the organ.
The tempering of an organ is especially critical because:
Except for the reeds, the tone is quite pure - free of complex harmonics.
The volume of the tone does not decay, as on the piano.
The solution to the problem - tempering the scale
There are approximately 150 tempering schemes that have been advanced over the centuries, with probably no more than 20 being practical for general use. Of these, about 10 are practical for keyboard instruments, and only one - equal temperament - practical for the modern piano.

For more useful information on this subject, go to this linky:

History of Temperament
 

bildozr

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And to think.. I'm still wearing pants.
 

Nicky

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ditonic comma would be a good name for a band. Thanks
Well, then, you owe me 24 cents.:D

BTW, Gibby, I sent an email to the owner of the offending F**der Forum and told them they may have suspended you without due process.
(referring to another thread by Mr. Gibson)
 


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