[Harp-L] Evolution of Temperaments

[icemanle@xxxxxxx]

 From: Michelle LeFree <mlefree@xxxxxxxxxxxxxxxxxxxxxx>
I'll posit that in a 
band setting, the average listener's ear would not be able to tell the 
difference plus or minus a cent or two between harps tuned to ET if they 
were played using the pucker embouchure. My feeling is, on the other 
hand, that those same listeners could easily discern the dissonant 
nature of chords played on those same ET-tuned harps compared to chords 
played on a JI- or Compromise-tuned harmonica. Therefore, if you 
predominantly play single notes, my opinion is that ET would be your 
tuning of choice. To give a point of reference here, according to Rick 
Epping, modern Marine Bands have certain notes tuned as many as 12-cents 
flat from ET, an amount that even an untrained ear might well be able to 
hear. 
 

In lab tests, the human hear doesn't hear a difference until 3 cents or
more. The test consisted of playing two tones - one after the other. It
was only at 3 cents and beyond that a difference was noticed by the
test subjects.



In nature, pure intervals are what you would naturally sing, or what a
violin player uses, when unaccompanied. It's kinda the natural order of
things. When keyboard instruments entered the picture and composers
wanted the freedom to modulate into any key they chose in their
compositions, Nature ran smack dab into Human Desire and something had
to give.
 

Having been a piano tech/tuner for over 30 years, and also having had an avid interest in acoustic science during my formative years, I've been fascinated by this subject. As a piano tech, tuning temperaments are part of what you learn - the history, evolution and differences between them..

INTERVALS 101

A perfect (pure) 5th, can also be considered an inverted perfect (pure) 4th. Middle "C" up to "G" is a perfect 5th. Take that "G" and move it down an octave. Middle "C" to this new "G" is now a perfect 4th. When you take a perfect 5th, created by going from a reference note UP, and invert it, it becomes a perfect 4th going down.

TEMPERAMENT 101

You are given one octave of notes to play with - visually imagine a keyboard and ignore any thought of temperaments - middle C and an octave up - all 12 chromatic notes. Here is how you get to each note one at a time using the interval of a perfect 5th. It's just like the Circle of 5ths. Start on middle "C". Move up a perfect 5th to "G". Instead of moving up another perfect 5th to "D", which would put you OUTSIDE of the one octave you are given to use, move DOWN a perfect 4th, inverting that perfect 5th. Now you've arrived at a "D" which is within your one octave. Go up another perfect 5th to "A". Instead of going up ANOTHER perfect 5th (which would once again take you out of your one octave), invert your interval and go down a perfect 4th to "E". 

Can you see a pattern developing? Go up a perfect 5th, down a perfect 4th, up a perfect 5th, down a perfect 4th, up a perfect 5th, down a perf......uh, I think you get it by now.

In this way, you will "create" all 12 notes of the chromatic scale without duplication until you finally arrive at your starting point, "C", which will be up one octave from where you began, but still within the one octave you get to play with.

NOW, if you transpose this "C" down an octave and compare it with the"C" you started with, the last "C" will be 24 cents SHARPER than the first one.

This is Nature at work in her mysterious ways. Using perfect (pure) intervals seems to add 24 cents to the octave. With the advent of keyboard instruments, this discrepancy just wouldn't do.

Oh my, what to do, what to do. Somehow 24 cents had to be subtracted from all these intervals so that your ending note would be the same as your starting one. It's these pesky extra 24 cents that created all the complications. One solution was to divide these 24 cents by two and subtract 12 cents from two of those perfect 5ths. This resulted in a really sweet sounding keyboard tuning with chords ringing richly UNTIL you got to those two intervals that were squeezed by 12 cents each. When these notes were used in chords or in the melody, it sounded horrible. One solution was to make sure compositions just didn't go there. This worked for a minute, but not for very long. Other solutions had to be explored.

This excess 24 cents was sliced and diced up many different ways and subracted from many different intervals. All solutions had advantages and disadvantages. What was finally agreed upon as a standard was to treat all notes equally by dividing the 24 cents by 12, coming up with 2 cents for each chromatic note. So, when creating all 12 notes through that cycle of upward 5ths and inverted intervals of downward 4ths, these perfect intervals were SQUEEZED smaller by 2 cents each - perfect 5ths were contracted by 2 cents and perfect fourths, being inversions, were expanded by 2 cents each. The result was that your ending note now matched perfectly your starting one. The intervals used became PERFECT IMPERFECT intervals.

Since the human hear can't really hear a difference until 3 or more cents, this 2 cent shaving was not really apparent and did solve the problem of weird ugly intervals and the inability to freely transpose music into any key. It equalized the playing field in a compromise with Mother Nature.

This is the background to the entity of equal temperament.

How it applies to a specific instrument, the harmonica, and its vibrating reed, is a whole other story that includes the stiffness of the reed changing the overtone series relationships and how upper partials interact with each other.

But, that's enough schoolwork for one day. 

Time for recess.

The Iceman





=