Re: [Harp-L] Evolution of Temperaments
Thank you, Iceman, for that clear, articulate explanation of
the need for equal temperament and how it arose.
Now, for the mathematically inclined, (if any), here are
some formulas that will relate frequency to haltone notes
and cents. They will work in Xcel, Basic, or on any
scientific pocket calculator.
Definitions:
N = musical interval in number of halftones, a + or -
number
C = musical interval in number of cents, a + or - number
Fo = known starting frequency/pitch in Hertz (cycles per
second)
f = unknown frequency/ pitch in Hz at the other end of an
interval.
^ indicates exponentiation
* indicates multiply
/ indicates divide
+ and - indicate add and subtract
log indicates the base-ten logarithm
To find the frequency (f) at the end of any interval as a
function of the starting frequency Fo and N halftones.
f = Fo * 2 ^ ( N/12)
Note that when N =12 halftones, then N/12 =1 and f = Fo * 2,
an octave.
If you assign Fo the value of 440 Hz, then f is the
frequency of any musical note where N is the number of
halftones counting from the note A4, (A above middle-C)
Example: N = 12 at A5, 14 at B5, etc.
To find the number of halftones N in an interval from Fo to
f:
N = 12 / log(2) * ( log(f) - log(Fo) )
A cent is perceived by the ear as 1/100th of a halftone.
Small deviations from musical notes in discussions of tuning
are usually quantified in cents.
To find the frequency at the end any interval as a function
of the starting frequency Fo and the mumber of cents:
f = Fo * 2 ^ ( C / 1200)
Example: 24 cents of pitch deviation has been mentioned in
Iceman's email.
f = 440 * 2 ^ (24 / 1200) = 478.16 Hz. or 38.16 Hz out of
tune.
At an octave up Fo = 880, then f = 956.32 Hz or 76.32 Hz out
of tune.
To find the number of cents C in the interval Fo to f:
C = 1200 / log(2) * ( log(f) - log(Fo) )
Now that's probably a lot more than anyone wanted to know
about equal temperament. That is what the "DELETE" key is
for. ;o)
Vern
----- Original Message -----
From: <icemanle@xxxxxxx>
To: <harp-l@xxxxxxxxxx>
Sent: Friday, October 30, 2009 6:38 AM
Subject: [Harp-L] Evolution of Temperaments
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
=
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