Re: [Harp-L] excuse me for reposting for clarity

[Arthur Jennings <timeistight@xxxxxxxxx>]

This "stack of fifths" tuning is called Pythagorean tuning after the Greek
mathematician who supposedly discovered it. While 11 out of 12 fifths i this
system are perfect, the thirds aren't very good:
http://en.wikipedia.org/wiki/Pythagorean_tuning



On Fri, Oct 30, 2009 at 7:33 AM, <icemanle@xxxxxxx> wrote:

>
> The original posting was a bit confusing as to who said what. This should
> help clarify........
>
>
> <<<<< 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
>
>
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-- 
Arthur Jennings
http://www.timeistight.com