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• ## 3:1 and 2:1 of the overtone-series more fundamental than 3/2 := (3:1)/(2:1)

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• ... Hi Paul, ... Simply consider all given values there as frequencies in Hz of absolute pitches, that are subjects of 3 possible sequential operations:
Message 1 of 5 , Jun 27, 2008
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--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:

Hi Paul,
> Start using just ONE clear and easy to understand method for
> indicating temperaments.
Simply consider all given values there
as frequencies in Hz of absolute pitches,
that are subjects of 3 possible sequential operations:

Algorithm for synchroneous well-temperaments:
1. Step 19-times an octaves down, by halfing the pitch-frequency
2. Go 12 times to partial 3:1, by multipying with facor 3.
occasional
3. Decrement frequncy by -1Hz down, when intend tempering flattend.

but only if you insist in "wide-5ths" then allow also too:
(4. Increment by +1Hz upwards, for an sharper "French"-5th.)

Comeback condition:
Choose the chain of flow in the operation sequence
so that the circle of a dozen 5ths returns back to the initial
start after 12times 3:1 and 19times 1:2 while fitting the
decrements so, that they yield an distribution of the PC=3^12:2^19
into
http://en.wikipedia.org/wiki/Superparticular_ratio
s.

> All that scala mismash and wierd stuff like
> multiplying frequencies by 3 instead of 1,5 just makes it all not
> worth the time.

That ratio of 3/2 = 1.5 arises operationally from taking the
quotient of the 3rd partial (3:1) over an octve (2:1),
when realting that both overtones #2 and #3 to theirs fundamental
(1:1) base.

http://en.wikipedia.org/wiki/Harmonic_series_(music)
"...allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, etc. times of the
fundamental."

but on strings there never appear 2/3 = 1:(3/2) due to the lack of
http://en.wikipedia.org/wiki/Subharmonics
in pianos:
http://www.sfu.ca/sonic-studio/handbook/Subharmonic.html
"Subharmonics do not normally occur in natural sounds, although the
subharmonic f/2 may be generated by the cone of a LOUDSPEAKER."

That makes an 5th (3:2 = 1.5) less fundamental than the ratio
inbetween the overtones 3:1 and 2:1 within the harmonic series.

Hence an 5th is composed by an
division of an 12th (3:1) as nominator
over an octave (2:1) as denominator by the calculation

(3:2) := (3:1):(2:1)

In other words:
any 5th (3:2) consists terms of
http://en.wikipedia.org/wiki/Harmonic
as composed of the difference of an '12th'-'8th'.
"3 just perfect fifth P8 + P5 1902.0 702.0"
when both do refer to the same (1:1) base or
http://en.wikipedia.org/wiki/Fundamental_frequency

hope that helps,
why i do prefer the multiplication by the "harmoic" factor 3
in order to stay wihin the partial-series.

Even Brad understood that in his:
http://www-personal.umich.edu/~bpl/larips/bachtemps.html
"...in the line of fifths A-E-B-F#-C#-G#-D#-Bb-F-C-G-D-A to reduce the
next note by 1 Hz, i.e. introducing a beat rate of 1 per second
against the preceding fifth. The fifths F#-C#-G#-D# and D-A are kept
pure. The other eight are adjusted by different geometric amounts,
based on the superparticular ratios described in his algorithm.
(Arithmetically, it amounts to subtracting 1 Hz from the top of most
of the columns, in his chart, wherever there are values in
parentheses.)"...

"Sparschuh's mathematical algorithm resembles the classic unproven
"Collatz Conjecture" from 1937, except that Sparschuh's iterated
function uses (3n-1) rather than (3n+1). [And see Eric Roosendaal's
3x+1 web site, along with this page by Frits Beukers demonstrating and
comparing the numerical sequences....]"

Yours Sincerely
A.S.

Yours Sincerely
A.S.
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