--- 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.)"...

Brad contiues or the experts:
"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.