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.