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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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  • Andreas Sparschuh
    ... 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.
      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

      > 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.

      "...allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, etc. times of the

      but on strings there never appear 2/3 = 1:(3/2) due to the lack of
      in pianos:
      "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
      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

      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:
      "...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

      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

      Yours Sincerely
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