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Music Software projects. New york university Adjunct Instructor Scott Burton. The Pythagorean Scale Adjusted to One Octave. Pythagorean vs. Harmonic S eries. Built on successive 3/2 ratios only vs. taken directly from upper ratios in the naturally occurring harmonic series

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music software projects

Music Software projects

New york university

Adjunct Instructor Scott Burton

pythagorean vs harmonic s eries
Pythagorean vs. Harmonic Series
  • Built on successive 3/2 ratios only vs. taken directly from upper ratios in the naturally occurring harmonic series
  • The next unique interval in the harmonic series after 3/2 is 5/4
  • This step in our modern scale is known as the “major third” (E in a C scale which we will see more of later).
    • “Unstable” interval of 81/64
    • Slightly higher than the closest interval of 5/4 which is present in harmonic series
    • Difference is 81/80
    • 81/80 = 81/64 ÷ 5/4
  • “Perfect Intervals”
    • Fourth
    • Fifth
pure or just intervals sound good but
“Pure” or “Just” Intervals sound good but…
  • C 1/1 x 528 = 528 Hz
  • D 9/8 x 528 = 594 Hz
  • E 5/4 x 528 = 660 Hz

Now build off of the "D" note:

  • D 1/1 x 594 = 594 cycles per second
  • E 9/8 x 594 = 668.25 cycles per second

Should the "E" be tuned to 660 or 668.25 Hz???

Not a problem for singers, string players (non-fixed pitch instruments)

Problem for pianos or other fixed pitch instruments!

some terminology
Some Terminology
  • “Pythagorean”
    • Built from the 3rd harmonic in the harmonic series (3/2)
    • The 3/2 ratio is also known as the “fifth” since it is the 5thdegree in the Pythagorean 7 note scale
    • All scale degrees/intervals are derived from by multiplying by successive fifths
  • “Just”
    • Usually synonymous with “Natural” or “Pythagorean” or other scales that use integer ratios to build their scale degrees
  • “Natural”
    • Taken from harmonic series integer ratios occurring in nature (e.g, the vibrating string)
    • Also used to refer to a scale without sharps and flats (more on this later)
  • “Equal-tempered” or “Even-tempered”
    • Octave is divided into equal steps
    • Makes all keys sound the same
    • Can be 12 or more intervals ( more than 12 is considered “microtonal” )
  • “Well-tempered”
    • Some intervals are more usable than others
    • “Mean-tone” is primary example
      • Optimize the thirds and and fifths in selected keys at expense of the rest
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