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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

New york university

Adjunct Instructor Scott Burton


The Pythagorean ScaleAdjusted to One Octave


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…

  • 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

  • “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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