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Music Software projects New york university Adjunct Instructor Scott Burton

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Comparing ET / Just / Harmonic Series Pythagoras established the steps per octave in western music which we still use today but… The 7-note natural scale third was dissonant and the 12 tone chromatic scale degrees were unusable. The Sixth and Seventh intervals in the natural scale were also dissonant. An important goal of music engineering was essentially to get Pythagorean dissonant intervals closer to harmonic series (or at least to smaller super-particular ratios) Ptolemy adjusted the third, sixth and seventh degrees (and kept ALL degrees Just) ET adjusted ALL intervals and thus ALL deviated from Just (some more than others…) We have implemented ET but not Ptolemy’s scale (yet)… Harmonic Series vs. ET: The Major Third is present in the Harmonic Series but the Minor Third is not (the 19 th series is close). In ET scale and others we will implement: Minor third + Major third = Fifth “Cents” – A more granular unit of measuring and comparing intervals: 100 cents = 1 half-step = semi-tone = smallest interval in our 12 tone Event-tempered scale Add capability to measure Cents Calculate Cents between intervals (and intervals between intervals) Cents = (1200/LOG(2,10)) * LOG(IntervalFactor,10) Example: Let IntervalFactor = 1.5 (the 3/2 “FIFTH”) Cents = 701.995 (ET Fifth of 700 Cents is 1.995 different than the just fifth of 3/2) Cents are a simple way to compare intervals across tuning systems (w/o assigning frequencies)

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Programming Phase Add Cents to your “interval” class Store letter names in your note/interval object for the notes (key of C 528Hz) NOTE: you can assume our base frequency stays the same so “C” is therefore meaningful. But you still have to parameterize by starting frequency Using letter note names will be useful for us later Add Interval names - m3, M3, etc. See column F for standard strings to use See even_temperment_sheet_phase4.xlsx Some of you had problems with playing two notes simultaneously. Playing the chord due next week. Arpeggio this week (the chord notes in sequence instead of simultaneously)

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Programming Phase cont... Produce three outputs: For base frequency of 528hz For ET, Harmonic Series and Pythagorean scales Use the 5 th octave of the Harmonic Series Using the M3 interval in the ET scale Print out the frequencies and cents of the closest interval to the M3 in the ET scale to: Harmonic series Pythagorean scale Output format: Units IntervalName ET HarmonicSeries Pythagorean HZ M3 XXX YYY ZZZ Cents M3 XXX YYY ZZZ

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