Music Software Projects New York University Adjunct Instructor Scott Burton

Comparing ET / Pythag / Harmonic Series Pythagoras established the number of steps per octave in western music which we still use today but… The 7-note natural scale third was dissonant 12 tone chromatic scale degrees were unusable The Sixth and Seventh intervals were also dissonant Goals of scale engineering after Pythagoras: Adjust dissonant intervals to be closer to harmonic series Adjust ratios to be super-particular Make them sound better For ex., Ptolemy adjusted the 3rd, 6 th and 7 th degrees Irony is that ET adjusted ALL intervals away from the series...

Comparing Scale Intervals The primary measure we will use to compare is “Cents” 100 cents per smallest interval in the 12 ET tones scale Cents = (1200/LOG(2,10)) * LOG(IntervalFactor,10) Example: Let IntervalFactor = 1.5 (the 3/2 or “Fifth”) = (1200/LOG(2,10)) * LOG(1.5,10) By Definition the Fifth in ET is 700 cents ET is different than the Just or “pure” fifth of 3/2 Cents are very handy when comparing intervals across tuning systems.

Programming Phase Add Cents calculation parameterized by either: 1. Any two intervals (e.g, “m2” and “5”) 2. Interval factor (e.g., 1.059) Store letter names in your note/interval object You can assume our base frequency stays the same at 528hz and we’ll call that “C” But you still have to parameterize by starting frequency Using letter note names will be useful for us as we build more tuning systems and compare them Add Interval string names “m3”, “M3”, etc. See column F in DegreeNaming.xlsx

Programming Phase cont... Produce three output rows per scale to stdout ET, Harmonic Series and Pythagorean Use first 7 notes only from the Pythagorean scale No accidentals from ET No sharps/flats Use the 5 th octave of the Harmonic Series Using the Major Third – the “M3” interval Print out the frequencies and cents of the just that interval from: ET scale Harmonic series Pythagorean scale (7 note, 8 including the octave) Output format: Units IntervalName ET HarmonicSeries Pythagorean HZ M3 XXX YYY ZZZ Cents M3 XXX YYY ZZZ

After the break: Prepare for Rhythm Learn how to read the “Riddim” sheet “Nanofly” by Billy Martin