1. Music Software projects New york university Adjunct Instructor Scott Burton

2. The Pythagorean System Invented around 600 B.C. All intervals based on naturally occurring 3 rd degree in the harmonic series The 3:2 ratio or factor of 1.5... Now what? Construct as a geometric series - successive multiplication by 1.5 How many intervals to create – e.g., how to divide up the octave? Answer = 12 and still holds true today (for western music anyway) Interesting that we don’t use his system anymore but standardized on 12 So why 12? Pythagoras thought that: 12 intervals created by successive multiplication of 3/2 should equal 7 octaves Our living spec spreadsheet illustrates this...

3. Pythagorean vs. Harmonic Series Built on successive 3/2 ratios only vs. taken directly from other ratios 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). Closest interval in Pythag’s scale is “Unstable” interval of 81/64 Slightly higher than the closest interval of 5/4 in harmonic series Difference is 81/80 81/80 = 81/64 ÷ 5/4 The “Third” is a distinctive, rich sounding interval but was considered “imperfect” back in the day due to above... “Perfect” Intervals: Fourth Fifth

4. 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 This difference is known as the “Syntonic Comma” (Google it) “Perfect Intervals” Fourth Fifth

5. “Pure” or “Just” Intervals sound great but… Consider the following: C 1/1 x 528 = 528 Hz D 9/8 x 528 = 594 Hz E 5/4 x 528 = 660 Hz Now let’s build an interval one pythag whole tone (9/8) from the "D" note: D 1/1 x 594 = 594 cycles per second E 9/8 x 594 = 668.25 cycles per second Which one is right? 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 when “modulating”

6. Some Terminology “Pythagorean” Built from the 3 rd harmonic in the harmonic series (3/2) The 3/2 ratio is also known as the “fifth” since it is the 5 th degree 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 interval 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 (“accidentals”) “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

7. For next week: Implement “Pythagorean” scale Refer to living spec spreadsheet “pythag_sheet_phase2.xlsx” Implement methods: Float hz = get_frequency(int interval_position_sorted) Return the frequency after being “octave adjusted” aka in the same octave as the starting frequency. Example: 704, 891,... Char* get_interval_ratio(int interval_position_sorted) example: "3/2", "9/4", "27/8”... Play scale ascending sort on frequency 1 second duration, 1 second silence between Happy Coding!