1. Music Software Projects New York University Adjunct Instructor Scott Burton

2. Harmony Let’s start with two notes sounding simultaneously using pure “just” ratios Unison 1:1 (we’ve been assigning 528Hz and calling C) Octave 2:1 Fifth3:2 Fourth4:3 Third (major)5:4 (M3) Third (minor)6:5 (m3) Each pair has a unique sound... Unison + Fifth Unison + Fourth Unison + M3 (happy or bright sounding) Unison + m3 (sad or mysterious sounding) Interesting that a major third and a minor third = perfect fifth: 5/4 * 6/5 = 3/2

3. Harmony... Two notes really define the character – especially unison + third We can have more than two notes sounding simultaneously To add color or richness let’s add a third note “Triad” : unison + third + fifth

4. Medieval Harmony Thirds were a problem thanks to Pythagoras (81:64 vs 5:4) Thus Thirds were avoided in pre-15 th century music Harmony used the consonant 5 th and/or it’s inversion the 4 th Very pure sounding but a little “sterile” Supposedly singing in harmony using 4 th and 5 th intervals necessitated the creation of the first flat or accidental Harmonizing in 3rds is quite nice but only if you don’t use the Pythag 3rd

5. Medieval Harmony Play the scale in the first_flat.xlsx sheet in this sequence: 1. play the first row (we’ll call the “base” scale) 2. play the "harmony 1" scale simultaneously with the base scale 3. play the "harmony 2" scale simultaneously with the base scale *This is known as parallel motion Singing in harmony using the 3 rd was avoided at the time because the Pythag third (81/64) was dissonant. We had to wait a few centuries for a better sounding 3 rd to be engineered Interesting to note that the Third would turn out to be a very sweet sounding and important interval - but only after it was adjusted from the pythag 81/64 It was further adjusted when ET was established...

6. Break

7. Modulation Refer to sheet: sharps_flats_in_keys.xlsx Play three note chord starting on C – the 1, 3 rd, 5 th intervals simultaneously This is called a “Triad”. Play a chord once for each key going around the circle of fifths. The starting or “root” note for each Triad is each point around the circle Clockwise: C G D A E B F# Counter-clockwise: C F Bb Eb Ab Db Gb 1 second duration no silence between chords Play using: SystemThirdFifth Just5/43/2 Pythag81/643/2 ET=POWER(POWER(2,1/12),4)=POWER(POWER(2,1/12),7)