1. Tuning Systems By Jamila Smart

2. Just Scales Made from integer ratios Pentatonic Pentatonic Pythagorean Pythagorean Natural Natural

3. Pythagorean Diatonic Diatonic –Addition of third and seventh –Different C4 frequency –Syntonic comma Dodecaphonic Dodecaphonic –Intervals created by raising the fifth to a power –Creates circle of fifths –Different ratios for the tritones

4. Natural Chromatic Chromatic –Just and Pythagorean M3, m3, M6 and m6 off –Syntonic diatonic scale –Wolf tones Major Major –Uses integers 1-6 –Two different semitone intervals

5. Tempered Mean-tone – optimize thirds and fifths in certain keys Mean-tone – optimize thirds and fifths in certain keys Well-tempered – All keys usable Well-tempered – All keys usable Equal-tempered – make all keys sound the same Equal-tempered – make all keys sound the same

6. Mean-tone Tempered Uses whole numbers as well as irrational numbers Uses whole numbers as well as irrational numbers Perfect M3 Perfect M3 Fifth not correct Fifth not correct

7. Microtone Small scale degree Small scale degree More chance that note will fall close to a ratio More chance that note will fall close to a ratio f (k,v) = f R x 2 v+k/N f (k,v) = f R x 2 v+k/N

8. 53 Tone N = 53 N = 53 3.615 cents apart 3.615 cents apart Fifth still flat Fifth still flat

9. Quarter Tone N = 24 N = 24 50 cent note interval 50 cent note interval Three Quarter-tone pieces for Two Pianos ii. Allegro - Ives Three Quarter-tone pieces for Two Pianos ii. Allegro - Ives

10. 43 Tone Invented by Harry Partch Invented by Harry Partch Just Just G Symmetric Symmetric

11. Bohlen-Pierce 13 tones A-H, J 13 tones A-H, J Tritave Tritave B-P tenth B-P tenth

12. Wendy Carlos Octaves must not contain whole number of notes Octaves must not contain whole number of notes More important to get ratios correct More important to get ratios correct Alpha Alpha Beta Beta Gamma Gamma