1. Bryce Canyon

2. CalPerformances, Jan 31, 2016
St. Louis Symphony performs: Oliver Messiaen ( ) “From the Canyons to the Stars” ( ) modern music supported with images and videos

3. Translatory Symmetries (Regularity)
Florida 1999
CS 39R
Translatory Symmetries (Regularity)
in Musical Scales
What is this ?

4. (One Octave of) Today’s Piano Scale
Who defines the frequencies?

5. Many Different Scale Systems !
Granada 2003
Many Different Scale Systems !
Twelve equal-ratioed steps: 12th root(2)

6. Equal Tempered Piano Scale
Twelve equal-ratioed steps: 12th root(2)
Johann Sebastian Bach

7. How Well Does This Work?
Frequency Mismatch (%)

8. Who Defines Which Tones to Use?
Many Scales:

9. Some Select Scales:
Which ones have the best translatory symmetry?

10. Whole-Tone Music
Let’s use mostly full steps!
Claude Debussy

11. 12-Tone Music
Let’s use all the tones equally!
Arnold Schoenberg

12. Other “Tone Symmetries”?
… inspired me to look at scales
with high translatory symmetry. Found: “Diminished Scale”
Oliver Messiaen

13. What Scale Do YOU Propose?
Select a subset of tones …
Check it out: