1. New York University Adjunct Instructor Scott Burton
MSP
New York University
Adjunct Instructor Scott Burton

2. Let’s Hear Some Modes... Initialize with the base frequency of 540Hz.
Play each note 1 second
No silence between notes
Play 8 tones to end on an octave

3. Let’s Hear the 12 step ET scale...
Initialize with the base frequency of 540Hz.
Play each note 1 second
No silence between notes
End on an octave

4. The Intervals of the Pythagorean Scale Spacings
What are the intervals between each interval?
Example starting with C:
C to D : 9/8 divided by 1/1
D to E: 81/64 divided by 9/8 = 9/8
E to F: 4/3 divided by 81/64 = 256/243

5. The Intervals of the Pythagorean Scale Spacings
This is why there are two “half-steps” or “semi-tones” in the western scale we use today…
See the black and white key patterns on the piano keyboard
Space between notes B/C and E/F is smaller than others
The fact that there are different distances between notes in a scale is what allows to us differentiate and identify a “key” or “tonal center”
A property of a key is a certain combination of whole and half steps
See spreadsheet “pythag_sheet_phase4.xls”

6. The two “half steps” or “semitones”
Where the half steps appear in a scale helps us identify a key.
Equal steps don’t create “gravity”
A vestige of the system of Pythagoras...

7. Pythag sheet review
Refer to living spec
Break

8. Even Temperament
With the introduction of fixed pitch instruments coupled with the desire to modulate freely something had to be done!
Temperament
3 general categories of adjustments to the Pythagorean scale:
Substitute some rational number fractions to more closely match the harmonic series (Ptolemy)
“Well” : some intervals tweaked at the expense of others
“Equal” : all intervals tweaked to be uniform, same spacings between all intervals
Each scale step is multiplied by a constant factor
12√2 = “semi-tone” =
Geometric series
Lands perfectly on an octave
“Perfectly impure” – no intervals are “Just”
Refer to even_temperment_sheet.xls living spec…

9. Terminology Review
“Just” mean integer ratios are used to build the scale degrees
“Pythagorean” aligns with the harmonic series for some intervals – for example with the “third” 81/64 vs. 5/4 (differs by 81/80 – the “syntonic comma”)
Why “octave”, “fifth”, “fourth”, etc. when we have 12 tones?
These terms took hold before the 7 tone scale was extended with “accidentals” (more granularity created extending out the 3/2 geometric series)
Semi-tone or half-step is smallest interval in conventional scale
Micro-tonal means more than 12
Whole tone = two semi-tones
“Minor” refers to 1 semi-tone less than “Major”
Example: A “Major Third” is 4 semi-tones, Minor Third is 3 semi-tones…
“Perfect” vs. “Imperfect” : Believed to be “perfect” if is part of the harmonic series, “imperfect if not”
Fifths are perfect, Thirds were not in Pythagorean scale…

10. Assignment for Next Week
Implement calculations of space between the scale degrees. We will be analyzing and comparing them.
Add new capabilities to our scales
Calculate interval spacing as fractions, numbers, cents and Hz. See cell formulas in ET_sheet.xlsx and the tab “pythag” in pythag_sheet_phase4.xlsx )
After implementing above enhancements your “interval” collection will handle:
Harmonic series
Pythagorean ET
You should be able to calculate the distance between any interval combination in the above 3 systems

11. Submission Specifics While playing notes show Hz on the screen
Add new function to your collection – return the spacing between any two intervals
char* interval_string_name = get_spacing(int interval_number_left, int interval_number_right)
“9/8” = get_spacing(1, 2)
For ET case you would return an decimal number
For regression test implement the derived numbers you see on the living spec sheets

12. “Just” tuning limitations
Building off a starting frequency with rational number fractions produces inconsistent intervals.
This can be a problem when modulating (changing key) or using harmony. Western classical music explored modulation more - generally speaking…
Each scale had different frequencies
The same intervals could be built in different ways with different results
Key changes within a song can sound “rough” for some note combinations/intervals – especially during the transition
Some note combinations produce more noticeable “beating”

13. Classes ahead
Oct 11th two weeks of assignments, Oct 18th guest lecture, Oct 25th quiz
Harmony and rhythm
“Aleatoric” composition. Latin root “alea” = die or dice
Implement a few more notable scale types post Pythagoras
2 quizes where you can use your software and sheets
Start formulating an idea for your product
The final app should incorporate melody, harmony & rhythm
While your app is playing you will have to use graphics synchronized with your music in some way
It can be an actual performance or a demonstration of music principles (teaching tool) or ?

14. Intro to Rhythm Component
We will use an simple notation invented by a lesser known but excellent percussionist
Practice it in class then code it!
“Nanafly” by Billy Martin from Medeski, Martin and Wood