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

2. Let’s Hear Some Modes...

3. 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

4. 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”

5. “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)… 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”

6. Pythag sheet review Refer to living spec Break

7. 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” = 1.0594631... Geometric series Lands perfectly on an octave “Perfectly impure” – no intervals are “Just” Refer to even_temperment_sheet.xls living spec…

8. 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…

9. Assignment for Next Week Implement calculations of space between the scale degrees. We will be analyzing and comparing them. Implement the Even-tempered or “ET” scale Implement the “chromatic” pythag scale After implementing above enhancements your “interval” collection will handle: Harmonic series Pythagorean Even Tempered You should be able to calculate the distance between any interval combination in the above 3 systems

10. Submission Specifics Print Even Tempered scale frequencies to stdout Add ET regression test harness using spreadsheet living spec to generate test cases 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 number For regression test implement the derived numbers you see on the living spec sheets

11. Classes ahead Form teams of 2 We’ll cover harmony and rhythm in the next several lectures 2 tests on subject domain The final will incorporate melody, harmony, rhythm to play your piece of music While your piece is playing you will have to use graphics that will be synchronized with your music in some way It can be a performance or a demonstration of music theory principles (teaching tool)