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

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

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

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

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”

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...

Pythag sheet review Refer to living spec Break

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…

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…

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

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

“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”

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 ?

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