Download MSP_lecture3

Music Software projects
New york university
Adjunct Instructor Scott Burton
The Pythagorean System
• Invented around 600 B.C.
• All intervals based on naturally occurring 3rd degree in the harmonic series
• The 3:2 ratio or factor of 1.5...
• Now what?
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•
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Construct as a geometric series - successive multiplication by 1.5
How many intervals to create – e.g., how to divide up the octave?
Answer = 12 and still holds true today (for western music anyway)
Interesting that we don’t use his system anymore but standardized on 12
• So why 12?
• Pythagoras thought that:
• 12 intervals created by successive multiplication of 3/2 should equal 7
octaves
• Our living spec spreadsheet illustrates this...
Pythagorean vs. Harmonic
Series
 Built on successive 3/2 ratios only vs. taken directly from other ratios naturally
occurring harmonic series
 The next unique interval in the harmonic series after 3/2 is 5/4
 This step in our modern scale is known as the “major third” (E in a C scale
which we will see more of later).
 Closest interval in Pythag’s scale is “Unstable” interval of 81/64
 Slightly higher than the closest interval of 5/4 in harmonic series
 Difference is 81/80
 81/80 = 81/64 ÷ 5/4
 The “Third” is a distinctive, rich sounding interval but was considered
“imperfect” back in the day due to above...
 “Perfect” Intervals:


Fourth
Fifth
Pythagorean vs. Harmonic
Series
 Built on successive 3/2 ratios only vs. taken directly from upper ratios in the
naturally occurring harmonic series
 The next unique interval in the harmonic series after 3/2 is 5/4
 This step in our modern scale is known as the “major third” (E in a C scale
which we will see more of later).
 “Unstable” interval of 81/64
 Slightly higher than the closest interval of 5/4 which is present in harmonic series
 Difference is 81/80
 81/80 = 81/64 ÷ 5/4
 This difference is known as the “Syntonic Comma” (Google it)
 “Perfect Intervals”


Fourth
Fifth
“Pure” or “Just” Intervals
sound great but…

Consider the following:

C
1/1 x 528 = 528 Hz

D
9/8 x 528 = 594 Hz

E
5/4 x 528 = 660 Hz
Now let’s build an interval one pythag whole tone (9/8) from the "D" note:

D
1/1 x 594 = 594 cycles per second

E
9/8 x 594 = 668.25 cycles per second
Which one is right?
Should the "E" be tuned to 660 or 668.25 Hz???
Not a problem for singers, string players (non-fixed pitch instruments)
Problem for pianos or other fixed pitch instruments when “modulating”
Some Terminology

“Pythagorean”
 Built from the 3rd harmonic in the harmonic series (3/2)
 The 3/2 ratio is also known as the “fifth” since it is the 5th degree in the Pythagorean 7 note scale
 All scale degrees/intervals are derived from by multiplying by successive fifths

“Just”
 Usually synonymous with “Natural” or “Pythagorean” or other scales that use integer ratios to build their
interval degrees

“Natural”
 Taken from harmonic series integer ratios occurring in nature (e.g, the vibrating string)
 Also used to refer to a scale without sharps and flats (“accidentals”)

“Equal-tempered” or “Even-tempered”
 Octave is divided into “equal” steps
 Makes all keys sound the same
 Can be 12 or more intervals ( more than 12 is considered “microtonal” )

“Well-tempered”
 Some intervals are more usable than others
 “Mean-tone” is primary example

Optimize the thirds and and fifths in selected keys at expense of the rest
For next week:
 Implement “Pythagorean” scale
 Refer to living spec spreadsheet
 “pythag_sheet_phase2.xlsx”
 Implement methods:
 Float hz = get_frequency(int interval_position_sorted)
 Return the frequency after being “octave adjusted” aka in the same octave
as the starting frequency.
 Example: 704, 891, ...
 Char* get_interval_ratio(int interval_position_sorted)
 example: "3/2", "9/4", "27/8” ...
 Play scale ascending sort on frequency
 1 second duration, 1 second silence between
 Happy Coding!