Harry Partch Ratio Representation Project


by Brian Harlan and Arun Chidambaram



II. Musical Ratios

Musical ratios are simply another way to denote pitch.  In Western music we typically denote pitches by using the letters A, B, C, D, E, F, G.  If we need to refer to a pitch in between one of these note names, we can use a sharp (#) or flat (b) symbol based on the context.  There are, however, other ways in which to denote pitch.  The most objective, and thus the one used most often by acousticians, is frequency.  Sound is transmitted in waves, and frequency is a measurement of the number of complete waves (or cycles) that occur in one second.  The number of cycles per second determines the pitch of atone.  For example, 440 cycles per second (or 440hz) is recognized in the West as the pitch ‘A.’  

    It is important to note that his is an arbitrary assignment, and that, although we have used the letters A-G as note names from hundreds of years, ‘A’ = 440hz was only standardized in the 20th century.  It was because of the arbitrary status of A-G that prompted Partch to prefer the use musical ratios. In a sense, musical ratios are also arbitrary, as they depend on a referential point that can be changed.  Yet, ratios have the advantage of revealing relationships that are otherwise less apparent.         

      In order to help his audience conceptualize musical ratios, Partch used the ancient technique of dividing the string of a monochord (a single-stringed instrument).  If the sting on a monochord with a tone of 100hz is stopped in the middle, it will produce a new tone of 200hz.  The ratio 200/100 can then represent an octave above the source.  To make matters simpler, Partch always reduced his ratios to the lowest common denominator; thus, 200/100 = 2/1. He also recognized the phenomenon of the “sameness “ of pitches separated by an octave. Similar to the idea of a pitch class, Partch denotes 200/100, 600/300, 800/400, and so on, with a single identity: 2/1.  If the string is stopped at a third division, and only one third is sounded, it will produce 300hz.  The 300hz placed in relationship to the previous division then become 300/200, or 3/2.   

 

                 It is common practice to build ratios upwards by putting the smaller number on the bottom, and Partch adheres to this.  Yet, he points out that ratios can also be built downwards.  If we descend from the source tone (rather than ascend from the sources tone, as in the example above) we obtain a result that is more like a simple division of the string length.  In fact, the string length is one of the key factors in the pitches that it generates.  If we decide that the 1/1 is the note ‘G,’ then the 3/2 above ‘G’ is ‘D.’  The 2/3 below ‘G’ is also a ‘D,’ but an octave below the previous ‘D.’  The fact the these two ratios represent the “same” pitch , while at the same time reveal a different relationship to 1/1, is a crucial principle in Partch’s theory of “Monophony.”  

1/1 = “unison”

2/1 = “octave”

3/2 = “perfect fifth”

4/3 = “perfect fourth”

5/4 = “major third”

8/5 = “minor sixth”

6/5 = “minor third”

5/3 (10/6) = “major sixth”

9/8 = “whole tone”

16/9 = “minor seventh”

16/15 = “semitone”

15/8 (30/16) = “major seventh”

       Another significant aspect of musical ratios is that they represent both a pitch, and an interval.  In other words, the 3/2 above 1/1 tells us not only which note we have encountered, but also its relationship to (or the distance from) the source tone (1/1). It was explicitly this dual function of ratios that Partch believed to be more meaningful than letter note names.  While a ‘D’ can indeed be recognized as a perfect fifth, it is only so in relation to ‘G.’  A 3/2, on the other hand, is always a3/2, regardless of its context.  

                        Below are ratios “equivalents” for intervals within an octave in equal temperament (the Western twelve-tone system that tunes all pitches an equal distance apart, or 12√2/1). In Partch’s music the ratios and equal temperament intervals will not sound the same because Partch uses a different tuning system called, just intonation.      

             

1/1 = “unison”

2/1 = “octave”

3/2 = “perfect fifth”

4/3 = “perfect fourth”

5/4 = “major third”

8/5 = “minor sixth”

6/5 = “minor third”

5/3 (10/6) = “major sixth”

9/8 = “whole tone”

16/9 = “minor seventh”

16/15 = “semitone”

15/8 (30/16) = “major seventh”

                         The system of tuning pure (or “just”) intervals predates the equal temperament system by thousands of years.  Through out Western musical history, many theorists and composers—including Partch—have preferred just intonation because it is more closely aligned with the natural “growth” of a tone.  When a tone is sounded our hearing is immediately drawn to the fundamental tone, which can be described as the pitch produced by the entire string on the monochord.  At successively higher frequencies, however, smaller segments (1/2, 1/3, 1/4,etc.) of the monochord’s string are also producing their own pitches at the same time.  This succession of frequencies is generally referred to as the overtone series (also called the harmonic series, or upper partials). The chart bellow shows that, if a string is vibrating at 110 cycles per second, each 1/2 will also vibrate at 220 cycles per second.  In a inter-related play of numbers, successive segments of the string (1/3, 1/4, 1/5 etc.) can be expressed as a successive integers multiplied by the fundamental (110 x 3 = 330, 110 x 4 =440, 110 x 5 = 550), and with each new integer, a new overtone is produced.

   

1/1

110hz

A2

Unison

2/1

220hz

A3

P8 (1st overtone)

3/2

330hz

E4

P5 (2nd overtone)

4/3

440hz

A4

P4 (3rd overtone)

5/4

550hz

C#5

M3 (4th overtone)

6/5

660hz

E5

m3 (5th overtone)

   

In just intonation the intervals above are tuned as they naturally occur.  Equal temperament tunings, in contrast, must squeeze intervals slightly in order to make them synthetically equal.  Although the octave and perfect fifth are nearly as pure as they are in just, on average all the other intervals in equal temperament are slight smaller than nature intended. What Partch also found intriguing about tuning in just intonation is that, if one continues with successive integer as in the method above, intervals—and thus pitches—begin to appear that do not exist in equal temperament.  A 7/6, for instance, produces a tone slightly smaller than a minor third, and an 8/7produces a tone slightly larger than a major second.  One can theoretically continue in this way to produce an infinite number of “new” pitches.  Partch decided to stop with the number eleven, but he continued to create new microtonal resources by adding ratios together.  His music is decidedly focused on forty-three tones to an octave, but its important to understand that this was his decision as a composer, not the theoretical limit of his system.

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Introduction  2 3 4  | Ratio Notation  |  Partch’s Theory of" Monophony”  |  Implementation  | Examples