Just Intonation

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Excerpt from Chapter One of The Natural Harmonic Series as a Practical Approach to Just Intonation, (c) 1991 by Denny Genovese

JUST INTONATION

      In the general sense, Just Intonation is any system  of musical  intervals that may be described in terms  of  whole number  ratios  of  frequencies. For  each  frequency,  one computes  its ratio to a fixed frequency, called the  fundamental, or tonic. The ratio of the fundamental frequency  to itself  is,  of course, 1/1. Just intonation should  not  be confused with any particular historical scale. It should  be thought  of instead as a means of organizing  intervals  according  to this structural principle. By  this  definition, just  intonation  includes Pythagorean  tuning,  all  ethnic scales  that are tuned by ratios of tones, and many  unique, modern systems of tuning. A scale made up of just  intervals may  have a variable number of degrees, and these may be  of many different sizes (distances between each other, or  from the  tonic). The exact number of intervals in a  just  scale may  be dependent upon several factors, including  the  complexity of the music to be played and the instrumentation to be used.

ABOUT  RATIOS

     In  just intonation, a musical interval is described  in terms   of the ratio of its frequency to that of  the  tonic  note of  a  scale. This provides a precise means of  identifying  each interval, not only in terms of its place in  the order  of the scale, but also by describing its size,  relative  consonance,  and  its relationship  to  various  other intervals in the scale. The interval described by the  ratio is derived by multiplying the frequency  of the tonic by the top  number in the ratio  and  then dividing the  result  by the  bottom  number  to produce the frequency of  the  upper pitch.  (Example: If C has a Ratio of 1/1 and G has a  Ratio of 3/2,  it means that G is half of three times the frequency of C. )

     Even  though  these ratios are  actually  formulas  for deriving  the frequencies of intervals, they  are  generally more  useful and descriptive than would be  the  frequencies themselves.  One reason for this is that while  actual  frequencies  may  be essential for certain  tuning  operations, they do not in themselves convey the additional  information that  is implied by ratios. The following sections  describe some of the additional information that ratios provide about intervals, and their relationships to each other, as well as important  concepts that help in the process of  using  this information.

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THE NUMERARY NEXUS

     A basic concept in working with musical ratios is  that of  the  Numerary  Nexus. Partch describes this  as  "  the number common to all identities in the ratios of one tonality  - the common anchor; the characteristic of a  series  of ratios that determines them as a tonality."  For example, in the  following sequence of intervals, the bottom  number  in each  ratio is 4. This is what each of these ratios  has  in common with each other:  

                        4    5    6   7

                       --   --   --  --

                        4    4    4   4 

     Since  all of the above ratios have 4 as  their  bottom number, we know that the top numbers of each ratio represent different multiples of the constant quantity represented  by the  number  4.  Similarly, the top number of  each  of  the following ratios is 7:

                          7    7   7

                         --   --  --

                          6    5   4 

     By  this,  we know that the bottom numbers in  each  of these  ratios represent different divisions of the  constant quantity represented by the number 7.

     It should be noted that the larger the top number (in a ratio that is part of a set of intervals with a nexus in the bottom), the larger will be the relative size of that interval.  Conversely, the larger the bottom number (in  a  ratio that is part of a set of intervals with a nexus in the top), the smaller the relative size of that interval will be.

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OTONALITY AND UTONALITY

     To simplify discussion of ratios, as well as scales and chords  which are built up from them, Partch introduced  the terms  Otonality, Utonality, Odentity and Udentity. The  top number in a ratio is defined to be the interval's  ODENTITY, meaning  the identity of the number which is OVER  or  above the line in the center. It is simply easier to say  "odentity" than "the number in the top position of the ratio."  The bottom number is similarly defined as the interval's UDENTITY, which is UNDER the line. Otonality refers to a  sequence of  intervals  with  a nexus in the  bottom  numbers,  while Utonality  refers  to  a sequence with a nexus  in  the  top numbers.  Thus,  the top (or OVER) numbers of  an  Otonality represent  the proportional relationships of its  intervals, while the bottom (or UNDER) numbers of a Utonality represent inversely proportional relationships.

RATIO ARITHMETIC

     In just scales, all ratios are typically measured  from a particular pitch, the ratio 1/1. At times it is useful  to know the relationship of certain intervals to some component of  the  scale, other than 1/1. At other times,  it  may  be necessary  to  determine what ratio may be  best  suited  to serve  a particular function. In each case, the solution  is obtained  through  the application  of  simple  mathematical procedures.

     Just musical scales can be built up by adding  together intervals  and groups of intervals with various  Otonal  and Utonal relationships. To add two just intervals (to create a new  interval as large as the two combined), multiply  their ratios.  This is done as follows: the top numbers are  first multiplied together, then the bottom numbers are multiplied. For instance, to add the intervals 10/9 and 9/8 the  following procedure would be used:

First, multiply the top numbers of the two ratios:

                      10  X  9  =  90

Then, multiply the bottom numbers:

                      9  X  8  =  72

Then, reduce the fraction:

                 90 / 9  =  10  / 2  =  5  

                 --         --          -

                 72 / 9  =   8  / 2  =  4

To subtract one interval from another, the first  ratio is divided by the ratio that is to be subtracted from it. To do  this, the top number of each ratio is multiplied by  the bottom  number of the other. Then, the fraction is  reduced. In  this example, a 5/4 interval is subtracted from  a  3/2. First, 3 is multiplied by 4. The product (12) is the resulting  Over number. Then, 2 is multiplied by 5 to  derive  the Under  number (10). Both numbers in the resulting ratio  are divisible by two, so the final form of the new ratio is 6/5:

                3       5     12       6

                -   /   -  =  --   =   -

                2       4     10       5

     Here  is  an  example of how these  procedures  can  be useful.  With the following scale, smooth major  triads  are possible on C, F and G. Smooth minor triads are also  possible on E and A. However, the minor triad on D sounds rough:

          1    9    5    4    3    5    15    2

          -    -    -    -    -    -    --    -

          1    8    4    3    2    3     8    1

          C    D    E    F    G    A     B    C

     We already know that the major third between F and A is smooth, because of its suitability in the F major chord. The problem is to determine what interval is required for another  D,  from which the existing F will constitute  a  smooth minor  third (6/5), and from which the existing A will  constitute a smooth perfect fifth (3/2). To do this, we  simply subtract 6/5 from 4/3, which yields the ratio 10/9. Now,  we have  all  of the resources of the original  scale,  plus  a smooth D minor chord. The new scale is as follows:

        1   10   9    5    4    3    5    15    2

        -   --   -    -    -    -    -    --    -

        1    9   8    4    3    2    3     8    1

        C    D-  D    E    F    G    A     B    C

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PROTOCOL FOR REPRESENTATION OF RATIOS

     Thus far, musical ratios have been shown with their top numbers  larger  than their bottom numbers. This  makes  the relationship  of Odentities more apparent to the  eye,  when the  first  interval in the sequence is  the  reference,  or tonic  of  the sequence. In the following sequence,  4/4  is taken as the tonic (since it may be reduced to 1/1).  Therefore,  5/4 is easily seen as 5 times the frequency  of  that tonic, divided by 4:  

  Odentities:           4      5     6

                        -      -     -

Udentities:             4      4     4   

Frequencies:           100    125   150

     Here, the nexus 4 represents 4 times the base frequency of  25,  and  the relative sizes of the  Odentities  may  be clearly  seen to be in ascending order. An opposite  convention  exists,  wherein the Udentities are  shown  as  larger numbers than the Odentities. This protocol suits Utonalities in a similar way that the former approach suits Otonalities:

Odentities:             4      4     4

                        -      -     -

Udentities:             6      5     4   

Frequencies:          133.3   160   200

     Here the ascending order of interval size is represented by the descending order of Udentities, since these represent divisions of the frequency of the 4/4 unity.

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OCTAVE EQUIVALENCE

     Since  it is customary to represent scales in ascending order,  starting with the tonic, and it is easier to  recognize  the tonic of a scale when there is consistency in  its placement,  and  the method is used  almost  universally  by modern writers, I normally represent just scales and  intervals  with higher numbers in the top of the ratios  than  in the  bottom.  The practice of reducing frequency  ratios  to their lowest common denominator, with the highest number  in the top place is termed OCTAVE EQUIVALENCE (or more  recently: DUPLE EQUIVALENCE). This is because it allows scales to be  constructed and organized within the limits of  a  duple (octave).

     To  illustrate how this is done and why it  is  useful, let  us  construct  a Pythagorean scale.  Starting  with  a single  tone (C=1/1), we begin by adding a 3/2  (G).  Adding another  3/2 to this G yields a D in the next  higher  duple (3/2  + 3/2 = 9/4). We know it is more than a  duple  higher than 1/1,  because 9 is more than 2 times 4. This D can only be  used  as a 2nd scale degree if it is in the  same  duple with  the  1/1 and 3/2. Since 9 cannot be divided by  2,  we multiply the bottom number by 2. This results in a ratio  of 9/8,  which  is a major 2nd in the same duple as  the  other tones  of the scale. From this 9/8, we now add another  3/2. this  gives us an A, within the same duple (27/16).  Another 3/2  interval is added to A, resulting in the  ratio  81/32. Since 81 is more than 2 times 32, we know that we have again exceeded  the limits of the duple. This is remedied  in  the same way as in the case of D: 32 is multiplied by 2, resulting in the interval 81/64, which is a Pythagorean major 3rd, in the same duple with the other tones.

So far, our scale looks like this:

                 1   9   81   3   27   2

                 -   -   --   -   --   -

                 1   8   64   2   16   1

                 C   D    E   G    A   C

     If we stopped here, we would have the historic scale of Ling  Lun, a Pythagorean type Pentatonic scale, which  would be  useful in playing a great variety of music. This  scale has been the basis of numerous musical traditions around the world.  However,  extending this scale further  (which  was also  done by Ling Lun) will give us an opportunity  to  see how  duple equivalence may be achieved from  intervals  that are  smaller than 1/1 (that is, located in a duple which  is lower  than  that  in  which  the  rest  of  our  scale   is organized).

     The  classic Pythagorean scale not only  has  intervals related  by 5ths, but by 4ths also. To obtain a perfect  4th from  C, we must measure a 3/2 downward from 1/1.  A  simple way  to do this is to invert the 3/2 ratio. A ratio  may  be inverted  by simply switching its upper and  lower  numbers. The inversion of 3/2 is therefore 2/3, which is less than  1 and  therefore in a lower duple than 1/1. Since 3 cannot  be divided  by  2, we multiply the upper number(2)  by  2.  The resulting  interval  is 4/3 (F), which is a  perfect  fourth higher than 1/1. We now continue this process, adding another 4/3 to F: 4/3 + 4/3 = 16/9 (Bb). We know that this interval is in the desired duple, because 16 is larger than 9 but not  as  large as 2 X 9. We now have a  7  note  Pythagorean scale,  wherein all intervals are derived by successions  of perfect 4ths and 5ths:  

                1  9  81  4  3  27  16  2

                -  -  --  -  -  --  --  -

                1  8  64  3  2  16   9  1

                C  D  E   F  G  A   Bb  C

     Further extension of this scale is possible by continuing this process in either direction: up by 3/2's or down by 4/3's. If a succession of twelve 3/2's were built upon  each other, an interval would be reached that would be 23.5 cents (about  1/4  of a semitone) sharper than the  interval  that would be reached by seven 2/1's. This interval is called the Pythagorean comma. Its ratio is 531441/524288. More will  be said  about  Pythagorean  scales in  another  chapter.  This digression is for the purpose of demonstrating duple equivalence.