
© 2004 Tonalsoft Inc. 
TABLE OF CONTENTS
AN EQUALTEMPERED UNIT OF MEASUREMENT: THE SEMITONE
A LOGICAL MATRIXBASED NOTATION
I have independently developed a theory almost identical in concept to Ben Johnston’s, but with some important differences in the notational system. The objections I have to Johnston’s notation were neatly summarized recently in 1/1 by Daniel J. Wolf, whose notation, although it uses different symbols, accepts as a solution the same structural layout as my JustMusic notation.
AN EQUALTEMPERED UNIT OF MEASUREMENT: THE SEMITONE
2 is the proportion which represents the "octave", so what we perceive as being linearly 1/12 of this  or the JustMusic Semitone  is an irrational proportion:
1 : =  1 : 2^{(1/12)} =  approx. 1 : 1.0594631 =  100 cents =  1.00 Semitone 
(1) 
This is the basic measurement used in the standard 12EQ scale: each degree is 1.00 Semitone apart, and it is very useful for comparing justintonation intervals and relating them to our familiar scale.
A LOGICAL MATRIXBASED NOTATION
Ratios Notated As A Prime Series
A ratio may be written and manipulated as a fraction, and each of its terms may be factored according to the Fundamental Theorem of Arithmetic into a series which consists of the series of positive prime integers, each of which is considered as a base raised to zero or positive integer exponents; these are then multiplied together to arrive at the product which expresses a term in the proportional number
Since
=  (2) 
the fraction can be eliminated, and the series of primebases raised to zero, positive, and negative integers can be used. This was stated succinctly by Douglas Keislar:
Any just interval is expressible as the product of powers of prime numbers:
2^{a} * 3^{b} * 5^{c} * L^{n}  (3) 
where a thru n are integers,
and L, called the "limit" of the system, is the largest prime number in the series.
For example:
2^{1} 3^{1} 5^{0} 7^{0}... = 1/2 * 3 * 1 * 1... = 3/2 (This is a 3Limit ratio)
The fundamental unit is the octave, which has the unique property that its two notes are felt in some indefinable way to be the same, though in pitch level they are recognizably different.
Any frequency f compared to itself is expressed as the unity ratio:
=
(4)
(for example: 256 Hz : 256 Hz = 1 : 1).
Any frequency f multiplied by any integer power of 2 results in a musical pitch which has the same aesthetic properties as f :
2^{p} * f f  (5) 
where p {...1, 0, 1,...}
In current terminology, this is called "octave equivalence".
Eliminating the "octaves": Pitchclass
For any integer x, since x * 1 = x, and x^{0} = 1 and, from equation (2), 2^{x} 1, any expression equivalent to 1 may be omitted from the notation, and only prime integers greater than 2 and with a nonzero exponent need be used as bases.
Besides its elegance, another advantage of this notation is that it discloses important information about:
JustMusic is based on the set of prime integers {3, 5, 7, 11, 13, 17, 19} as bases of the ratio terms. This arbitrary upper limit on the set of prime integer bases is referred to as the 19Limit (analagous to Partch's terminology of odd limits). Whether referring to a tone, scale, composition, or entire musical system, the primelimit is indicated by the highest prime base which has a nonzero exponent.
To simplify notation, the bases may be omitted, and the exponents alone presented as a series, in which case zeroexponents must be included as placeholders:
a b c d e f g  (6) 
where {a,...g} {...1, 0, 1,...}
and a = 3^{a}, b = 5^{b}, c = 7^{c}, d = 11^{d}, e = 13^{e}, f = 17^{f}, and g = 19^{g}
For example:
2 1 0 0 1 0 0 = 3^{2} 5^{1} 7^{0} 11^{0} 13^{1} 17^{0} 19^{0} = 9 1

51 1 1

131 1 = 9

5 * 13= 9

65* 2^{3} = 72

65Therefore, 2 1 0 0 1 0 0 º 72/65
(This particular ratio falls within the 13Limit.)
The representational procedure used by this notation is somewhat analogous to that used in our standard mathematical numbering system. Here, the least significant digit (the exponent of primebase 3) is on the left. By assuming the series of primebases as a given, and notating only the exponents, the two important parameters of frequency relationship, namely sonance and tonal proximity, are obviously exposed by indicating the primebases involved and by the "weight" of each prime as revealed by the absolute value of its exponent.
This notational scheme also forms the basis for JustMusic mathematical operations, such as calculating intervals (see below). I believe that sophisticated systemic mathematical procedures can be applied to the exponents in the prime series in ways similar to those applied by contemporary composers and theorists to the set of integers {0,...11} which represent pitchclasses of the 12EQ system.
In addition, this notational scheme is exactly analogous to the JustMusic Planetary Graph, described in my book.
Because this notational system can be used with any combination of primebases, I usually find it less cumbersome to use the bases as well as the exponents in the notation, thus omitting primes which have a zero exponent (primes which are º 1). These series of basesandexponents, along with naturals, sharps, flats, and doublesharps and doubleflats, constitute the accidentals which are printed just before each note in the JustMusic notation.
The seven pitchclasses of the 3Limit Diatonic minor scale are the only notes in the system which never need an accidental; for 3Limit pitchclasses with sharps and flats, which use only primebases 2 and 3, the numerical part of the accidental can be omitted if desired, because the basic lettername and sharp/flat parts of JustMusic notation are based on a 3limit cycle, exactly as the notation evolved historically in music theory. Also, to portray the special place of 1/1 as a reference, I use n^{0} as its accidental.
[As was stated earlier, our perception of adding and subtracting pitches must be calculated with ratios as multiplication and division; therefore, I will use quotes around "+" and "" when discussing the same.]
There are 2 methods for calculating rational intervals: multiplication of fractions, and vector addition of exponents.
To add or subtract intervals by vector addition of exponents, simply add or subtract by vector addition the set of exponents for each tone or interval, using 0 where necessary as a placeholder:
Examples:
"Major Third + Minor Third" =
5^{1} (= 5/4) * 3^{1}5^{1} (= 6/5) =
Vector Addition  actual calculation  

[ 0 1]  3^{0} 5^{1}  
+  [ 1 1]  +  3^{1} 5^{1}  
_________  =  ________  
[ 1 0]  3^{1} 5^{0}  = 3^{1}  (= 3/2)  
"Perfect Fifth  Major Third" =
3^{1} (= 3/2) ¸ 5^{1} (= 5/4) =
Vector Addition  actual calculation  

[ 1 0]  3^{1} 5^{0}  
  [ 0 1]    3^{0} 5^{1}  
_________  =  ________  
[ 1 1]  3^{1} 5^{1}  (= 6/5)  
And as always, two negatives make a positive:
"Major Third  Perfect Fourth" =
5^{1} (= 5/4) ¸ 3^{1} (= 4/3) =
Vector Addition  actual calculation  

[ 0 1]  3^{0} 5^{1}  
  [1 0]    3^{1} 5^{0}  
_________  =  ________  
[ 1 1]  3^{1} 5^{1}  (= 15/8)  
BUILDING THE MATRIX
It is my belief (extrapolating on Partch's observations) that musicians have adopted the use of each successive higher prime gradually throughout musical history. In my book, I show how standard lettername notation with sharps and flats arose within a 3Limit (or Pythagorean) tuning system.
The historical outline in my book is presented with an admitted bias toward Western theory; however, I do believe that my ideas can be applied with great success to the musical systems of other cultures. This could correct the mistake that has occurred of forcing nonWestern music into the mold of our familiar system simply because it was the best way that could be found to represent these unfamiliar musics to Western readers. I explore the ancient Greek and Indian scale systems for just this reason, since they have little bearing on modern European musictheory.
In JustMusic notation, the powers of 3 which are factors in the ratio can be represented by the usual letters and noteheads, accompanied when necessary by the standard sharp/flat accidentals. As each following prime number is introduced, it must be indicated along with its exponent, as part of the accidental. (Simply for the sake of completeness, I generally indicate primebase 3 also.)
Starting Point: A Reference Tone
The approximate range of human hearing is from 20 to 20,000 Hz. Described as powers of 2, this is roughly 2^{4} (=16) to 2^{14} (= 16,384) Hz. Since 2^{n} º 1, and since 2^{8} Hz= 256 Hz  which is quite close to "middleC" in the standard 12EQ scale based on A440 Hz  the basic reference tone which I use is n^{0} = C1 Hz. Since my whole system is based on proportional numerical measurement, I feel this is the most logical starting point.
Alternatively, "middleC" may be equated with n^{0}, with lower "octaves" descending through the negative exponents of 2, and higher "octaves" ascending through the positive exponents.
Thus, the exponents of primebase 2 will indicate the beginning of each new ascending "octave", represented in notation as follows:
[note from November 2003: as of now, i have quite decisively come to prefer the version with "middleC" as n^{0}.
If 256 Hz ( 2^{8} Hz) is retained as "middleC" in the version where "middleC' is n^{0}, then 1 Hz would be equated to 2^{ 7}. The graph shows that that at the low end of the audible range, 16 (= 2^{4}) Hz is equated to 2^{ 4}. At the high end, 2^{14} (= 16,384) Hz would be equated to 2^{6}.
Another benefit of "middleC" centricity is that the historically important pitches of trebleG and bassF, which are the references for their respective clefs and whose letters actually became transformed into those clefs, are exactly a "perfect5th" above and below "middleC" respectively.]
The first dimension: primebase 3
Starting with the first prime base in the series which is not equivalent to unity, namely 3, we can build a scale of musical materials by using exponents which increase and decrease from 0, and by multiplying one of the terms continuously by 2 until the ratio falls between 1 and 2, as follows:
3^{0} = 1 = 1/1
3^{1} = 3 = 3/1, * = thus 3^{1} º 3/2
.
.
.
3^{1} = 1/3, * = thus 3^{1} º 4/3
.
.
.
On a graph, the xaxis will represent the integer exponents of prime base 3, and letternames and Semitone values are supplied along with the ratios:
4 D 4.98
 1 A 0.00
 3 E 7.02

3^{1}  n^{0}  3^{1} 
The primefactor notation represents these pitchclasses in all their "octaves", with the regular staff notation displaying the "octave" registration.
The tonal relationships of 3^{1} : 3^{0} and 3^{1} : 3^{0} will be recognized by musicians as the important ones of "subdominant" and "dominant" respectively.
The Pythagorean Diatonic (minor) scale.
All ancient Pythagorean accounts of the diatonic genus of the Greek system are given thus:
ratio  Semitones  note name 
3^{1}  7.02  Nete diezeugmenon 
3^{1}  4.98  ParaNete diezeugmenon 
3^{3}  2.94  Trite diezeugmenon 
3^{2}  2.04  ParaMese 
n^{0}  0.00  MESE 
3^{2}  9.96  Lichanos 
3^{4}  7.92  ParHypate 
3^{1}  7.02  Hypate 
This scale was easily tuned by ear by means of a series of the "Perfect 4ths" [= 3^{1}] and "Perfect 5ths" [= 3^{1}] outlined above.
Arranged into table whose xaxis represents powers of 3, and giving letternames and Semitone values, we get:
ParHypate  Trite  Lichanos  ParaNete  MESE  Nete Hypate  ParaMese 
128 F 7.92
 32 C 2.94
 16 G 9.96
 4 D 4.98
 1 A 0.00
 3 E 7.02
 9 B 2.04

3^{4}
 3^{3}
 3^{2}
 3^{1}
 n^{0}
 3^{1}
 3^{2}

If the pitchclasses are given letternames and rearranged to fit within the "octave" A to A, it forms what became the standard Pythagorean diatonic "natural minor" scale, which was briefly described above:
(A n^{0}  0.00) 
G 3^{2}  9.96 
F 3^{4}  7.92 
E 3^{1}  7.02 
D 3^{1}  4.98 
C 3^{3}  2.94 
B 3^{2}  2.04 
A n^{0}  0.00 
This is the collection of pitchclasses which became the basis of music theory in ancient Greece, and it was transmitted from the Greek literature to medieval Europe by Boethius around 500. His theory became the standard in Europe for about 1000 years.
Most of the standard terminology of harmony derives from this scale: each pitchclass is assigned an integer degreenumber starting from 1, and intervals are calculated simply by counting degreeintegers inclusively (note that this meaning of "degree" is slightly different, and less precise, than that previously defined in the EQ equations: in this scale "degree" does not imply equal steps).
By allowing each of the seven degrees in the standard scale to be used as either mi or fa in the musica recta system, auxilliary degrees are obtained which are higher or lower in pitch than their namesake tone. These are indicated with the use of "accidental" signs, resulting in a gamut of 17 tones as determined by Prosdocimus:
32768 Gb 8.82
 8192 Db 3.84
 4096 Ab 10.86
 1024 Eb 5.88
 256 Bb 0.90
 128 F 7.92
 32 C 2.94
 16 G 9.96
 4 D 4.98
 1 A 0.00
 3 E 7.02
 9 B 2.04
 27 F# 9.06
 81 C# 4.08
 243 G# 11.10
 729 D# 6.12
 2187 A# 1.14

3^{9}
 3^{8}
 3^{7}
 3^{6}
 3^{5}
 3^{4}
 3^{3}
 3^{2}
 3^{1}
 n^{0}
 3^{1}
 3^{2}
 3^{3}
 3^{4}
 3^{5}
 3^{6}
 3^{7}

(A n^{0}  12.00) 
G# 3^{5}  11.10 
Ab 3^{7}  10.86 
G 3^{2}  9.96 
F# 3^{1}  9.06 
Gb 3^{9}  8.82 
F 3^{4}  7.92 
E 3^{1}  7.02 
D# 3^{6}  6.12 
Eb 3^{6}  5.88 
D 3^{1}  4.98 
C# 3^{4}  4.08 
Db 3^{8}  3.84 
C 3^{3}  2.94 
B 3^{2}  2.04 
A# 3^{7}  1.14 
Bb 3^{5}  0.90 
A n^{0}  0.00 
The same process can be followed with primebase 5:
5^{1} 
5 C# 5^{1} 3.86 
5^{0} 
1 A n^{0} 0.00 
5^{1} 
8 F 5^{1} 8.14 
3^{0} 
Ratios which include primebase 5 as a factor were first used as more consonant "corrections" for 3Limit ratios which are nearby in frequency. Originally, the reconfiguration ran thus:
5^{y} 
0 
128 F 3^{4} 7.92 
32 C 3^{3} 2.94 
16 G 3^{2} 9.96 
4 D 3^{1} 4.98 
1 A n^{0} 0.00 
3 E 3^{1} 7.02 
9 B 3^{2} 2.04 

4  3  2  1  0  1  2  
3^{x}  
Became:
5^{y}  0 
4 D 3^{1} 4.98 
1 A n^{0} 0.00 
3 E 3^{1} 7.02 
9 B 3^{2} 2.04 


1 
8 F 5^{1} 8.14 
6 C 3^{1}5^{1} 3.16 
9 G 3^{2}5^{1} 10.18 

4  3  2  1  0  1  2  
3^{x}  
5^{y} 
0 
1 A n^{0} 0.00 
3 E 3^{1} 7.02 
9 B 3^{2} 2.04 


1 
8 F 5^{1} 8.14 
6 C 3^{1}5^{1} 3.16 
9 G 3^{2}5^{1} 10.18 
27 D 3^{3}5^{1} 5.20 

4  3  2  1  0  1  2  3  
3^{x}  
Although making "C" the central reference of the whole system [n^{0}] changes all of the numerical values, both ratios and Semitones, the relationships between all tones remain exactly the same:
5^{y}  1 
5 A 3^{1}5^{1} 8.84 
5 E 5^{1} 3.86 
15 B 3^{1}5^{1} 10.88 


0 
4 F 3^{1} 4.98 
1 C n^{0} 0.00 
3 G 3^{1} 7.02 
9 D 3^{2} 2.04 

5  4  3  2  1  0  1  2  
3^{x}  
Because of the strong feeling of tonality inherent in this collection of pitchclasses, the "Cmajor" scale in the above graph became the standard reference scale and "key" of commonpractice music in Europe. It is also the basis of Ben Johnston’s notation, which is my chief criticism of his notational system.
I agree with Wolf that it is better to use a system which is based on the 3Limit, with each higher primebase adding its own unique symbol to the accidental. In my case, the symbols are nothing more than the primebase with its appropriate exponent.
While normally actually using such tuning systems as 12EQ, meantone, or well temperaments, "standard" Eurocentric music theory is based mostly on a 5Limit justintonation conception of harmony.
In practice, musicians often use or imply ratios with higher prime factors. Theorists have also specified much higher primes in their descriptions of tuning systems. In order to portray these systems visually, I have abandoned the matrix format presented here and instead now use Lattice Diagrams, which give the ability to represent any number of prime factors, each with their own unique dimension in the diagram.
Examples of this approach can be found in the other papers published on this website.
1998 June 13
revised 1998 November 4
and 1998 November 21
References
Wolf, Daniel. 1996. Letter in 1/1, the journal of the JustIntonation Network, 9:3 [Summer], p 15.
Babbitt, Milton. 1960. TwelveTone Invariants as Compositional Determinants
The Musical Quarterly, vol 46, p 246259.
Barker, Andrew, ed. 1989. Greek Musical Writings, volume 2: Harmonic and Acoustic Theory.
Cambridge University Press, Cambridge.
Boretz, Benjamin and Cone, Edward T., ed. 1972. Perspectives on Contemporary Music Theory.
W. W. Norton & Co., New York.
Clynes, Manfred, ed. 1982. Music, Mind, and Brain: the Neuropsychology of Music.
Plenum Press, New York.
Encyclopaedia Britannica. 1981. Music article.
Fonville, John. 1991. Ben Johnston's Extended Just Intonation: A Guide for Interpreters.
Perspectives of New Music, vol 29, no 2 [Summer], p 106137.
Helmholtz, Hermann. [1863] 1954. On the Sensations of Tone
Trans., Alexander Ellis. Reprint ed., Dover, New York.
Johnston, Ben. 1964. Scalar Order as a Compositional Resource.
Perspectives of New Music, vol 2, no 2 [SpringSummer], p 5676.
Keislar, Douglas. 1987. The History and Principles of Microtonal Keyboards.
Computer Music Journal, vol 11, no 1 [Spring], p 1828.
Kraehenbuehl, David and Schmidt, Christopher. 1962. On the Development of Musical Systems.
Journal of Music Theory, vol 6, p 3265.
Monzo, Joseph L. 1997. JustMusic: A New Harmony  Representing Pitch as Prime Series.
(forthcoming).
updates: