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 letter-name notation with sharps and flats arose within a 3-Limit (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 non-Western 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 music-theory.

In JustMusic notation, the powers of 3 which are factors in the ratio can be represented by the usual letters and note-heads, 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 prime-base 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 2⁴ (=16) to 2¹⁴ (= 16,384) Hz. Since 2ⁿ º 1, and since 2⁸ Hz= 256 Hz - which is quite close to "middle-C" in the standard 12-EQ scale based on A-440 Hz - the basic reference tone which I use is n⁰ = C-1 Hz. Since my whole system is based on proportional numerical measurement, I feel this is the most logical starting point.

Alternatively, "middle-C" may be equated with n⁰, with lower "octaves" descending through the negative exponents of 2, and higher "octaves" ascending through the positive exponents.

Thus, the exponents of prime-base 2 will indicate the beginning of each new ascending "octave", represented in notation as follows:









The first dimension: prime-base 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:

3⁰ = 1 = 1/1

3¹ = 3 = 3/1, * = thus 3¹ º 3/2

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3^-1 = 1/3, * = thus 3^-1 º 4/3

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On a graph, the x-axis will represent the integer exponents of prime base 3, and letter-names and Semitone values are supplied along with the ratios:

4 -- 3 D 4.98	1 -- 1 A 0.00	3 -- 2 E 7.02
3-1	n0	31

The prime-factor notation represents these pitch-classes in all their "octaves", with the regular staff notation displaying the "octave" registration.

The tonal relationships of 3^-1 : 3⁰ and 3¹ : 3⁰ 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
31	7.02	Nete diezeugmenon
3-1	4.98	ParaNete diezeugmenon
3-3	2.94	Trite diezeugmenon
32	2.04	ParaMese
n0	0.00	MESE
3-2	9.96	Lichanos
3-4	7.92	ParHypate
31	7.02	Hypate

This scale was easily tuned by ear by means of a series of the "Perfect 4ths" [= 3^-1] and "Perfect 5ths" [= 3¹] outlined above.

Arranged into table whose x-axis represents powers of 3, and giving letter-names and Semitone values, we get:

ParHypate	Trite diezeugmenon	Lichanos	ParaNete diezeugmenon	MESE	Nete diezeugmenon Hypate	ParaMese
128 -- 81 F 7.92	32 -- 27 C 2.94	16 -- 9 G 9.96	4 -- 3 D 4.98	1 -- 1 A 0.00	3 -- 2 E 7.02	9 -- 8 B 2.04
3-4	3-3	3-2	3-1	n0	31	32

If the pitch-classes are given letter-names 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 n0	0.00)
G 3-2	9.96
F 3-4	7.92
E 31	7.02
D 3-1	4.98
C 3-3	2.94
B 32	2.04
A n0	0.00

This is the collection of pitch-classes 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 pitch-class is assigned an integer degree-number starting from 1, and intervals are calculated simply by counting degree-integers 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).

Extended 3-Limit systems

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 ----- 19683 Gb 8.82	8192 ---- 6561 Db 3.84	4096 ---- 2187 Ab 10.86	1024 ---- 729 Eb 5.88	256 --- 243 Bb 0.90	128 --- 81 F 7.92	32 -- 27 C 2.94	16 -- 9 G 9.96	4 -- 3 D 4.98	1 -- 1 A 0.00	3 -- 2 E 7.02	9 -- 8 B 2.04	27 -- 16 F# 9.06	81 -- 64 C# 4.08	243 --- 128 G# 11.10	729 --- 512 D# 6.12	2187 ---- 2048 A# 1.14
3-9	3-8	3-7	3-6	3-5	3-4	3-3	3-2	3-1	n0	31	32	33	34	35	36	37

(A n0	12.00)
G# 35	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 31	7.02
D# 36	6.12
Eb 3-6	5.88
D 3-1	4.98
C# 34	4.08
Db 3-8	3.84
C 3-3	2.94
B 32	2.04
A# 37	1.14
Bb 3-5	0.90
A n0	0.00

Music with a 5-Limit basis

The same process can be followed with prime-base 5:

51	5 -- 4 C# 51 3.86
50	1 -- 1 A n0 0.00
5-1	8 -- 5 F 5-1 8.14
	30

Ratios which include prime-base 5 as a factor were first used as more consonant "corrections" for 3-Limit ratios which are nearby in frequency. Originally, the reconfiguration ran thus:

5y	0	128 ---- 81 F 3-4 7.92	32 ---- 27 C 3-3 2.94	16 ---- 9 G 3-2 9.96	4 --- 3 D 3-1 4.98	1 --- 1 A n0 0.00	3 --- 2 E 31 7.02	9 --- 8 B 32 2.04
		-4	-3	-2	-1	0	1	2
						3x

Became:

5y	0				4 --- 3 D 3-1 4.98	1 --- 1 A n0 0.00	3 --- 2 E 31 7.02	9 --- 8 B 32 2.04
	-1					8 --- 5 F 5-1 8.14	6 --- 5 C 315-1 3.16	9 --- 5 G 325-1 10.18
		-4	-3	-2	-1	0	1	2
						3x

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Later, as the development of harmony emphasized the juxtaposition of the "major" [otonal] and "minor" [utonal] modes, to the exclusion of all the rest of the old modes, "D" shifted to 3³5^-1, to give a 4:5:6 triad on G 3²5^-1, the "Dominant" (V):

5y	0					1 --- 1 A n0 0.00	3 --- 2 E 31 7.02	9 --- 8 B 32 2.04
	-1					8 --- 5 F 5-1 8.14	6 --- 5 C 315-1 3.16	9 --- 5 G 325-1 10.18	27 ---- 20 D 335-1 5.20
		-4	-3	-2	-1	0	1	2	3
						3x

Although making "C" the central reference of the whole system [n⁰] changes all of the numerical values, both ratios and Semitones, the relationships between all tones remain exactly the same:

5y	1					5 --- 3 A 3-151 8.84	5 --- 4 E 51 3.86	15 --- 8 B 3151 10.88
	0					4 --- 3 F 3-1 4.98	1 --- 1 C n0 0.00	3 --- 2 G 31 7.02	9 ---- 8 D 32 2.04
		-5	-4	-3	-2	-1	0	1	2
						3x

Because of the strong feeling of tonality inherent in this collection of pitch-classes, the "C-major" scale in the above graph became the standard reference scale and "key" of common-practice 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 3-Limit, with each higher prime-base adding its own unique symbol to the accidental. In my case, the symbols are nothing more than the prime-base with its appropriate exponent.

Higher Primes in Music

While normally actually using such tuning systems as 12-EQ, meantone, or well temperaments, "standard" Eurocentric music theory is based mostly on a 5-Limit just-intonation 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

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References

Wolf, Daniel. 1996. Letter in 1/1, the journal of the Just-Intonation Network, 9:3 [Summer], p 15.

Babbitt, Milton. 1960. Twelve-Tone Invariants as Compositional Determinants
The Musical Quarterly, vol 46, p 246-259.

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 106-137.

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 [Spring-Summer], p 56-76.

Keislar, Douglas. 1987. The History and Principles of Microtonal Keyboards.
Computer Music Journal, vol 11, no 1 [Spring], p 18-28.

Kraehenbuehl, David and Schmidt, Christopher. 1962. On the Development of Musical Systems.
Journal of Music Theory, vol 6, p 32-65.

Monzo, Joseph L. 1997. JustMusic: A New Harmony - Representing Pitch as Prime Series.
(forthcoming).

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For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here or try some definitions.

I welcome feedback about this webpage: corrections, improvements, good links. Let me know if you don't understand something. return to my home page return to the Sonic Arts home page