DECODING GANN's HARMONIC WHEEL AFTER MAKING THE CONNECTION GANN ANGLES ARE CYCLICAL OCTAVES

 

The Tempered Scale

If you start from a particular note and ascend or descend in either perfect fifths or thirds, one never arrives at a note that is a whole number of octaves from the starting note. It is partly this reason that both the Pythagorean and Just scales have posed problems for recent Western musicians (in the last few hundred years) and adjustments have been made to these scales to find an acoustical compromise.

Leonard Euler's Equal Temperament

In this form of compromise, the first of these requirements is sacrificed for the good of the latter two requirements.

 

MUSIC DEFINED AS MATH IS A LOGICAL MATRIX-BASED 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 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

the fraction can be eliminated, and the series of prime-bases raised to zero, positive, and negative integers can be used.

"Octave" Equivalence

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:

(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

where p {...-1, 0, 1,...}

In current terminology, this is called "octave equivalence".

A Starting Point:
The Reference Tone In Equal Temperament and the Octaves

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.

SUMMARY:
"middle-C" may be equated with n0,
lower "octaves" descending through the negative exponents of 2,
higher "octaves" ascending through the positive exponents.
Therefore, the exponents of prime-base 2 will indicate the beginning of each new ascending "octave".

In the equally tempered scale all semitone intervals are exactly the same ratio, irrespective of frequency location. This implies that all whole-tone intervals are likewise equal. In order to accomplish this the octave is divided into twelve equal semitone intervals.

The two criteria then, are

1. That all semitone intervals be identical; i.e.
2. That the octave maintain its integrity; i.e. f(note 13) = 2 x f(note 1)
The first criterion means that the ratio of all the semitone intervals is the same. Call this ratio a.

The ratio of any two semitones is therefore the same:

Furthermore, since f(note13) = af(note 1),
we have a= 2
or a (12th root of 2)

The twelfth root of two is approximately 1.059463094.


Using this fact, the following table shows frequency ratios of intervals within one octave.
This gives actual approximate frequencies that result if we begin with Concert A, A4 = 440 Hz (the standard of pitch).

 

Observe that....

The following table allows a comparison between some 12-tone equal temperament and theoretical ratios:
From these results it is clear that equal temperament mistunes all intervals except the octave. It drastically mistunes thirds and sixths.

 

1. The Harmonic Series

The harmonic series is a series of frequencies that are whole number multiples of a fundamental. For example, taking the tone Middle C whose fundamental frequency is approximately 261 Hz, the harmonic series on this frequency is:

Harmonic Series = f , f2 , f3 , f4 , ......... fn.
where f = 261 Hz.,
f= 261 Hz , f2=522 Hz , f3=783 Hz, etc................... 261 Hz x n.

An interval can be expressed as a ratio of the frequencies of the two tones:
The higher of the two tones (f2) is placed over the lower (f1) because we are considering an upward interval.
Example: For the perfect fifth C - G would be...


1. The Start of the Pythagorean Scale

To the ratios... add the letter-names, semitone values, and integer exponents of prime base 3:


The perfect 5th has the exact ratio, 3/2... or 1.5.
A fourth has the exact ratio 4/3 or 1.33.

This is the start of the Pythagorean Scale.

2a. Construction of the Pythagorean Diatonic Scale

The First Dimension: Prime-Base 3


IMPORTANT: Note that the powers of 3 are no longer linear when the correct semitone values are arranged.

 

If the pitch letter-names are rearranged to fit within the "octave" A to A, it forms what became the standard Pythagorean diatonic "natural minor" scale:


The names for the notes derive from Pythagoras

Recognize that the Pythagorean mathematical interpretation
of the Universe was:

Celestial Motion and Acoustical Motion have the same mathematical basis because only propositions proven by mathematics were accepted as universal truths. The letter names of the notes came about from the following...


Pythagorean
Name for the Note		 Earth was the unmoving center of the Universe... Translations are... In Greek... the first letter is....(Pitch)  
Hypate

Highest up (Hypaton) - Saturn
E
slowest moving planet from earth...given lowest pitch
Parhypate
 Next to the Highest - Jupiter
F
  
Hypermese
or Lichanos		 Above the Mese
or middle finger- Mars
G
  
Mese Middle - Sun
A
  
Paramese Next to Middle - Mercury
B
  
Paranete Next to Lowest - Venus
C
  
Nete Lowest (Neaton) - Moon
D
 swiftest planet...given highest pitch



"Music of the Spheres"

-530 ... Pythagoras studies propositional geometry and vibrating lyre strings

Pythagorean Doctrine: Is the union of harmonics and astronomy; All things must abide by the universal laws of harmonics...as dictated by divine arithmetic proportion.

(Pythagoras added the eighth string to the LYRE to complete an octave.)

Pythagoras is also given credit for first discovering Phi, or the Golden Ratio 1.618

You may wish to explore Just-Intonation which followed Pythagorean Intonation.


References Wolf, Daniel. 1996. Letter in 1/1, the journal of the Just-Intonation Network.

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.

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.