The Keys of Atlantis
A Study of Ancient Unified Numerical and Metrological Systems
by Peter Wakefield Sault
Copyright © Peter Wakefield Sault 19732016
All rights reserved worldwide
Chapter 1.
The Ring of Truth
Mathematical Derivation of The Musical Intervals
SectionsIllustrations
1. A Difference Tone
2. The TonicOctave Difference Tone
3. Effect of Harmonics upon Waveform
4. The Circle of Natural Intervals
5. Marking out Fret Positions with A Natural Interval Wheel
6. Natural Intervals and Regular Polygons
Tables
1. Dodekaphonic Intervals
2. Derivation of Dodekaphonic Intervals by Triplications and Halvings
3. Derivation of Dodekaphonic Intervals by Trisections and Doublings
4. Dodekaphonic Comb Filter
5. Naturally Tempered Frequency Ratios
6. Correlative Intervals of Just Intonation
also available as an Interactive Computer Program
7. Lowest Common Denominators of The Ratios of Just Intonation
11. A Note on Musical Terminology
U.S. readers should give special attention to the usage here of the words note and tone. In this text, the meanings which they bear derive from the English musical tradition and differ from contemporary U.S. usage. The differences are explained fully in the ‘Oxford Companion to Music’ to which all those uncertain about these or any other musical terms are referred. Briefly, as used in this book, the word ‘note’ refers to a pitched sound and to neither the interval of a major step nor to a mark upon a written score. A ‘tone’ is the interval of a major step, equal to two successive semitones. This definition of the tone does not, however, apply to combination tones, also called resultant tones and comprising difference tones and summation tones, all of which belong to the science of acoustics and refer to physical vibrations.
British  American  Meaning 

Note (1)  Tone  Pitched sound 
Tone  Note (1)  Major step 
Note (2)  Note (2)  Written mark 
12. Introduction to The Dodekaphonic Intervals
Prior analysis of the derivation of the dodekaphonic scale is a necessary underpinning to the exposition of the Babylonian unit of rotation, the degree of arc. This analysis, whilst dependent only upon arithmetic calculation, is unavoidably complicated. Those readers who do not wish to tackle it should go straight to Section 15, bearing in mind that the conclusion of that section depends upon the arguments laid out in the preceding ones. Anyone wishing to dispute these conclusions must demonstrate that there is a flaw in a preceding section. The author welcomes such challenges; the aim of this book is to establish the truth of the matter and if it is otherwise than stated here then the (false) arguments must necessarily be modified or withdrawn along with any dependent conclusions.
* * *
The musical dimension of interest here is that of melody and harmony with particular regard to the structure and derivation of the dodekaphonic (twelvenotesperoctave) musical scale from which melodies are wrought. The notes of the scale are placed at twelve distinct intervals, in the vibration frequency domain, from the root, or bass note, up to and including the octave (see Table 11 below).
The upwardly delimiting interval is defined by a vibration frequency ratio of 1:2. This is the simplest possible numerical ratio that defines a separation in the vibrationfrequency domain (1:1 defines only a point location in the frequency continuum) and, for reasons which will be made clear in Section 13, is the progenitor of all the intervening intervals.
Classification  Pythagorean Concord 
Natural Major Scale  

Pitch  Tonal  Degree  Name  
0  Unison  I  Tonic, Root  
1  Semitone  
2  Tone  II  Supertonic  
3  Minor 3^{rd}  
4  Major 3^{rd}  III  Mediant  
5  4th  Diatessaron  IV  Subdominant 
6  Tritone  
7  5^{th}  Diapente  V  Dominant 
8  Minor 6^{th}  
9  Major 6^{th}  VI  Submediant  
10  Minor 7th  
11  Major 7^{th}  VII  Leading note  
12  Octave  Diapason  VIII  Tonic, Final 
Key to Table 11.  

1.  Classification  Pitch 
2.  Classification  Tonal 
3.  Pythagorean Concords 
4.  Natural Major  Degree 
5.  Natural Major  Name 
* * *
A musical note is a sound in which a given regular vibration predominates to the extent that a listener obtains a clear impression of pitch. Although a standard pitch is defined, the actual vibration frequency of this is entirely arbitrary and no more than a convenience for musicians who are to play together and must do so ‘in tune’ with one another. The vibration of the pitch known as ‘Middle A’ is fixed by modern convention at a frequency of 440 Hertz (cycles per second). However, the convention has not always been thus and the ‘standard’ frequency of ‘A’ has in previous times been higher or lower. Mozart is reputed to have used a piano on which ‘A’ was tuned to 421.6 Hertz. Socalled ‘philharmonic pitch’ was based upon an ‘A’ of 452.5 Hertz.
More fundamental than the standardization of any single pitch is that of the intervals between different pitches. Where two notes are of the same pitch the resulting interval is that of unison (‘one sound’). This is not truly an interval and there is really only the one note, no matter how many different simultaneous sources it may have. Additional sources are harmonically redundant; their presence increases only power (experienced as loudness) while introducing neither separation nor contrast. Unison has no extension in the frequency domain.
Although sounding two notes an octave apart creates a separation in the frequency domain, there is no contrast such as is experienced in varying degrees of pungency with all the other intervals. The sensation of an octave interval has a quality analogous to the taste of pure water. This characteristic neutrality, unique to the octave interval, results from the production of a difference tone equal to one of the constituents, the lower frequency, or bass note, of the pair. Difference tones were first observed by Giuseppe Tartini (16921770) and their frequencies can be calculated by subtracting that of the bass note (the lower of the two frequencies, or vibration numbers) from that of the upper note (the higher vibration number). The physical nature of a difference tone is not the same as that of the two vibrations giving rise to it but is regular amplitude modulation brought about by the summation of the two constituent vibrations.
Nonetheless, providing its frequency lies within the range of human hearing, a difference tone is quite audible as a note in its own right with the single exception of two vibrations separated by the interval of diapason, where the difference tone coincides precisely with the bass note. This condition is unique to diapason. Thus two notes an octave apart, while different, are considered to be harmonic identities of one another. The octave, or ‘final’, note of any scale is regarded, figuratively, as a ‘return’ to the root. Where a scale, such as the natural major, is a subset of the dodekaphonic the pattern of distribution of the notes is repeated in each octave ascended. Hence only a single octave need be described in order to describe all octaves.
Figure 12. The TonicOctave Difference Tone
Each octave ascended represents a doubling of the frequency of the heard pitch. However, each ascension of an octave sounds to be an interval of the same size to the ears. Therefore the arithmetic differences in frequency are not meaningful. The significance lies in the proportions; for any two intervals to sound the same to the ears, it is the ratio of the frequency of one constituent note to the other which must be the same. Thus the sense of pitch proceeds linearly (in arithmetic progression) where the frequencies of the physical vibrations proceed logarithmically (in geometric progression).
The recognition of the harmonic identity of the octave with the tonic led Jean Philippe Rameau (16831764) to the theory of chordal inversion, the foundation of modern harmony which has been developing ever since. Until Rameau, different inversions of the same chord were regarded as different harmonies from one another and therefore as not at all the same chord. The repertoire of symphonic music is a consequence of Rameau's insight into the equivalent harmony of chordal inversions.
This identity has also been explained with harmonic theory, most notably by Hermann Helmholtz (18211894). This must not be confused with harmony theory as documented by Arnold Schoenberg (18741951) inter alia, which is concerned with such matters as chord progressions.
Overtones are secondary vibrations occurring in any naturally oscillating system. Their frequencies are always whole number multiples of that of the fundamental, or first harmonic. Thus the frequency of the second harmonic, or first overtone, is always twice that of the fundamental; that of the third harmonic, or second overtone, is always three times that of the fundamental and so on. If the second harmonic is played as a note in its own right, then it has its own set of harmonics, in similar order. Von Helmholtz theorized that the degree of harmony between two notes of different pitch is a function of the coincidence of loworder harmonics. For example, to take just the first seven harmonics of each of two notes with respective fundamental vibrations of 200 Hertz and 300 Hertz:
Note  Harmonics (frequencies in Hertz)  

1^{st}  2^{nd}  3^{rd}  4^{th}  5^{th}  6^{th}  7^{th}  
1  200  400  600  800  1000  1200  1400 
2  300  600  900  1200  1500  1800  2100 
The vibration numbers of the 3^{rd} and 6^{th} harmonics of the first note (200 Hertz) are the same as those of the 2^{nd} and 4^{th} harmonics respectively of the second note (300 Hertz). Thus, according to von Helmholtz, the two notes have a strongly harmonious, or consonant relationship; which indeed they do, for the second note is pitched a perfect 5^{th} above the first, and the two comprise the concord of diapente. The universal concord of diapason (the interval of an octave) is similarly demonstrated:
Note  Harmonics (frequencies in Hertz)  

1^{st}  2^{nd}  3^{rd}  4^{th}  5^{th}  6^{th}  7^{th}  
1  100  200  300  400  5000  600  700 
2  200  400  600  800  1000  1200  1400 
Helmholtz's theory does not, however, explain why there are twelve notes in the scale, nor the distribution of those twelve notes. Take the following pair:
Note  Harmonics (frequencies in Hertz)  

1^{st}  2^{nd}  3^{rd}  4^{th}  5^{th}  6^{th}  7^{th}  
1  500  1000  1500  2000  2500  3000  3500 
2  700  1400  2100  2800  3500  4200  4900 
Here the vibration number of the 7^{th} harmonic of the first note is the same as that of the 5^{th} harmonic of the second note. The frequency ratio 5:7 does not, however, correspond to any interval of the dodekaphonic scale. Therefore there must be some other principle at work in the selection of intervals for inclusion in the scale.
The mixture of harmonics in a sound supply its timbre, or ‘coloration’, and is the reason why notes blown on a trumpet sound different to the same notes bowed on a viol; each instrument produces a different and characteristic mixture. The power of each harmonic is generally in inverse proportion to its number; the higher the number, the lower the power. Figure 13 shows how differing mixtures of harmonics effect the shape, or waveform, of a fundamental.
Figure 13. Effect of Harmonics upon Waveform
Key to Figure 13.  

a.  1^{st} harmonic (or ‘fundamental’). 
b.  2^{nd} harmonic (or ‘1^{st} overtone’), twice the frequency and half the amplitude of the 1^{st} harmonic. 
c.  3^{rd} harmonic (or ‘2^{nd} overtone’), thrice the frequency and a third of the amplitude of the 1^{st} harmonic. 
d.  Sum of the 1^{st} and 2^{nd} harmonics. 
e.  Sum of the 1^{st} and 3^{rd} harmonics. 
f.  Sum of the 1^{st}, 2^{nd} and 3^{rd} harmonics. 
g.  Sum of all the harmonics from the 1^{st} through the 12^{th}. 
* * *
It is apparent from Figure 13 that the fundamental frequency is never lost in the melange of other harmonics. It is upon the fundamental that the pitch of a note depends. Thus can the sound of one type of musical instrument be differentiated from another, even while each plays the same note at the same time (i.e. in unison).
There are two distinct methods of distributing the notes of the dodekaphonic scale, each giving approximately the same result. These are called temperaments, or tunings, and the first of these, historically, is just intonation, or natural temper, introduced into Hellenic culture by Pythagoras of Samos (6^{th} Century BC). This employs a set of whole number ratios to define the pitch separations of the intervals. These ratios preclude transposition and, in consequence, melodies can be played in only a single key, that of the single note which forms the fixed tonic. The socalled modes of the natural major scale shift not the tonic but only the compass of melodies about that tonic.
The second tuning is equal temper, first proposed by Eudoxos of Knidos (408353 BC) but not implemented until the 15^{th} Century, when Spanish guitar makers started producing equally tempered instruments. Equal temper, or something very closely approximating it, was adopted and popularized by Johann Sebastian Bach (16851750) who exploited its simple advantage over just intonation in his classic exposition ‘The Well Tempered Clavier’ (1722). That advantage is unlimited modulation to any key, for it is a defining characteristic of equal temper that every semitone step is equal to every other. Hence the tonic is transposable to any note without any change in the character of the resulting scale.
The disadvantage of equal temper is that no interval except the octave is perfect. For it is only when the frequency ratio of two notes is one of whole numbers that there are harmonics of exactly the same frequency in each. Because they are enculturated to it, modern listeners do not notice the very slight loss of harmony which equal temper entails, unlike many in Bach's day who found it unpalettable because they had been brought up to the sounds of naturally tempered musical instruments.
13. Derivation of The Dodekaphonic Intervals
The Pythagorean concords (see Table 11) comprise the defining intervals for the rest of the scale and are derived from the first four harmonics. The prior necessity for a first harmonic (fundamental) initializes the system by forcing the arbitrary selection, anywhere within the frequency range of human hearing, of a starting pitch; the root, or tonic.
The second harmonic provides the delimiter of the compass (extension in the vibration frequency domain) of the scale from the root note and thereby provides the universal concord, diapason, at an interval of an octave from the root. The vibration frequency ratio of the root to the octave is 1:2. Diapason is the only interval whose difference tone is equal to the bass note. For this reason diapason does not create an impression of contrast despite its displacement, in the frequency domain, from the root. Hence in the theory of harmony it is regarded as an identity of the tonic.
The third and fourth harmonics give diapente (the perfect 5^{th}) and thereby diatessaron (the perfect 4^{th}). A perfect 5^{th} from the root results from the transposition downwards by an octave of the note defined by the third harmonic. This transposition is effected in order to bring the note within the compass of the scale (i.e. diapason). Diapente is also the difference tone between the notes defined by the second and third harmonics. Diatessaron is the difference tone between the third and fourth harmonics. The frequency ratio of the second, third and fourth harmonics (2:3:4) is that which defines the position of diapente within the compass of diapason (where 2:4 = 1:2 = diapason). This threemember ratio defines both diapente (2:3) and diatessaron (3:4). Diatessaron is therefore the interval from the perfect 5^{th} to the octave. If this interval is transposed downward by a perfect 5^{th} such that the bass note coincides with the root of the scale, then the interval of a perfect 4^{th} from the root is defined. The interval which then separates the perfect 4^{th} from the octave is another perfect 5^{th}.
The third harmonic also reinforces the structure derived so far by virtue of the fact that the difference tone between the first and third harmonics is equal to the second harmonic (i.e. to diapason). The fourth harmonic is really just a doublediapason and for this reason naturally terminates the series of harmonics which play any part in the derivation of the scale. The only other interval defined by the concords is that of the tone, or major step, which Pythagoras defined as the step from mese to paramese. The tone is not itself a concord. The frequency ratio of the tone is calculated as the quotient of 2:3 (diapente) divided by 3:4 (diatessaron), and is equal to 8:9.
It might be thought that, since the primary pitches which initialize the system (the Pythagorean concords) are derived from the first four harmonics, the remainder of the dodekaphonic scale would be derived from further successive harmonics. But this is not the case for the simple reason that transposition of harmonics above the fourth into the compass of the scale does not produce an even distribution of notes. Nor does it reach a natural end (i.e. a finite number of notes per octave) because the series of harmonics is theoretically infinite.
Because the major 3^{rd} is defined in just intonation by the frequency ratio 4:5, it might also be thought that the fifth harmonic is the defining principle of this interval; that the major 3^{rd} is the transposition downward by two octaves of the fifth harmonic. However, this is a merely coincidental structural reinforcement of the justly intoned major 3^{rd} which, as will be seen, has a prior and more forceful derivation. However, it is nonetheless true that the same process that is being applied here to the primes 2 and 3 can be applied to the primes 2 and 5 and that this gives rise to a cycle of major 3^{rds}, corresponding to dodekaphonic interval classes 4, 8 and 12 (i.e. major 3^{rd}, minor 6^{th} and 8^{ve}, the augmented triad).
To obtain the general locations of the remaining notes of the scale two mutually symmetrical series are defined by diapason and diapente. Stacking twelve diapentes (i.e. ascending through twelve successive perfect 5^{ths}) gives a note that is almost exactly seven diapasons (octaves) above the root of the series. Conversely and in other words, ascending seven octaves gives a note which is almost exactly twelve perfect 5^{ths} above the root. Looked at another way (and bearing in mind that a perfect 5^{th} is a 3^{rd} harmonic already transposed downward by one octave), starting with a given note, a second note can be derived from its third harmonic, a third note can be derived from the third harmonic of the second note and similarly a fourth note from the third note and so on. The thirteenth note so derived (taking the starting note as the first) almost coincides with the twentieth note derived by ascending in similar manner through successive second harmonics. Put simply, 3^{12} (or 531,441) is approximately equal to 2^{19} (or 524,288), the two differing by about 1.36%.
Thus the first series is obtained by transposing the ascended 5^{ths} downwards into the compass of the first octave. The octave note so obtained is imperfect, being a little above a true octave. However, the scale so derived possesses an approximately logarithmic distribution of frequencies (geometric progression), which means of course that the distribution of pitches is subjectively linear (arithmetic progression). This is the scale of traditional Chinese bells and hammered harps, which does not correct the imperfect octave. In fact the Chinese extended cyclic tuning, as the method is called, into successive octaves such that, relative to the initial root, each succeeding octave became progressively more imperfect. The process of ascending in successive third harmonics is laid out in Table 12 below.
Powers of 3  Transpositions  Correspondences  

Octaves  Divisors  Quotients  Pitch  Tonal  
3^{0}  1  0  1  1  0  Unison 
3^{1}  3  1  2  1.5  7  5^{th} 
3^{2}  9  3  8  1.125  2  Tone 
3^{3}  27  4  16  1.6875  9  Major 6^{th} 
3^{4}  81  6  64  1.2656  4  Major 3^{rd} 
3^{5}  243  7  128  1.8984  11  Major 7^{th} 
3^{6}  729  9  512  1.4238  6  Tritone 
3^{7}  2,187  11  2,048  1.0678  1  Semitone 
3^{8}  6,561  12  4,096  1.6018  8  Minor 6^{th} 
3^{9}  19,683  14  16,384  1.2013  3  Minor 3^{rd} 
3^{10}  59,049  15  32,768  1.8020  10  Minor 7^{th} 
3^{11}  177,147  17  131,072  1.3515  5  4^{th} 
3^{12}  531,441  18  262,144  2.0273  12  Octave 
Key to Table 12.  

1.  Powers of 3  Exponents

2.  Powers of 3  Values

3.  Transpositions  Octaves to Descend

4.  Transpositions  Divisors

5.  Transpositions  Quotients

6.  Correspondences  Pitch

7.  Correspondences  Tonal

* * *
The two series of powers of two and powers of three can never have an exactly coincident value. The series of twelve ascending perfect 5^{ths} gives a note whose frequency is approximately 129.746 times that of the root note, where the frequency of the seventh octave note is precisely 2^{7} (2 raised to the power of 7), or 128, times that of the root. Hence the ratio of the frequency of the seventh successive octave to that of the twelfth successive perfect 5^{th} is 128:129.746 which evaluates to approximately 1.0136. The frequencies of the two notes arrived at by separately ascending twelve 5^{ths} and seven 8^{ves} differ from one another by roughly 1.36% or slightly less than a quarter of a semitone.
The second series is obtained by descending through twelve perfect 5^{ths}. To be symmetrical, the transpositions upward should be into the octave below the root of the series of ascending 5^{ths}, taking that root as the symmetry point. The transpositions, however, must be into the same octave as the first series. Therefore all the descending 5^{ths} are transposed upwards by one octave more than the first series of ascending 5^{ths} is downwardly transposed. To achieve symmetry it would be necessary to ascend from the root but descend from the octave, placing the symmetry point halfway between them at pitch class 6 (the tritone). This, however, complicates the calculations by introducing the square root of two. Since the end result is the same irrespective of whether the two series are generated essentially asymmetrically by employing only integer numerators and denominators or symmetrically by introducing the square root of two, the former approach has been adopted in the tables shown here for the sake of simplicity and because it is conventional to define musical scales relative to the root rather than to the tritone. The process of descending in successive third subharmonics is laid out in Table 13 below. The result in this case is that the root coefficient is imperfect, being a little less than the true root coefficient which is unity.
Inverse Powers of 3 
Transpositions  Correspondences  

Octaves  Multipliers  Products  Pitch  Tonal  
3^{0}  1  1  2  2  12  Octave 
3^{–1}  ^{1}/_{3}  2  4  1.3333  5  4^{th} 
3^{–2}  ^{1}/_{9}  4  16  1.7778  10  Minor 7^{th} 
3^{–3}  ^{1}/_{27}  5  32  1.1852  3  Minor 3^{rd} 
3^{–4}  ^{1}/_{81}  7  128  1.5802  8  Minor 6^{th} 
3^{–5}  ^{1}/_{243}  8  256  1.0535  1  Semitone 
3^{–6}  ^{1}/_{729}  10  1,024  1.4047  6  Tritone 
3^{–7}  ^{1}/_{2,187}  12  4,096  1.8729  11  Major 7^{th} 
3^{–8}  ^{1}/_{6,561}  13  8,192  1.2486  4  Major 3^{rd} 
3^{–9}  ^{1}/_{19,683}  15  32,768  1.6648  9  Major 6^{th} 
3^{–10}  ^{1}/_{59,049}  16  65,536  1.1099  2  Tone 
3^{–11}  ^{1}/_{177,147}  18  262,144  1.4798  7  5^{th} 
3^{–12}  ^{1}/_{531,441}  19  524,288  0.9865  0  Unison 
Key to Table 13.  

1.  Powers of 3  Exponents

2.  Powers of 3  Values

3.  Transpositions  Octaves to Ascend

4.  Transpositions  Multipliers

5.  Transpositions  Products

6.  Correspondences  Pitch

7.  Correspondences  Tonal

* * *
These two complementary series are matched by interval class, the second being mapped into direct onetoone correspondence with the first. It turns out that the intervals in each pair are separated by the exact same ratio, 1:1.0136, that separates the 19^{th} power of 2 (524,288) from the 12^{th} power of 3 (531,441), these being the most proportionately proximate values in the two series of powers of 2 and 3. Twelve equal passbands (see Table 14 below), which together form a virtual comb filter in the vibration frequency domain, have been defined. All the intervals of both just intonation and equal temper fall within the regions defined by these passbands. This is the derivation of the dodekaphonic scale. All that remains to be shown is how fixed positions within each passband can be defined. There are two methods for doing this; just intonation and equal temper.
Classification  Passband Boundary 
Just Intonation  Equal Temper  

Pitch  Tonal  Lower  Upper  Ratio  Coefficient  Exponent  Coefficient  
0  Unison  0.9865  1  1:1  1  0  1  
1  Semitone  1.0535  1.0679  15:16  1.0667  ^{1}/_{12}  1.0595  
2  Tone  1.1099  1.125  8:9  1.125  ^{1}/_{6}  1.1225  
3  Minor 3^{rd}  1.1852  1.2014  5:6  1.2  ^{1}/_{4}  1.1892  
4  Major 3^{rd}  1.2486  1.2656  4:5  1.25  ^{1}/_{3}  1.2599  
5  4^{th}  1.3333  1.3515  3:4  1.3333  ^{5}/_{12}  1.3348  
6  Tritone  Augmented 4^{th}  1.4047  1.4238  32:45  1.4063  ^{1}/_{2}  1.4142 
Diminished 5^{th}  45:64  1.4222  
7  5^{th}  1.4798  1.5  2:3  1.5  ^{7}/_{12}  1.4983  
8  Minor 6^{th}  1.5802  1.6018  5:8  1.6  ^{2}/_{3}  1.5874  
9  Major 6^{th}  1.6648  1.6875  3:5  1.6667  ^{3}/_{4}  1.6818  
10  Minor 7^{th}  1.7778  1.8020  9:16  1.7778  ^{5}/_{6}  1.7818  
11  Major 7^{th}  1.8729  1.8984  8:15  1.875  ^{11}/_{12}  1.8877  
12  Octave  2  2.0272  1:2  2  1  2 
Key to Table 14.  

1.  Classification  Pitch

2.  Classification  Tonal

3.  Passband Boundary  Lower

4.  Passband Boundary  Upper

5.  Just Intonation  Ratio

6.  Just Intonation  Coefficient

7.  Equal Temper  Exponent

8.  Equal Temper  Coefficient

14. Derivation of Equal Temper
Although, historically, the employment of equal temper came after that of just intonation, it is somewhat easier to understand than just intonation and is in any case included here only to show that all the equally tempered intervals fall within their respective passbands as defined above in Section 13.
In order that the ratio of vibration frequencies in each semitone step be the same as that of every other semitone step, the coefficient derived from the ratio must be fixed such that when multiplied by itself twelve times the coefficient of an octave interval is produced. Since the coefficient of the octave interval is two, that of each semitone step must be equal to the twelfth root of two, approximately 1.05946. Thus the frequency ratio of every semitone step is 1:1.05946.
The twelfth root of two is the same as the sixth root of the square root of two and since the square root of two is an irrational number (i.e. the decimal part is infinitely long and nonrecurring), so must be the twelfth root of two. Hence no equally tempered intervals except unison (1:1) and the octave (1:2) can be expressed as ratios of whole numbers.
The coefficients of the other intervals can be calculated accordingly. For example, the coefficient of a 5^{th} (pitch class 7) is equal to the twelfth root of two multiplied by itself seven times, which is say that the exponent of two which gives this coefficient is equal to seven twelfths ( ^{7}/_{12} ), or 0.58333. Now, two raised to the power of 0.58333 is equal to 1.49831 and it can be seen that this is not the coefficient of a perfect 5^{th}, which is 1.5 (i.e. ^{3}/_{2} ). However, the value of the equally tempered 5^{th} does fall within the passband for 5^{ths} and all the other equally tempered intervals also fall within their respective passbands (see Table 14).
15. Derivation of Just Intonation
The reason for defining the dodekaphonic intervals as vibration frequency ratios of small whole numbers is to maximize their consonance, as defined by von Helmholtz and described above in Section 12.
There is a definite rationale behind the selection of the particular ratios used; they are not simply any old ratios of small whole numbers which happen to fall within the passbands; the available choice of such ratios is rich and varied for each passband. A representative list of alternatives is shown in Table 15 below. The particular ratios which comprise just intonation are those which mutually maximize the quantity of correlative intervals which are exactly equal to corresponding members of the tonic set. The tonic, or definitive, set comprises those intervals from the root to the other notes of the scale. The correlative set comprises those between any two notes neither of which is the root.
1 Semitone 
2 Tone 
3 
4 
5 4^{th} 
6a 
6b 
7 5^{th} 
8 
9 
10 
11 

15:16  8:9  5:6  4:5  3:4  32:45  45:64  2:3  5:8  3:5  9:16  8:15 
16:17  9:10  16:19  19:24  20:27  17:24  12:17  27:40  12:19  19:32  5:9  17:32 
17:18  17:19  21:25  27:34  23:31  22:31  31:44  31:46  17:27  25:42  19:34  9:17 
18:19  25:28  27:32  35:44  37:50  27:38  19:27  25:37  22:35  16:27  14:25  19:36 
31:33  33:37  37:44  43:54  43:58  29:41  41:58  29:43  27:43  22:37  37:66  33:62 
All the ratios shown above have values which place them within their respective dodekaphonic passbands. The list is not comprehensive, but only a representative sample of the alternatives to the ratios of just intonation which occupy the top line. Ratios are paired off horizontally in this table such that the product of each pair is equal to two. For example, the product of the second ratio given for pitch class 2 (major step), 9:10, and the respective ratio for the complementary interval of pitch class 10 (minor 7^{th}), 5:9, is 1:2 or a perfect octave. Some of the ratios not used in just intonation, such as the complementary pair comprising 29:41 and 41:58 for the tritones, have values closer to the median or to equal temper. In the example given, the coefficients evaluate to 1.4138 and 1.4146 respectively, which are much closer to the square root of two than the respective ratios of just intonation. The reasons why 32:45 and 45:64, evaluating to 1.4063 and 1.4222 respectively, are used in just intonation are explained below.
* * *
The tonic set is subject to a further constraint in that the ratios of each complementary pair must have a product of exactly two. Thus, for instance, the ratios of the intervals of a tone and a minor 7^{th}, 8:9 and 9:16 respectively, multiply together to form a ratio of 1:2 (an octave). In consequence, the interval from pitch 0 to pitch 2 (a tone) is equal to the interval from pitch 10 to pitch 12. Similarly, the interval from pitch 0 to pitch 10 (a minor 7^{th}) is equal to that from pitch 2 to pitch 12. The tone and the minor 7^{th} form a complementary pair. Descending by a minor 7^{th} from the octave note gives that note which is a tone (or major step) above the root. Similarly, descending by a tone from the octave note gives that note which is a minor 7^{th} above the root.
Because of the above constraint, it is not enough that a ratio per se fall within a passband; its complement must also do so or else both are unacceptable.
There are two reasons why each pair of complementary intervals must multiply up to two. Firstly, as stated above, these intervals must stack to form a perfect octave. Secondly, the relationships must hold true no matter whether the vibration ratios refer to frequency or wavelength. Where a frequency is doubled the wavelength is halved. Therefore each extreme of diapason can be expressed as either a doubling or a halving. The root (bass pitch) is both half the frequency and twice the wavelength of the octave. Conversely, the octave is both twice the frequency and half the wavelength of the root. In order to express the intervals as vibration coefficients with values between one and two irrespective of the terms (frequency or wavelength) of the expression, then in terms of wavelength, the intervals are products of the octave wavelength rather than quotients of the root wavelength. Thus the product of frequency and wavelength ratios is two for every interval.
For example, a tone up from the root can be expressed as ^{9}/_{8} of the root frequency, or as ^{16}/_{9} of the octave wavelength. Both coefficients then have values between unity and two and their product is two ( ^{9}/_{8} × ^{16}/_{9} = 2). Conversely, a tone down from the octave (i.e. a minor 7^{th} above the root) is ^{9}/_{8} of the octave wavelength and ^{16}/_{9} of the root frequency. Therefore the frequency and wavelength products of the complementary ratios of the tone and the minor 7^{th} are also both equal to two.
Although only a single tritone ratio is shown in all except Table 15, there are, in consequence of the need for complementary pairing, two such intervals defined in just intonation. For the interval from the root (pitch 0) to pitch 6 defined thus far, 45:64 (the diminished 5^{th}), does not correspond to the interval from pitch 6 to the tonic (pitch 12) despite which both are of interval class 6. That interval, the secondary tritone, has a ratio of 32:45 (the augmented 4^{th}). It can be seen that the product of these two different tritones is then a perfect octave; 32:45 × 45:64 = (32 × 45):(45 × 64) = 1440:2880 = 1:2. This means that neither is truly a tritone such as exists within equal temper, where the tritone ratio is 1:√2, which when squared gives 1:2. For this reason, within just intonation, neither is in actual fact called a tritone. The ratio of 32:45 is referred to as an augmented 4^{th} and that of 45:64 as a diminished 5^{th} and their product is the octave. The duality of the justly intoned tritone is, however, a purely theoretical necessity. Justly intoned guitars, flutes and claviers cannot be set up so that both are playable. In practice only one can be used and that which minimizes the lowest common denominator of all the intervals, the diminished 5^{th}, is selected for this purpose. Hence the justifiable omission of the augmented 4^{th} from general consideration and elsewhere than Table 15, where its presence is necessary in order to show that the practical tritone (the diminished 5^{th}) fulfills the requirement that its complement also fall within the limits of the passband.
Among the correlative intervals it will be found that, for instance, that between the notes of the perfect 4^{th} and the perfect 5^{th} is exactly equal to a tone (8:9). The population of such correlative intervals is maximized by the particular ratios which comprise just intonation (see Table 16 below).
The process of selecting a definitive tonic set which maximizes these equalities is, like many others involving whole (or ‘natural’) numbers, not susceptible to algebraic computation. The desired result can be achieved only by empirical means.
When both tritones are employed (the theoretical ideal) the 44 correlative intervals are drawn from a total of 66 derivative intervals. In other words, the maximum population of correlative intervals is exactly twothirds (⅔) of the total number of derivative intervals.
Table 16. Correlative Intervals of Just Intonation
A correlative interval is a derivative interval which exactly equals an interval of the definitive tonic set (shown at right). Such intervals are marked ‘+’ at the corresponding intersections. Excluding the tonic set and all unisons, there are 37 such intervals. A change to any member of the tonic set will reduce this total, which is the maximum possible. Inclusion of the augmented 4^{th} extends the count by 7, to 44. Derivative intervals are evaluated by dividing the upper definitive interval by the lower. For example, to evaluate the derivative interval between pitch 4 and pitch 7 (relative interval class, 7  4 = 3) the definitive interval of pitch 7 (2:3) is divided by that of pitch 4 (4:5) in the following manner:
(2:3) ÷ (4:5) = (2:3) × (5:4) = (2 × 5):(3 × 4) = 10:12 = 5:6
The quotient, 5:6, is exactly equal to definitive interval 3, the minor 3^{rd}, thus making it correlative. The intersection of the horizontal row from pitch 4 and the vertical row from pitch 4 is marked ‘+’ accordingly. As a further example, definitive interval 10 (9:16) is divided by definitive interval 7 (2:3), also a derived interval of class 3 (10  7 = 3):
(9:16) ÷ (2:3) = (9:16) × (3:2) = (9 × 3):(16 × 2) = 27:32
This value does not correspond to any definitive interval and so is marked accordingly, as an irrelative (i.e. noncorrelative) interval.
An interactive program version of Table 16 is available.
16. Derivation of The Degree of Arc
Having established the rationale behind the particular vibration frequency ratios of just intonation, a simple arithmetic transformation is performed upon them to ascertain their lowest common denominator. The ratios are expressed firstly as fractions. Thus, for instance, the tone ratio, 8:9, becomes the fraction ^{9}/_{8}, or 1^{1}/_{8}. The denominator of this fraction is eight. Similarly, the ratio of the minor 3^{rd}, 5:6, becomes ^{6}/_{5} (1^{1}/_{5}), with a denominator of five. Hence the lowest common denominator of the tone and the minor 3^{rd} is the product of five and eight, or forty (5 × 8 = 40). Expressing both fractions in terms of this denominator gives ^{45}/_{40} for the tone and ^{48}/_{40} for the minor 3^{rd}. Extending this process to the full set of intervals, but excluding the augmented 4^{th}, gives a lowest common denominator of 360, as shown in Table 17 below. The process is demonstrated graphically in Figure 14. Hence this is the true derivation of the Babylonian unit of rotation, the degree of arc. Because of this, it also may be concluded that Pythagoras did not invent the system of just intonation but learned of it during his 12year sojourn in Babylon (see Appendix B).
Together, the inverse functions of frequency and wavelength give both the diminished 5^{th} and the augmented 4^{th} from the single ratio of 45:64. The former results from multiplication of the root frequency by this ratio and the latter by multiplication of the octave wavelength.
Classification  Vibration Ratio 
Numerator (N)  

Pitch  Tonal  Frequency ^{N}/_{360} 
Wavelength ^{N}/_{720}  
0  Unison  1:1  360  720  
1  Semitone  15:16  384  675  
2  Tone  8:9  405  640  
3  Minor 3^{rd}  5:6  432  600  
4  Major 3^{rd}  4:5  450  576  
5  4^{th}  3:4  480  540  
6  Tritone  Augmented 4^{th}  32:45  (506^{1}/_{4})  512 
Diminished 5^{th}  45:64  512  (506^{1}/_{4})  
7  5^{th}  2:3  540  480  
8  Minor 6^{th}  5:8  576  450  
9  Major 6^{th}  3:5  600  432  
10  Minor 7^{th}  9:16  640  405  
11  Major 7^{th}  8:15  675  384  
12  Octave  1:2  720  360 
Key to Table 17.  

1.  Classification  Pitch

2.  Classification  Tonal

3.  Vibration Ratio

4.  Numerator  Frequency

5.  Numerator  Wavelength

Figure 14. The Circle of Natural Intervals
Diapason results from a doubling of either the root frequency or the octave wavelength. There must, then, be a nonzero value to be doubled. In Figure 14, the root frequency (or conversely, the octave wavelength) is therefore represented by a unit circle. Diapason is achieved by doubling the rotation, giving two superimposed circles. Between root and octave are distributed the practical intervals of the dodekaphonic scale. Hence, in terms of frequency, the tone is represented by a rotation equal to nine eighths ( ^{9}/_{8} ) of the unit circle. In terms of wavelength, ^{9}/_{8} brings about a descent from the octave to the minor 7^{th}. Thus the pitch numbers, outermost in the figure, represent both an ascending scale from the root in terms of frequency or a descending scale from the octave in terms of wavelength. The remaining intervals follow in similar order. To express all the angles separating the twelve radii as whole numbers, the circle must be divided into 360 equal sectors.
17. Derivation of The Minute of Time
Applying the same process as described in the preceding section but including the augmented 4th gives a lowest common denominator of 1440. This eliminates the bracketed figures shown among the numerators of Table 17, which contain quarters, and is the number of minutes in each and every day. Interestingly, the Vedic day has inverted the number of hours and minutes per hour and comprises 60 hours of 24 minutes each. The number of minutes per day is, however, unchanged.
Further evidence of the nonarbitrary nature of the minute of time is presented in Chapters 3 and 4.
18. Corollaries and Curiosities
A Practical Application
The Circle of Natural Intervals has a practical application as a fret position markout wheel in the manufacture of pythagorean fretted musical instruments. If each radius is given a point the wheel can then be rolled lengthwise along the unfretted fingerboard piece leaving a mark at each fret position, as illustrated in Figure 15 below, representing a considerable saving in labor and a matching improvement in repeatable precision. Take for example an instrument with 12 frets. The twelfth fret, being that of the octave note, lies halfway along the strings leaving half the length of the strings unfretted. Although this may seem a trivial point, the unfretted half is represented by the first, undivided, revolution implied in Figure 14 and the fretted half by the second, divided, revolution. The wheel must be rolled such that the closest fret spacing is toward the center of each string. The fret numbers shown in Figure 15 below run in the opposite direction to the interval class numbers shown on the fret markout wheel because the wheel displays relationships in terms of vibration frequencies, whereas fret positions reflect those of the wavelengths. Vibration frequency and wavelength have a reciprocal relationship and, since all the intervals are complementary by definition, a wavelength wheel is the mirror image of what would be a frequency wheel if such a name had any meaning (although one may “mark time”, one does so only in an abstract sense).
Figure 15. Marking Out Fret Positions with A Natural Interval Wheel
* * *
Natural Harmony
As shown below in Figure 16, three vertices of each of the first three regular polygons align with radii of the Circle of Natural Intervals. Each group of three coincident pitches forms a simple major triad chord.
Figure 16. Natural Intervals and Regular Polygons
Key to Figure 16.  

A.  Triangle 
B.  Square 
C.  Pentagon 
The coincident pitches of triangle and square fall in the same natural major scale, with pitch 0 being the keynote. Those of the triangle give the IV triad and those of the square give the I triad. In this scheme of things the coincident pitches of the pentagon are, however, in a key very distant from either of the other two.
* * *
Natural Melody
Listen to ODEION Natural No.1
This piece of music was generated in less than a millisecond, as a MIDI datastream, entirely by a computer program* and is constrained to melodic steps (inclusive of all immediate, alternate and interlinear relations) comprising only definitive and correlative (i.e. not irrelative) intervals, as shown in Table 16.
Music Copyright © Peter Wakefield Sault 2000. All rights reserved worldwide. Commercial use in part or in whole expressly forbidden. All other use freely licensed.
* ODEION.EXE (V0.60, ©1996) for IBM PC under DOS 6.22 with Roland MPU401 MIDI interface. Program invented, designed and coded by Peter Wakefield Sault.
References and Bibliography
1.  JeanPhillipe Rameau, Treatise on Harmony, 1722.  
2.  Johann Sebastian Bach, The WellTempered Clavier, 1722.  
3.  Guiseppe Tartini, Trattato di Musica, 1754.  
4.  Hermann Helmholtz, On the Sensations of Tone, 2nd Ed., 1885. (1st Ed., 1862.)  
5.  Arnold Schoenberg, Theory of Harmony, 1911.  
6.  Sir James Jeans, Science & Music, 1937.  
7.  Percy A. Scholes, Ed. John Owen Ward, The Oxford Companion to Music, 10th Ed., 19381970.  
8.  Arthur H. Benade, Fundamentals of Musical Acoustics, 1976.  
9.  Wayne Bateman, Introduction to Computer Music, 1980. 
Copyright © Peter Wakefield Sault 19732016
All rights reserved worldwide