Ratios

73.1 Ratios, Latin / Greek

(2 : 1) Dupla	Diapson
(3 : 1) Tripla	Diapson Diapente
(4 : 1) Quadrupla	Bis Diapson
(3 : 2) Sesquialtera	Diapente
(4 : 3) Sesquitertia	Diatessaron
(5 : 4) Sesquiquarta	Diatonus Semitonus
(8 : 3) Duplasuperbipartiens	Diapson Diatesseron
(9 : 8) Sesquioctava	Tonus

Sesquialter = 1 tone + 1/2.
Sexquaternarius = 1 tone + 1/4.
Sexquiquarta = 1 tone + 1/3.
Sexquiterterius = tone + 1/3.
Sexquartano = increase by 1/4.
Sexquinterno = increase by 1/5.

Latin Terms.......................Greek Terms

.	2 Di
Terti, tern	3 Tri
Quadri, quart, quin	4 Tetra
Quinqu, quint, quin	5 Penta
Sex, sext	6 Hexa
Sept	7 Hepta
Octo, octave	8 Octa.
Deci, decim	9 Nona.
Duodeci, douden	10 Deca.

73.2 Diagram. Music of the Spheres Pythagorean Sourcebook and Library

"Starting from the bottom, the diagram includes the four elements in spatial arrangement: terra, aqua, aer, ignis. Then rising in order are the spheres of the seven planets, and the sphere of fixed stars at the top, making a total of eight spheres in order to accommodate the eight musical notes of the diapson. The planets are labeled in the right-hand margin by both names and astronomical symbols, and are also indicated mythologically by the circular vignettes of the appropriate god or goddess. The interval between planets are marked "tone" or "half-tone" in accordance with the statement of Pliny (Historia naturalis, II.xx; quoted p. 179). A musical mode for each planet is also indicated; for example, Mars plays in the Phrygian mode, Jupiter in the Lydian, and Saturn in the Mixolydian... Each planet is also assigned a musical note, marked to the left of the three-headed dragon (which, though it doesn't have its tail in its mouth, symbolizes Time according to a passage from Macrobius' Saturnalia). Each celestial sphere is further identified with one of the Muses, depicted in the circular vignettes on the far left. To provide the necessary number of nine, earth is identified with Thalia at the bottom. Reigning over all, in the appropriate position of deity, is Apollo, attended by the three Graces and advertised by a banner which proclaims, "The power of the Apollonian mind completely controls these Muses". The intention is clear: each Muse, each note, each planet, though playing an individual part, contributes concordantly to a larger whole, represented in the single figure of Apollo".

Franchino Gafori,
Practica Musice .

73.3 Diagram. The Music of the Spheres

73.4 Diagram. "De musica"

"The diapson is derived mathematically from the ratios between the first four integers: 1, 2, 3, 4... The ratio 12 : 6 is equivalent to 2 : 1, the whole diapson, with a double (dupla) proportion. The ratios 12 : 9 and 8 : 6 are equivalent to 4 : 3, the diatesseron, with a 1 1/3 (sesquitertial) proportion. The ratios 12 : 8 and 9 : 6 are equivalent to 3 : 2, the diapente, with a 1 1/2 (epitrital) proportion. The ratio 9 : 8, the susquioctaval proportion, determines the unit known as the "tone", two of which are then inserted between 12 and 9 and two more between 8 and 6, thereby making eight notes, an octave".

73.5 Diagram. Ratios

73.6 Diagram. The Harmonic System of 15 chords in the Diatonic Mode

"Morley provides an ample explanation of his diagram: "There be three things to be considered: the names, the numbers, and the distances. As for the names, you must note that they be all Nounes, adjectives, the substantive of which is chorda , or a string (i.e., the lowest note is properly proslambanomene chorda ) ... The numbers set on the left side declare the habitude (which we call proportion) of one sound to another, as for example, the number set at the lowest note Proslambanomene is sesquioctave, to that which is set before the next: and sesquitertia to that which is set at Lychanos hypaton,and so by consideration of these numbers, may be gathered the distance of the sound of the one from the other: as sesquioctave produceth one whole note. Then betwixt Proslambanomeme, an hypate hypaton, is the distance of one whole note. Likewise sesquitertia, produceth forth: therefore Proslambanomene and Lychanos hypaton are a forth, and so the others".

Thomas Morley, A plaine and easie introduction to practicall musicke (London, 1597).

Moreover, the diapson, which represents the 2 : 1 proportion, can be repeated an infinite number of times along the open ended continuum of sound- the proportions 3 : 1, 4 : 1, 5 : 1 and so forth ad infinitum are possible- so music provides a convenient way of relating the finite to the infinite, or better yet, of knowing the infinite through the finite.

Touches of Sweet Harmony

73.7 Diagram. The Divine Monochord The Cosmographical Glass by S.K. Heninger, 520.94 H3886c

Fludd gives this diagram this caption: "We set forth here quite precisely the monochord of the universe with its proportions, consonances, and intervals; as we show that its motive force is extra-mundane (i.e. a celestial hand)". The essential feature is a monochord stretching from the lowest to the highest in the universe,with the hand of God reaching from a cloud to tune it. There are fourteen intervals on the monochord which produce fifteen musical notes, corresponding to the "harmonic system of 15 chords" which, for example, Thomas Morley had described... From the bottom there are the first four elements (earth, water, air and fire), then the seven planets (Moon, Mercury, Venus , Sun, Mars, Jupiter, Saturn), and the sphere of the fixed stars, and finally the three angelic hierarchies...

The "material" diapson stretches from the earth to the sphere of the Sun, the "formal" (i.e. conceptual) diapson from the sphere of the Sun to the summit of the empyrean. These two taken together form a double diapson. Within each of these , the forth (diatesseron) and the fifth (diapente) are indicated. The "material" forth stretches from the bottom through the four elements. The "material" fifth stretches from the sphere of fire to the sphere of the Sun. The "formal" forth stretches from the sphere of the Sun to the sphere of fixed stars, including the planets Mars, Jupiter, and Saturn. The "formal" fifth stretches from the sphere of fixed stars through the empyrean. The appropriate musical notes are indicated by letters beside the monochord itself, and the intervals are marked as a full tone (tonus) or a half-tone (semitonus).

On the left are labeled the mathematical proportions_ i.e., the sesquitertial proportion for the forth, the sesquialteral proportion for the fifth, the double proportion for the diapson, the triple proportion for the diapson plus a fifth, and the quadruple proportion for the double diapson".

Robert Fludd, Utriusque cosmi majoris scilicet et minoris metaphysicia... 4 vols. (Oppenheim, 1617-19).

"Universal harmony as a musical paradigm of all creation is graphically depicted by Robert Fludd, which illustrates his Utriusque cosmi majoris scilicet et minoris metaphysica, physica atque technica historia . The categories of nature are arranged vertically on a monochord in rough correspondence to their physical stratification as perceived by our senses: the 4 elements at the bottom, then the 8 heavenly spheres, and finally the 3 angelic hierarchies which comprise the empyrean. The diagram encompasses 15 notes- 2 complete",

(Note: Notice the double or binary nature of nature here: 10 decimal becomes 20 Mayan vegesimal and 4 becomes 8),

..."1 "material"(seen) and the other "formal"(unseen). The sun sits appropriately in the middle (mediator between god and man), marking the highest note of the material diapson, which stretches upward from the lowest note played by earth. The sun also marks the beginning of the formal diapason, which stretches upward to the highest note of the monochord played by the seat of the Epiphanies. The implication, of course, is that both the "formal" and the "material" diapasons are tuned by the same harmonies".

73.8 Diagram. Misica Humana The Cosmographical Glass by S.K. Heninger, 520.94 H3886c

"The harmonies of microcosmic man are set forth as 3 diapasons. There is a "material" diapason comprising the 3 elements above earth, a "middle" diapason comprising the 9 heavenly spheres, and a 'spiritual" diapason comprising the 9 angelic hierarchies. At the side a label informs us: "Three times the diapason marking the three-fold division of the human soul". Another label proclaims: "The essential harmony by which the human soul takes for its own arrangement the division of any cosmos, just so it has three parts". [The numerical building blocks in this structure are 3 and 9, and the effective harmony is the ratio 9/3. Fancifully, the diapason is determined by the proportion 3/1; the diapente is marked by the proportion 2/1 and the diatesseron as 3/2.]"

(Note: The bracketed [ ] text is noted to be wrong proportion).

"Decreasing degrees of spirituality are indicated by letters as the soul descends from the deity at the top to the human at the bottom, and a table in the lower right identifies each step:

A. Pure mind; the spirit of God.
B. The intellect setting in motion the topmost portion or primum mobile of the mind.
C. Mind and intellect in the rational spirit, which allows reason or intellect.
D. The rational spirit, with mind and intellect in the middle soul.
E. The middle soul swimming in ethereal fluid; or the vital light within the mind.
F. The body, which is the receptacle of all things".

Robert Fludd, Utriusque cosmi majoris scilicet et majoris metaphysica, physica atque technica
historia, 4 vols. (Oppenheim 1617-19)

73.9 Nichomachus

"The interval, then which is between the fifth and the forth, that is, the interval by which the fifth is greater than the forth, was confirmed to be a sesquioctaval ratio, which is as 9 is to 8. And either way it is proven that the octave is a system of a fifth and a forth in conjunction. Just as the double ratio (2 : 1) consists of the sesquialter (3 : 2) and the sesquitertian
(4 : 3), as for example, 12, 8 and 6; or conversely, it (sc.octave) consists of the forth and the fifth, just as the double ratio consists of the sesquitertian and the sesquialter, as for example, 12, 9 and 6 in such order.

Then Euclid proves mathematically the impossibility of such a resolution as this Nichomachean reasoning in Sectio Canonis, Theorem 9, (Jan 157. 5-158.7). he begins with the postulate: "Six sesquioctaval intervals are greater than one interval in the double proportion".

73.9.1

"In order to set up this progression in the most economical way, Euclid first determined the smallest rational number in the six susquioctave intervals (or 7 terms) so that the entire series would have 8 as a common factor",

73.10.1 (Correspondences, Fuller, Synergetics 905.49).

"Accordingly, 8 is squared and the resultant term 64",

(Correspondences, Fuller, Synergetics, 1106.12).

"is multiplied by 8, five more times yielding 262144 (2^18). To this term one-eighth of its total value is added, the process being reapplied to each successive result, producing thereby a series of rational number in the relationship 8 : 9. Thus, it is proven by Euclid that an octave (2 : 1) cannot consist of six whole-tones-- as Nichomachus would have it-- but of some fractional interval less than its total range".....

73.11

73.12 Nichomachus Rendering of the Text:

..."so that within each interval there were two means, the one exceeding and being exceeded by the extremes of the same fraction, the other exceeding and being exceeded by the extremes by the same number. And the Demiurge(God) filled up the distance between the hemiolic and epitritic intervals with the left over interval of the epogodic ratio".

hemiolic = sesqualter (3 : 2)
epitritic = sesquitertian (4 : 3)
epogodic = sesquioctave (9 : 8)

73.13 Pythagoras, Whole number

"Harmonic sounds, said the Pythagoreans, are produced by ratios expressible as whole numbers, and the simpler the ratios (the smaller the whole numbers expressing them) the more consonant the sound".

The Vibrating String of the Pythagoreans.
Scientific American
Dec. 1967, pg. 93.

73.14 Pythagoras and Numerical Cosmology

"There is no necessary connection between the discovery of the musical ratios and the knowledge of the nature of sound, of vibration or wave movement of the air".

..."Pythagorean musical theory tried to derive as much as possible from a priori considerations, and to refer as seldom as possible to experience and experiment. Even the basic facts are - apparently - derived from speculation, and everything else is derived from calculation of ratios. The basic principle, at least from the time of the Sectio Canonis and the Aristotelian Problemata, was that musical intervals are expressed in the form of "superparticular" or "multiple" proportions.

The reasons for the position of the superparticular proportions are not immediately obvious. It is based partly on the fact that all these proportions were designated in Greek by a single word, but at the same time every (omitted Greek word) represents the connection of an odd and an even number, and thus exemplifies the harmony of Limit and Unlimited. So Pythagorean musical theory is intimately related to numerical cosmology, and the importance of superparticular proportion comes from its relation to number speculation in general".

Lore and Science in Ancient Pythagoreanism
By Walter Berkert
182.2 B9174 l(el)

73.15 Pythagorean Scale Ratio

..."In order to arrive at whole number solutions, we will use the octave of 6 : 12.

The first step is one of arithmetic mediation. To find the arithmetic mean we take the 2 extremes, add them together, and divide by 2. The result is a vibration of 9, which, in relation to 6, is in the ratio 2 : 3. This is the perfect 5th, the most powerful musical relationship.
The second form of mediation is harmonic. It is arrived at by multiplying together the 2 extremes, doubling the sum, and dividing that result by the sum of the 2 extremes (i.e., 2AB / A + B). The harmonic mean linking together 6 and 12 then is 8. The proportion, 6 : 8 or 3 : 4, is the perfect 4th, which is actually the inverse of the perfect 5th.
Through only 2 operations we have arrived at the foundation of the musical scale, the so called "musical" or "harmonic" proportion, 6 : 8 :: 9 : 12, the discovery of which was attributed to Pythagoras".

73.16 Diagram. The Harmonic Nodal Points and Overtone Series on the Monochord Pythagorean Sourcebook and Library

73.17 Diagram. The Harmonic Proportion

73.18

"This arrangement of the perfect consonances of the octave, 5th and 4th needs to be played out, preferably on the monochord, in order to fully appreciate its significance. While we have not completed a complete scale, we have arrived at the architectural foundation on which it is based. By carefully observing the above arrangement, however, we shall discover enough information to complete the scale.

First of all, it would be well to notice the particular form of musical and mathematical "dialectic" which is occurring. That is to say, not only is 6 : 9 a perfect 5th, but 8 : 12 is as well; i.e., 6 : 9 :: 8 : 12. Nor is that all, for while 6 : 8 is a 4th, so too is 9 : 12; or, 6 : 8 :: 9 : 12. Again, the significance of this harmonic symmetry will be fully realized by playing these relations out. However, not only are the 4th and 5th manifested in these multiple ways, but the ratio 8 : 9 defines the whole tone as well.

The tone having been defined, the final creation of the scale is quite simple. The vibration of the tonic C is increased by the ratio 8 : 9 to arrive at D. D is increased by 8 : 9 to arrive at E. Now, if E were increased by that ratio, it would overshoot F; hence we must stop. The ratio between E and F ends up being 243 : 256, called the Greek leimma, or "left over," corresponding to our semi-tone".

Pythagorean Sourcebook and Library
B243. G88

73.19 Iamblichus "The most Perfect Proportion" Musical (Harmonic) Proportion

"It consists of 4 terms so combined that the 2 middle terms are the arithmetic and harmonic means between the extremes:

73.20.1

A : A + B / 2 = 2AB / A + B : B
^.............^
Whole formula reversed

73.21 Pythagoras

A. The 5th and 4th = The octave.

B. Whole tone= 9 : 8; the difference between a 4th and a 5th: 3 : 2 / 4 : 3 = 9 : 8.

C. 6 tones of 9 : 8, when added, equals slightly more than 2 : 1 (octave).

D. The Semi-tone is not really half a tone. The diatonic tetrachord contains, in descending order, 2 whole tones and a fraction of a tone, by which convention is called a semi-tone, but in "natural" as opposed to "tempered" tuning, is rather less that a half a whole-tone.

E. Scale in Tabular Form:

6 intervals in 9 : 8 ratio = 5 x (8^3)^2 = 262144. If each descending interval equals a whole-tone, the series = 262144, 294912, 331776, 373248, 419904, 472392, 531441 but:
262144 x 2 = 524288 which is less than 531441, therefore 6 whole-tones exceeds the octave.

73.22.1

F. If the proportion of the 4th (4 : 3),

73.23.1

is given in figures x 64 = ratio 256 : 192. If 192 is the mese or middle note taken as the base of the calculation, the 5th below it is a relative string length of 288 (192 x 3/2) = 288. The number of the intervals are in descending pitch: 192, 216, 243, 256, 288. 288/256 = 9 : 8 (32).

73.23.1

G. See Pythagorean Scale- 512 tone number = Last note in octave 2.

73.24 Diagram. Formation and Ratio of the Pythagorean Scale

73.24.1

73.25 Ratio

"For the method of conversion, and comparison of values, especially, of Aristoxenus and Archytas, see Tannery MSc Ill 98ff. Of course, equal division of the interval- "line" leads to ratios of vibration frequency that are almost exclusively irrational ("tempered tuning"), whereas frequency ratios using whole numbers lead to transcendental division of the interval- "line" ("natural tuning"), making impossible the establishment of a common unit and therefore precluding modulation. According to the Pythagorean music theory it is impossible to halve the octave, 5th, 4th or whole tone, and the tempered halftone simply does not exist."

Pythagorean Music Theory

Ratios, Old Scales

73.26 Diagram. Equal Temperament and Pythagorean Intervals Helmholtz

"The dissonances occasioned by the upper partial tones are consequently somewhat milder than those due to Pythagorean intervals, but the combinational tones are much more disagreeable.

But the combinational tones of the Equally Tempered thirds lie between those of the perfect and Pythagorean thirds, and are less than a semitone difference from those of just intonation. Hence, they correspond to no possible modulation, no tone of the chromatic scale, no dissonance that could possibly be introduced by the progression of the melody; they simply sound out of tune and wrong".

"In the Pythagorean intonation the combinational tones sound rather as if someone were intentionally playing dissonances":

73.26.0 Diagram. Differences of the Pitch Numbers of Combinational Tones

73.26.1

Impossible Correspondence Index