# pair

James Clerk Maxwell
"9 coefficients determine the relationship between flux and intensity
6 of these coefficients form 3 pairs of equal quantities
3 pairs of equal coefficients will self-conjugate." [James Clerk Maxwell]

The original quote from "https://svpwiki.com/pdffiles/A_Treatise_on_Electricity_and_Magnetism.pdf - vol. I".

"The case in which the components of the flux are linear functions of those of the force is discussed in the chapter on the Equations of Conduction, Art. 296. There are in general nine coefficients which determine the relation between the force and the flux. In certain cases we have reason to believe that six of these coefficients form three pairs of equal quantities. In such cases the relation between the line of direction of the force and the normal plane of the flux is of the same kind as that between a diameter of an ellipsoid and its conjugate diametral plane. In Quaternion language, the one vector is said to be a linear and vector function of the other, and when there are three pairs of equal coefficients the function is said to be self-conjugate." [James Maxwell, A Treatise on Electricity and Magnetism - vol. I]

Hughes
densed into a pair; as an example, we trace the notes and colours in the fundamental scale of C. [Harmonies of Tones and Colours, On Colours as Developed by the same Laws as Musical Harmonies3, page 20]

Notes and colours are thus condensed into a pair springing from the fountain, and mingling with each other in an endless variety. Although yellow as a colour is explained away as white, it is, nevertheless, the colour yellow in endless tints and shades throughout nature, and proves to us that the three great apparent primaries correspond to the tonic chord of the scale of Ci.e., C, E, G = red, yellow, blue; or more correctly, C and G correspond to red and blue with the central fountain of E, white and black mingled, from which all tones and colours arise. [Harmonies of Tones and Colours, On Colours as Developed by the same Laws as Musical Harmonies3, page 20]

The three lowest of the six tones are complementary pairs with the key-note and its two highest tones. Observe the curious order in which the tones sound, avoiding consecutive fifths. First, we have the key-note and its root, or fellow; next A; then D and its root; and then E, whose root, A, has already sounded between the first and the second pair. B, the fourth and central tone in depth, sounds seventh, and, finding no fellow within the compass of the harmony developing it, is isolated. Observe also how closely a key-note and its kindred tones are linked into each other. The Primaries spring from the key-notes, the Secondaries from the Primaries; the first pair comprises a key-note and a tone of the Primaries, the other two pairs have each a tone of the Primaries and a tone of the Secondaries. The key-note, after giving out its tones in trinities, or [Harmonies of Tones and Colours, Diagram II - The Twelve Keynotes1, page 23]

The Major Key-note of C is here shewn developing its trinities from within itself, veering round; C and the other 11 developing their trinities in musical clef. Below each is the order in which the pairs meet, avoiding consecutive fifths. Lastly, C# is seen to be an imperfect major harmony; and G?, with B as C?, make the same harmony as F#. The intermediate tones of sharps and flats of the 7 white notes are here coloured in order to shew each harmony, but it must be remembered that they should, strictly, have intermediate tints. [Harmonies of Tones and Colours, The Major Keynote of C, page 24c]

The Sevens of the Key-notes and their scales, the latter written also as they pair by fifths. [Harmonies of Tones and Colours, The Sevens of the Keynotes, page 25e]

AS an example of the twenty-four, compare A major, developing, in Diagram II., with A minor, Diagram IX., taking the notes in the order which they sound in trinities. The three notes of the primaries sounded by A minor are, first, the same root as the major; the two next are the fourth and seventh higher notes (in the major, the fifth and sixth); the secondaries only vary by the sixth and seventh notes being a tone lower than in their relative major. Observe the order in which the pairs unite; the fourth in depth, sounded seventh, isolated. A and its root do not rise from the chasms. The fundamental key-note C was seen not to be interfered with, neither is the fundamental minor key-note A; G# on the one side, and B? on the other, being the key-notes. The seven of each minor harmony embrace only seventeen tones. C major and A minor are the only two keys which sound the seven white notes of keyed instruments. The minor scale and chords of A are not included in this remark. [Harmonies of Tones and Colours, Diagram IX - The Minor Keynote A and Its Six Notes, page 34a]

The diagram represents the Minor Key-note A and its 6 notes veering round in trinities; A and the other 11 developing their trinities in musical clef. Below each is the order in which the pairs unite, avoiding consecutive fifths, Lastly, D? is shewn to be an imperfect minor harmony, and by employing B as C?, E? is seen to be the same harmony as D#. As before, it should be remembered that the sharp and flat notes should, strictly, have intermediate tints. [Harmonies of Tones and Colours, The Diagram Represents the Minor Keynote, page 34c]

The seven of each harmony, with its scale. Sharps or flats, which vary in the scales from the harmonies, are written to each note, and only govern that one note. The scales are written as they pair. [Harmonies of Tones and Colours, The Seven of Each Harmony, page 35e]

If we examine the line last quoted by the laws of life which regulate the foregoing scheme, we may compare it with the fundamental threefold chord of the scale of C and its relative colours,

 C E G C red rises RED. BLUE.

from the fountain key-note which contains in itself all tones. "Him first," the Son of God proceeding from the Almighty, and yet in Himself the Trinity in Unity. E, yellow or light. E is the root of B, ultra indigo, or black. "Him midst," the Almighty Father, the Fountain of life, light gradually rising and dispelling darkness. G, blue, "Him last," the Holy Spirit, proceeding from the Father and the Son, Trinity in Unity. The Son of God and the Holy Spirit are the complemental working pair throughout the universe; each containing "the seven spirits of life." Red and blue contain all colours in each. C and G are a complemental pair, C rising from the fountain key-note which contains in itself all tones, and C and G combine all tones in each. In Chapter III. it is explained that all varieties of tones and colours may be condensed into this pair, rising from and falling again into the fountain. [Harmonies of Tones and Colours, Reflections on the Scheme2, page 44]

"I esteem myself fortunate in being introduced to you, and becoming acquainted with your beautiful work on 'Tones and Colours.' I have, to the best of my ability, worked out your idea, by writing down in music the various discords in use amongst musicians, and resolving them according to the laws of Harmony, and I find in all cases the perfect triad agrees with what you term the trinities in colours. The way in which you find the whole circle of Major and Minor keys by pairs in colours is deeply interesting, and must be true. The only point of divergence between your system and that recognised by all musicians is the ascending Minor Scale. No musically trained ear can tolerate the seventh note being a whole tone from the eighth. The Minor second in the lower octave descending is very beautiful, and it is strange how all composers feel a desire to use it. To mention one case out of hundreds, I may cite Rossini's well-known air, 'La Danza.'
"Yours faithfully,
"W. CHALMERS MASTERS." [Harmonies of Tones and Colours, Supplementary Remarks and Diagrams, page 53]

DIAGRAM XVI.—The seven of each harmony, with its scale—The pairs of the trinities and scales. Harmonies of Tones and Colours, Additional Diagrams, page 57]

The seven of each Harmony, with its Scale. The pairs of the Trinities and Scales. [Harmonies of Tones and Colours, The Seven of each Harmony with its Scale, page 59]

In the Minor Scale, the Trinities and Scale develope five pairs; the last pair become the fifth higher key-note and its root, consequently the sixth pair would develope the higher key.[Harmonies of Tones and Colours, The Seven of each Harmony with its Scale, page 59]