from The Myth of Invariance by Ernest G McClain

page 1: Agni, savior and god of fire, is himself "the holy Singer who precedes the sages . . . waxen mighty by laudations" and the secret he alone can impart is a certain "lofty Hymn". (3.5.1-2 and 4.5.3)

page 6: [regarding tenfoldness] ... and ten is the number of piles of wood for the fire in which Agni is born as well as the number of his secret "dwelling places."

page 37: [regarding tone mandalas] But it is equally legitimate to consider D = 360 = 720 at the top of the circle as "1" also, in the sense of the geometric mean whose "appearance" has been changed to avoid fractions. Now a-flat lies somewhat askew from the symmetrical [Asvin] twins we have been studying; it is near the locus of the square root of 2 which would have put the equal tempered A-flat = G-sharp precisely at the bottom of the circle, directly opposite D. It is in this region of the of the circle, I believe, that Indra will be born later as Indra-Agni, god of fire, not as a "point" in the circle but as a segment of the circle. The Rg Veda describes this birth as that of a kind of "breach baby":  "Forth from the side obliquely will I issue" (R.V. 4.18.12). There is no integer at the locus in question. The great expansion of the number sets in later diagrams is motivated, I believe, by the effort to approximate as exactly as possible the irrational square root of 2 which is needed to locate a tone symmetrically opposite the mean on D, that is, precisely in the middle of our octave.

page 40: [regarding Vishnu's three steps]

Among the skillful Gods most skilled is he,
who made the two world-halves which
bring prosperity to all;
Who with great wisdom measured both the
regions out, and established them
with pillars that shall ne'er decay.


Alain Daniélou wrote Music and the Power of Sound, The Influence of Tuning and Interval on Consciousness and a footnote in this is of interest regarding the role of the tritone in India. The trigger was a request to review a new book about Harmonic Geometry by John Oscar Lieben, which quotes a footnote in Daniélou, page 156:

51. The use in Indian music of the augmented fourth, or tritone, at the criti­cal times of midnight and midday reminds us of the magical importance attached to those hours, and of the use of the tritone (diabolus in musica) by Western musicians for the representation of magic, which is nothing other than the possible intersection, at certain critical hours, of worlds that can­not normally communicate. It is used conspicuously in this way by Schumann for the character of Manfred, by Wagner every time a magi­cian appears, by Berlioz in the Symphonie fantastique, by Weber in Der Freischutz, and so on. In Chinese music the Iü rui bin, corresponding to the augmented fourth, represents the summer solstice, the critical mo­ment in the annual cycle when the masculine influx, hot and creative, gives place to the feminine influx, cold and destructive.

The tritone plays an important place in the work of Ernest G McClain's classic work The Myth of Invariance, informed by his contact with Antonio de Nicolas who studied the Indian tradition as habing been concerned with harmonic audition rather than extensive vision. Agni, the Indian god of fire, plays the role of tritone within harmony by enabling creativity as a liberating counter to cosmic and human egoity. I wrote this summary document; of what Ernest McClain wrote about Agni. With this in mind one can return to Daniélou's main text:

The Periods of the Day

In The Myth of Invariance, page 7, Ernest McClain wrote:

Throughout my study I shall focus on invariances, that is, on patterns which remain the same in different contexts. This theme of invariance was expressly formulated by Marius Schneider :

In view of the inconstancy of the world of form, primitive man questions the reality of static (spatial) phenomena and believes that transient (temporal) dynamic rhythms are a better guide to the substance of things.15

Schneider affirms that "sound represents the original substance of the world" for the historian of culture, and points out that the Indian tradition emphasizes the "luminous nature of sound" in the similarity between svar (light) and svara (sound).16 My mathematical presentation can be regarded as a very specialized development of Schneider's general view of the role of music in the spiritual history of man.

15 Marius Schneider, "Primitive Music," The New Oxford History of Music, vol. 1, (London : Oxford University Press, 1957) p. 43 [out of print].

16 Ibid., pp. 45-49. Schneider's essay anticipates many of the themes in my book. Especially significant is his appreciation of the male-female sexual imagery involved with the instruments and their playing, "bisexuality" proving to have an interesting arithmetical meaning. Schneider also emphasizes the "friction" necessary to sound a tone, and points to the sound-box as "a kind of sacrificial cavern" (pp. 46-51), ideas which will prove important in later chapters. I deeply regret that the magnum opus which he now has in progress, summarizing his life's work in musicology and ethnomusicology, is not yet available, for it is obvious that Schneider and I are concerned with the same topics, although from quite opposite perspectives which must eventually be coalesced.

As time permits, further extracts from this essay by Marius Schneider (see note 15 above), runs on below. Pages, separated by lines and numbered, may break differently to the original publication.