# Talk:Octave

**Unification Aspects:**

1. Octaves are related to prime number reciprocals

According to Adri de Groot (*Number Theory in Light of Unification Thought,* 2005), the phenomenon of octaves is related to the nature of reciprocals in (prime) numbers. The reciprocal of 81 (although not prime) provides a good example: 0.012345679012345679…, a forever repeating string of numbers, whose actual numerical value “grows” as 0.0-1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16-17-18-19-etc. (hyphens introduced to clarify that single digit space writing of 12345679… is a hidden form of the actual ever increasing numbers following the n+1 formula). Thus, on the one hand, the numbers become larger and larger, as seen in the hyphenated values, but also, in its written form, repeat themselves in different locations of the reciprocal and endless quantity of times. A musical scale, from the lowest possible to the highest possible audible note displays the same traits—the notes in one sense just become higher and higher, but in another sense they also “repeat,” even though the audible self-similarity occurs in a different location. This phenomena has led De Groot to name the repeats in prime number reciprocals noctaves (by joining “n” of “numbers” with “octaves”). Thus, octaves and numerical reciprocals display a dual structure—they seemingly repeat, which is at once real and an illusion, and they simultaneously decrease/increase forever in value or pitch (frequency).

Musical frequencies above and below the range of human or animal perception should perhaps be dismissed for possessing no practical value, just as enormously small portions of numerical values, let’s say after the 100th digit beyond the decimal point also do not possess much practical value.

Unification Thought takes the view that all of creation is patterned in some fashion after the model of humanity, which is the central model of the universe, and which is based on the Four Position Base and on the perpetuity of mankind through this foundation. As children can never be in the same location or position as their parents, neither can octaves occupy the same position or frequency as another octave.

2. One of the means of the restoration of mankind after the "fall" is the seeking of harmony between men and women. This concept can be analogized by using the octave which provides the options for a harmonious beginning and ending between dissonances and other consonances. The octave plays the role of being a tonal singularity for peaceful harmonious resolutions and can be likened to a tool in seeking harmony between men and women. Perhaps this is why the interval of an octave was and is still so important to both western and non-western music.teachers and researchers who wish to further pursue these topics from a unification perspective.