Projective tuning space

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Projective tuning space is the projectivization of ordinary tuning space. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.

Quotes

"Plot the number of steps of each ET's mapping of prime 2 on the x-axis, prime 3 on the y-axis, and prime 5 on the z-axis. Actually you divide the number of ET steps by the log of the prime to get the distance along the corresponding axis. Now put your eye at the origin and look along a line equidistant from all three axes. That's Projective Tuning Space." -- Paul Erlich

"You can think of this as a view of ETs, where the mapping of 2 is on one axis, the mapping of 3 on another axis, and the mapping of 5 on the third axis. The numerals only show the mapping of 2. We view or project so that "contorted" tunings (whose mappings are not in lowest terms) are hidden behind the lowest-terms versions. There's a mathematical duality relationship between this and projective Tenney ratio space (looking at the Tenney lattice so that powers of ratios are hidden behind the ratios to the first power -- your eye is at 1:1). These relationships are touched on in technical terms on Gene's pages but were discussed in much more detail on the tuning-math list. The important things to note here are that 2D temperament classes lie on straight lines (every comma corresponds to a straight line passing through tons of ETs in which it vanishes); the ETs on each line behave just like on the scale tree; and the concentric hexagons correspond to how damaged the intervals get compared with JI, with no damage in the center and pretty egregious damage on the outer hexagon." -- PE

pts-2-3-5-e2-twtop-tlin.jpg

Gallery of projective tuning space images