Projective tuning space
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.
"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
Note: to be clear, what is described in the above quote up to the point you place your eye at the origin is technically val space, not tuning space. Upon placing your eye at the origin, that step and everything afterward is the part where you perform the projection. Projecting val space and projecting tuning space give the same result, so the quote still describes what you see. But do not use it to infer what tuning space is. Tuning space is similar, but it is what you get when you plot actual tunings in space, expressed e.g. in cents, such as ⟨1200 1900 2800]. You could think of the difference this way: in val space, the val ⟨6 10 14] would be further away from the origin from ⟨3 5 7], hidden exactly behind it; however in tuning space both of these could be represented by ⟨1200 2000 2800] (as well as stretchings thereof, which would be hidden behind or directly in front of this, such that they project down to the same point, e.g. ⟨1200.12 2000.20 2800.28] or ⟨1199.88 1999.80 2799.72]).
"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
- Gallery of projective tuning space images
- Recreating an iconic map of tunings and temperaments, a YouTube video by Tony Durham about PTS (also part 2)