An equal-temperament which lies at the boundary between two temperament families.
--- In tuning@yahoogroups.com, "Carl Lumma"
> Why an et?
Because we need to draw a boundary.
> I would have thought they defined a temperament of
That gives the mother of the two sisters [in a family].
> > two 7-limit sisters a 5-limit et, and so forth. I was
You could for instance find it as the unique linear combination of the prime mapping vals> giving the node. In practice, it's normally obvious.
Here's an example. Suppose we have various versions of augmented:
We can find the node between any two of these by subtracting the two wedgies and taking the third, fifth, and sixth coefficent. We get
It's not hard to guess what the 7-limit extension is even without computing the answer, but in fact to simply draw the boundary we don't need to -- the node already tells us where that lies.
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> the same rank (or codimension or whatever we're calling it) as
> the sisters themselves. For example:
>
> (c1, c2, c3a)
> (c1, c2, c3b)
> gives node (c1, c2)
>
> No?
> > calling this the "nexus", but Monz got me to change that to
> > "node." This node can be uniquely extended to a p-limit
> > temperament in which the sisters have the same mapping and
> > tuning, and so define the pivot tuning which can be seen
> > as the boundary between them.
>
> Great. But how does this extension work?
-------------------- comma sequence -------------------
bival (wedgie) ratios monzos
aug1: << 3 0 9 -7 6 21 || [128/125, 28/27] [ | 7 0, -3 0 > , | 2 -3, 0 1 > ]
aug2: << 3 0 -6 -7 -18 -14 || [128/125, 64/63] [ | 7 0, -3 0 > , | 6 -2, 0 -1 > ]
aug3:= << 3 0 6 -7 1 14 || [128/125, 36/35] [ | 7 0, -3 0 > , | 2 2, -1 -1 > ]
aug4:= << 3 0 -3 -7 -13 -7 || [128/125, 21/20] [ | 7 0, -3 0 > , | -2 1, -1 1 > ]
val
aug1, aug2: < 15 24 35 |
aug1, aug3: < 3 5 7 |
aug1, aug4: < 12 19 28 |
aug2, aug3: < 12 19 28 |
aug2, aug4: < 3 5 7 |
aug3, aug4: < 9 14 21 |
. . . . . . . . .