# node

[Joe Monzo]

An equal-temperament which lies at the boundary between two temperament families.

. . . . . . . . .
[Gene Ward Smith, Yahoo tuning message 55150 (Mon Aug 2, 2004 8:46 pm)]

--- In tuning@yahoogroups.com, "Carl Lumma" wrote:

> Why an et?

Because we need to draw a boundary.

> I would have thought they defined a temperament of
> 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?

That gives the mother of the two sisters [in a family].

> > two 7-limit sisters a 5-limit et, and so forth. I was
> > 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?

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:

```                                         -------------------- 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 > ]
```

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

```               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 |
```

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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