Interval tree with nobles in cents

Here is a version of Erv's Scale tree specifically for use as a map of
rational and noble intervals. It shows the simpler noble numbers in
cents, in parenthesis, on the relevant edges of the tree. If I
considered a noble number unlikely to be musically significant then
I've shown it in square brackets, or omitted it.

I note that there are other strong metastable intervals near the
strong consonances (unison octave fifth fourth) that are not shown as
such on this tree, because they do not correspond to simple noble numbers.

It seems unlikely that those nobles significantly wider than an octave
will have any musical significance because such wide intervals are
usually consonant anyway, or perhaps a-sonant.

If you are viewing this on Yahoo's web interface the tree is likely to
be a complete mess, and you will need to choose the option on the
right: "Choose Message Option/Show Fixed Width Font". Even so, Yahoo
will probably have deleted some leading spaces, or some lines will be
too long, and you will have to paste it elsewhere and make the
(hopefully obvious) repairs.

[Copper] Nickel Chrome Titanium Silver Gold
1:1 \
8:7(\244c) / |
7:6 \ | .
13:11/ (284c) | .
6:5 \ | .
17:14\ / (339c) |
11:9 / \ |
16:13/ \ |
5:4 \ |
19:15\ / \ |
14:11\ / \ |
23:18/ \ / (422c) |
9:7 / \ |
22:17\ [448c] \ |
13:10/ \ |
17:13/ \ |
4:3 \ |
19:14\ / \ |
15:11\ / \ |
26:19/ [541c] / \ |
11:8 \ / \ |
29:21\ / \ / (560c) |
18:13/ \ / \ |
25:18/ \ / \ |
7:5 / \ |
24:17\ / \ |
17:12\ / \ |
27:19/ \ (607c) \ |
10:7 / \ |
23:16\ [630c] \ |
13:9 / \ |
16:11/ \ |
3:2 \ |
17:11\ / | |
14:9 \ / | |
25:16/ [771c] / | |
11:7 \ / | |
30:19\ / (792c) / | |
19:12/ \ / | |
27:17/ \ / | |
8:5 \ / | |
29:18\ / \ / | |
21:13\ / \ / | |
34:21/ \ / \ / | (833c)
13:8 / \ / | |
31:19\ [849c] \ / | |
18:11/ \ / | |
23:14/ \ / | |
5:3 / | |
22:13\ / | |
17:10\ / | |
29:17/ [923c] / | |
12:7 \ / | |
31:18\ / \ (943c) | |
19:11/ \ / | |
26:15/ \ / | |
7:4 / | |
23:13\ / | |
16:9 \ / | |
25:14/ \ (1002c) | |
9:5 / | |
20:11\ (1039c) | |
11:6 / | |
13:7 / \ |
[Copper] Nickel Chrome Titanium Silver Gold 2:1 Gold
13:6 \ /
11:5 \ |
20:9 / (1378c) |
9:4 \ |
25:11\ / (1424c) |
16:7 / \ |
23:10/ \ |
7:3 \ |
26:11\ / \ |
19:8 \ / \ |
31:13/ \ / (1503c) |
12:5 / \ |
29:12\ [1530c] \ |
17:7 / \ |
22:9 / \ |
5:2 \ |
23:9 \ / \ |
18:7 \ / \ |
31:12/ [1641c] / \ |
13:5 \ / \ |
34:13\ / \ / \ (1666c)
21:8 / \ / \ |
29:11/ \ / \ |
8:3 / \ |
27:10\ / \ |
19:7 \ / \ |
30:11/ \ (1735c) \ |
11:4 / \ |
25:9 \ [1772c] \ |
14:5 / \ |
17:6 / \ |
3:1 /
16:5 \ /
13:4 \ /
23:7 / [2055c] /
10:3 \ /
27:8 \ / [2109c] /
17:5 / \ /
24:7 / \ /
7:2 \ /
25:7 \ / \ /
18:5 \ / \ /
29:8 / \ / \ [2226c]
11:3 / \ /
26:7 \ [2276c] \ /
15:4 / \ /
19:5 / \ /
4:1 /
17:4 \ /
13:3 \ /
22:5 / [2558c] /
9:2 \ /
23:5 \ / \ [2649c]
14:3 / \ /
19:4 / \ /
5:1 /
16:3 \ /
11:2 \ /
17:3 / \ [2988c] .
6:1 / .
13:2 \ [3272c] .
7:1 / |
8:1 / \
[Copper] Nickel Chrome Titanium Silver Gold 1:0

-- Dave Keenan

To fix the mess that Yahoo has made, you need to delete the first
space from every line that begins with one.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> To fix the mess that Yahoo has made, you need to delete the first
> space from every line that begins with one.
>
> -- Dave Keenan

"View source" fixes it.

-Carl

I think I am missing something here
Every page of the 33 pages of the scale tree
http://anaphoria.com/sctree.PDF <http://anaphoria.com/sctree.PDF>
list Noble mediants. Since the tree can be taken down as far as one
wants, the nobles themselves will also form a continuum
One problem with listing the distances between members of the rational
series preseding down is that one might miss that they are all epimoric
intervals. http://anaphoria.com/epimore.pdf
<http://anaphoria.com/epimore.pdf>

For one reason might one assume that a noble might be insignificant and
unusable a priori?

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> Here is a version of Erv's Scale tree specifically for use as a map of
> rational and noble intervals. It shows the simpler noble numbers in
> cents, in parenthesis, on the relevant edges of the tree. If I
> considered a noble number unlikely to be musically significant then
> I've shown it in square brackets, or omitted it.
>
> I note that there are other strong metastable intervals near the
> strong consonances (unison octave fifth fourth) that are not shown as
> such on this tree, because they do not correspond to simple noble
numbers.
>
> It seems unlikely that those nobles significantly wider than an octave
> will have any musical significance because such wide intervals are
> usually consonant anyway, or perhaps a-sonant.
>
> If you are viewing this on Yahoo's web interface the tree is likely to
> be a complete mess, and you will need to choose the option on the
> right: "Choose Message Option/Show Fixed Width Font". Even so, Yahoo
> will probably have deleted some leading spaces, or some lines will be
> too long, and you will have to paste it elsewhere and make the
> (hopefully obvious) repairs.
>
> [Copper] Nickel Chrome Titanium Silver Gold
> 1:1 \
> 8:7(\244c) / |
> 7:6 \ | .
> 13:11/ (284c) | .
> 6:5 \ | .
> 17:14\ / (339c) |
> 11:9 / \ |
> 16:13/ \ |
> 5:4 \ |
> 19:15\ / \ |
> 14:11\ / \ |
> 23:18/ \ / (422c) |
> 9:7 / \ |
> 22:17\ [448c] \ |
> 13:10/ \ |
> 17:13/ \ |
> 4:3 \ |
> 19:14\ / \ |
> 15:11\ / \ |
> 26:19/ [541c] / \ |
> 11:8 \ / \ |
> 29:21\ / \ / (560c) |
> 18:13/ \ / \ |
> 25:18/ \ / \ |
> 7:5 / \ |
> 24:17\ / \ |
> 17:12\ / \ |
> 27:19/ \ (607c) \ |
> 10:7 / \ |
> 23:16\ [630c] \ |
> 13:9 / \ |
> 16:11/ \ |
> 3:2 \ |
> 17:11\ / | |
> 14:9 \ / | |
> 25:16/ [771c] / | |
> 11:7 \ / | |
> 30:19\ / (792c) / | |
> 19:12/ \ / | |
> 27:17/ \ / | |
> 8:5 \ / | |
> 29:18\ / \ / | |
> 21:13\ / \ / | |
> 34:21/ \ / \ / | (833c)
> 13:8 / \ / | |
> 31:19\ [849c] \ / | |
> 18:11/ \ / | |
> 23:14/ \ / | |
> 5:3 / | |
> 22:13\ / | |
> 17:10\ / | |
> 29:17/ [923c] / | |
> 12:7 \ / | |
> 31:18\ / \ (943c) | |
> 19:11/ \ / | |
> 26:15/ \ / | |
> 7:4 / | |
> 23:13\ / | |
> 16:9 \ / | |
> 25:14/ \ (1002c) | |
> 9:5 / | |
> 20:11\ (1039c) | |
> 11:6 / | |
> 13:7 / \ |
> [Copper] Nickel Chrome Titanium Silver Gold 2:1 Gold
> 13:6 \ /
> 11:5 \ |
> 20:9 / (1378c) |
> 9:4 \ |
> 25:11\ / (1424c) |
> 16:7 / \ |
> 23:10/ \ |
> 7:3 \ |
> 26:11\ / \ |
> 19:8 \ / \ |
> 31:13/ \ / (1503c) |
> 12:5 / \ |
> 29:12\ [1530c] \ |
> 17:7 / \ |
> 22:9 / \ |
> 5:2 \ |
> 23:9 \ / \ |
> 18:7 \ / \ |
> 31:12/ [1641c] / \ |
> 13:5 \ / \ |
> 34:13\ / \ / \ (1666c)
> 21:8 / \ / \ |
> 29:11/ \ / \ |
> 8:3 / \ |
> 27:10\ / \ |
> 19:7 \ / \ |
> 30:11/ \ (1735c) \ |
> 11:4 / \ |
> 25:9 \ [1772c] \ |
> 14:5 / \ |
> 17:6 / \ |
> 3:1 /
> 16:5 \ /
> 13:4 \ /
> 23:7 / [2055c] /
> 10:3 \ /
> 27:8 \ / [2109c] /
> 17:5 / \ /
> 24:7 / \ /
> 7:2 \ /
> 25:7 \ / \ /
> 18:5 \ / \ /
> 29:8 / \ / \ [2226c]
> 11:3 / \ /
> 26:7 \ [2276c] \ /
> 15:4 / \ /
> 19:5 / \ /
> 4:1 /
> 17:4 \ /
> 13:3 \ /
> 22:5 / [2558c] /
> 9:2 \ /
> 23:5 \ / \ [2649c]
> 14:3 / \ /
> 19:4 / \ /
> 5:1 /
> 16:3 \ /
> 11:2 \ /
> 17:3 / \ [2988c] .
> 6:1 / .
> 13:2 \ [3272c] .
> 7:1 / |
> 8:1 / \
> [Copper] Nickel Chrome Titanium Silver Gold 1:0
>
> -- Dave Keenan
>

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> I think I am missing something here
> Every page of the 33 pages of the scale tree
> http://anaphoria.com/sctree.PDF <http://anaphoria.com/sctree.PDF>
> list Noble mediants. Since the tree can be taken down as far as one
> wants, the nobles themselves will also form a continuum

Or at least approximate a continuum as closely as you like, yes. Seems obvious. What's your point?

> One problem with listing the distances between members of the rational
> series preseding down is that one might miss that they are all epimoric
> intervals. http://anaphoria.com/epimore.pdf
> <http://anaphoria.com/epimore.pdf>

Do you mean "One problem with _not_ listing the distances ..."?

Yes. An alternative statement of the condition mentioned earlier (already given 3 formulations) is that the phi-mediant of two ratios is only noble when the ratio _between_ the two ratios is epimoric, i.e. has a difference of one between numerator and denominator when in lowest terms.

> For one reason might one assume that a noble might be insignificant and
> unusable a priori?

Do you mean "For what reason ... "?

I don't believe I said anything about any nobles being "unusable". That would be silly, since clearly the entire audible continuum is usable.

But one can predict that some noble numbers will not be audibly harmonically significant in ordinary music with ordinary harmonic timbres, in much the same way one can predict it of some rational numbers. Namely that the numerator and/or denominator (in lowest terms) are too large to bear any relationship to any audible partials or audible sum or difference tones.

To say that it is not audibly harmonically significant is only to say that it cannot be distinguished by listening, from a neighbouring ratio or noble of similar numerical complexity.

-- Dave