Composite Numbers

Hi, List!

Do you have any name for the following phenomenon? Generally a
hebdomekontany has 70 tones but because of compositeness of 9 and 15
1-3-5-7-9-11-13-15 Hebdomekontany has only 58 distinct tones.
Similarly when one extends 7-limit sets of tones in the direction 3
one often gets some 9-limit stuff for free. Could this be considered
a form of enharmonicity?

And a related question: do you think that a chord like 8:10:12:15 is
more consonant than 8:10:11:12? What do you call this phenomenon?

Kalle

--- In tuning@y..., "kalleaho" <kalleaho@m...> wrote:
> Hi, List!
>
> Do you have any name for the following phenomenon? Generally a
> hebdomekontany has 70 tones but because of compositeness of 9 and 15
> 1-3-5-7-9-11-13-15 Hebdomekontany has only 58 distinct tones.

We call the phenomenon "degeneracy". And we say that this
Hebdomekontany is "degenerate". It's a math term and implies no
disrespect or devaluing.

> Similarly when one extends 7-limit sets of tones in the direction 3
> one often gets some 9-limit stuff for free. Could this be considered
> a form of enharmonicity?

I wouldn't call it enharmonicity, but I see what you mean. I'd call it
it "getting some stuff for free". :-)

> And a related question: do you think that a chord like 8:10:12:15 is
> more consonant than 8:10:11:12?

Yes.

> What do you call this phenomenon?

"dyadic consonance" or "simple dyads" or more generally subset
consonance" or "simple subsets"? Look first at all the dyads within
the chord and see if any can be simplified.

8:10:12:15
4: 5
5: 6
4: 5
2 : 3
2 : 3
8 : 15

The first chord (a major seventh) has only one dyad that _can't_ be
reduced to lower terms. In a recently explained notation we could
write it as
8[4:5:6|4:5]15 to show some of this consonance.

Try the other one

8:10:11:12
4: 5
10:11
11:12
8 : 11
5 : 6
2 : 3

Only a few reduced.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > What do you call this phenomenon?
>
> "dyadic consonance" or "simple dyads" or more generally subset
> consonance" or "simple subsets"? Look first at all the dyads within
> the chord and see if any can be simplified.

"Subset consonance" sounds good to me.

>
> 8:10:12:15
> 4: 5
> 5: 6
> 4: 5
> 2 : 3
> 2 : 3
> 8 : 15
>
> The first chord (a major seventh) has only one dyad that _can't_ be
> reduced to lower terms. In a recently explained notation we could
> write it as
> 8[4:5:6|4:5]15 to show some of this consonance.
>
> Try the other one
>
> 8:10:11:12
> 4: 5
> 10:11
> 11:12
> 8 : 11
> 5 : 6
> 2 : 3
>
> Only a few reduced.

Thanks, Dave! BTW, I like your habit of notating dyads the same way
as other chords! :)