Beep and bug

The 13-limit temperament 4&9 is a 13-limit extension of 7-limit beep
with TM basis {27/25, 21/20, 33/32, 65/64}. It has mapping given by

[<1 2 3 3 3 3|, <0 -2 -3 -1 2 3|]

We can fill in the gap, getting a prime to go with +1 generators, by
adding the comma 39/34, which is of dubious vintage, being a bit
larger than 8/7; however it gives a mapping with very nice properties:

[<1 2 3 3 3 3 4|, <0 -2 -3 -1 2 3 1|]

This assigns a unique number of contiguous generator steps to each odd
number from 1 to 17:

-5: 15
-4: 9
-3: 5
-2: 3
-1: 7
0: 1
1: 17
2: 11
3: 13

Any subset of a chain of nine generators, together with a selection of
one of the chain (which need not be an element of of the subset) as
root, determines a 17-limit otonal chord up to octave equivalence.
Utonal chords can then be notated as the inverses of otonal chords.

Since we seem to have two names, "beep" and "bug", I propose to call
this bug, unless there is an objection. It isn't important so much as
a temperament, but because it can be lifted so readily to 17-limit JI;
its usefulness in that department seems to make it deserving of a name.

Gene Ward Smith wrote:

> The 13-limit temperament 4&9 is a 13-limit extension of 7-limit beep
> with TM basis {27/25, 21/20, 33/32, 65/64}. It has mapping given by
> > [<1 2 3 3 3 3|, <0 -2 -3 -1 2 3|]
> > We can fill in the gap, getting a prime to go with +1 generators, by
> adding the comma 39/34, which is of dubious vintage, being a bit
> larger than 8/7; however it gives a mapping with very nice properties:
> > [<1 2 3 3 3 3 4|, <0 -2 -3 -1 2 3 1|]
> > This assigns a unique number of contiguous generator steps to each odd
> number from 1 to 17:
> > -5: 15
> -4: 9
> -3: 5
> -2: 3
> -1: 7
> 0: 1
> 1: 17
> 2: 11
> 3: 13
> > Any subset of a chain of nine generators, together with a selection of
> one of the chain (which need not be an element of of the subset) as
> root, determines a 17-limit otonal chord up to octave equivalence.
> Utonal chords can then be notated as the inverses of otonal chords. > > Since we seem to have two names, "beep" and "bug", I propose to call
> this bug, unless there is an objection. It isn't important so much as
> a temperament, but because it can be lifted so readily to 17-limit JI;
> its usefulness in that department seems to make it deserving of a name.

The simplest bug-compatible temperament by the combined-ET method or the fractional-generator method is:

1/5 4&5 [<1, 2, 3, 3, 4, 4, 4|, <0, -2, -3, -1, -3, -1, 0|].

So there could be better alternatives for approximating JI. Try this one:

1/7 [<1, 2, 2, 3, 4, 4, 4|, <0, -3, 2, -1, -4, -2, 1|]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> The 13-limit temperament 4&9 is a 13-limit extension
> of 7-limit beep with TM basis {27/25, 21/20, 33/32, 65/64}.
> It has mapping given by
>
> [<1 2 3 3 3 3|, <0 -2 -3 -1 2 3|]

this is great -- a model of how to describe a temperament.

i only wish there was a way to distinguish between
the periods and the generators without labeling them.

-monz

monz wrote:

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> > wrote:
> > >>The 13-limit temperament 4&9 is a 13-limit extension
>>of 7-limit beep with TM basis {27/25, 21/20, 33/32, 65/64}.
>>It has mapping given by
>>
>>[<1 2 3 3 3 3|, <0 -2 -3 -1 2 3|]
> > > > > this is great -- a model of how to describe a temperament.
> > i only wish there was a way to distinguish between
> the periods and the generators without labeling them.

If the octave is mapped to an exact multiple of periods without any generators (which is the typical case unless you're doing something like a BP scale), the first number in the generator mapping will always be zero.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> The simplest bug-compatible temperament by the combined-ET method or
the
> fractional-generator method is:
>
> 1/5 4&5 [<1, 2, 3, 3, 4, 4, 4|, <0, -2, -3, -1, -3, -1, 0|].

This is full of equivalences we don't want.

> So there could be better alternatives for approximating JI. Try this
one:
>
> 1/7 [<1, 2, 2, 3, 4, 4, 4|, <0, -3, 2, -1, -4, -2, 1|]

15/8 ~ 7/4 isn't ideal, and 9 is out in left field somewhere. I hadn't
looked at that one, but had considered

[<1, 2, 3, 3, 3, 3, 4|, <0, -2, -3, -1, 3, 2, 1|]

and

[<1, 1, 2, 2, 2, 2, 3|, <0, 2, 1, 3, 5, 6, 4|]

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > The 13-limit temperament 4&9 is a 13-limit extension
> > of 7-limit beep with TM basis {27/25, 21/20, 33/32, 65/64}.
> > It has mapping given by
> >
> > [<1 2 3 3 3 3|, <0 -2 -3 -1 2 3|]

> this is great -- a model of how to describe a temperament.
>
> i only wish there was a way to distinguish between
> the periods and the generators without labeling them.

The period maps 2 to a positive integer, and the generator maps it to 0.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> > >>The simplest bug-compatible temperament by the combined-ET method or
> > the > >>fractional-generator method is:
>>
>>1/5 4&5 [<1, 2, 3, 3, 4, 4, 4|, <0, -2, -3, -1, -3, -1, 0|].
> > > This is full of equivalences we don't want.

What I'm saying is that's the one that would logically get the name "bug" if we extend the 5-limit names to higher limits.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> What I'm saying is that's the one that would logically get the name
> "bug" if we extend the 5-limit names to higher limits.

But I want to steal either "beep" or "bug" for this musical analysis
purpose, leaving aside the question of actually using it as a temperament.