Meantone reduced blocks

It would be nice to classify 12-note, 5-limit Fokker blocks at least
up to meantone reduction. While pondering that, I thought I'd see how
an example which does not reduce to Meantone[12] worked out.

The "thirds" scale, the genus derived from 6/5 and 5/4, can be
analyzed as a Fokker block using the method I've given as

thirds[i] = (25/24)^i (128/125)^round(i/3) (648/625)^round(i/4+1/8)

In meantone, 25/24 maps to 7, and 128/125 and 648/625 to -12. The
meatone reduction therefore is

7i - 12(round(i/3) + round(i/4+1/8))

I don't see a better way offhand of writing this function. Graphing it
gives a result which looks somewhat erratic.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> It would be nice to classify 12-note, 5-limit Fokker blocks at least
> up to meantone reduction. While pondering that, I thought I'd see
how
> an example which does not reduce to Meantone[12] worked out.
>
> The "thirds" scale, the genus derived from 6/5 and 5/4, can be
> analyzed as a Fokker block using the method I've given as
>
> thirds[i] = (25/24)^i (128/125)^round(i/3) (648/625)^round(i/4+1/8)
>
> In meantone, 25/24 maps to 7, and 128/125 and 648/625 to -12.
>
> The
> meatone reduction therefore is
>
> 7i - 12(round(i/3) + round(i/4+1/8))

Please express this meantone scale in conventional letter-name-and-
accidental notation.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > The
> > meatone reduction therefore is
> >
> > 7i - 12(round(i/3) + round(i/4+1/8))
>
> Please express this meantone scale in conventional letter-name-and-
> accidental notation.

Here is the scale in terms of meantone fifths:

[0, 7, -10, -3, 4, -1, -6, 1, -4, 3, -2, -7]

Using an "f" for my flat symbol, here it is in sharps and flats:

[C, C#, Eff, Ef, E, F, Gf, G, Af, A, Bf, Cf]

I'm not sure yet how uncommon this sort of thing is, but probably not
very common. There is only one possible {128/125, 648/625} scale up to
transposition, and one question is what other comma pairs give us
something differing from Meantone[12] on reduction.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > The
> > > meatone reduction therefore is
> > >
> > > 7i - 12(round(i/3) + round(i/4+1/8))
> >
> > Please express this meantone scale in conventional letter-name-
and-
> > accidental notation.
>
> Here is the scale in terms of meantone fifths:
>
> [0, 7, -10, -3, 4, -1, -6, 1, -4, 3, -2, -7]
>
> Using an "f" for my flat symbol, here it is in sharps and flats:
>
> [C, C#, Eff, Ef, E, F, Gf, G, Af, A, Bf, Cf]
>
>
> I'm not sure yet how uncommon this sort of thing is, but probably
not
> very common. There is only one possible {128/125, 648/625} scale up
to
> transposition, and one question is what other comma pairs give us
> something differing from Meantone[12] on reduction.

Have you found any that don't reduce to something other than:

Meantone[12]
Diaschismic[12]
Augmented[12]
Diminished[12]

when tempered accordingly?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Have you found any that don't reduce to something other than:
>
> Meantone[12]
> Diaschismic[12]
> Augmented[12]
> Diminished[12]
>
> when tempered accordingly?

Thirds.scl qualifies.

>> Have you found any that don't reduce to something other than:
>>
>> Meantone[12]
>> Diaschismic[12]
>> Augmented[12]
>> Diminished[12]
>>
>> when tempered accordingly?
>
>Thirds.scl qualifies.

Is there an extra negation in your sentence there, Paul?

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Have you found any that don't reduce to something other than:
> >
> > Meantone[12]
> > Diaschismic[12]
> > Augmented[12]
> > Diminished[12]
> >
> > when tempered accordingly?
>
> Thirds.scl qualifies.

What were the unison vectors for that again?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Have you found any that don't reduce to something other than:
> >>
> >> Meantone[12]
> >> Diaschismic[12]
> >> Augmented[12]
> >> Diminished[12]
> >>
> >> when tempered accordingly?
> >
> >Thirds.scl qualifies.
>
> Is there an extra negation in your sentence there, Paul?
>
> -Carl

Yup -- good catch, carl! I think my brain's taking a permanent
vacation.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> >
> > > Have you found any that don't reduce to something other than:
> > >
> > > Meantone[12]
> > > Diaschismic[12]
> > > Augmented[12]
> > > Diminished[12]
> > >
> > > when tempered accordingly?
> >
> > Thirds.scl qualifies.
>
> What were the unison vectors for that again?

Diesis and diesis--128/125 and 648/625.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > > wrote:
> > >
> > > > Have you found any that don't reduce to something other than:
> > > >
> > > > Meantone[12]
> > > > Diaschismic[12]
> > > > Augmented[12]
> > > > Diminished[12]
> > > >
> > > > when tempered accordingly?
> > >
> > > Thirds.scl qualifies.
> >
> > What were the unison vectors for that again?
>
> Diesis and diesis--128/125 and 648/625.

But doesn't that reduce to Augmented[12] when 128/125 is tempered out
and Diminished[12] when 648/625 is tempered out??

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> But doesn't that reduce to Augmented[12] when 128/125 is tempered
out
> and Diminished[12] when 648/625 is tempered out??

Ooops. Had my wiring crossed.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > But doesn't that reduce to Augmented[12] when 128/125 is tempered
> out
> > and Diminished[12] when 648/625 is tempered out??
>
> Ooops. Had my wiring crossed.

The above follows from the Hypothesis, of course.

So, what's your answer to my original question now? Or should I
wait 'till you're done generating all these blocks?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> So, what's your answer to my original question now? Or should I
> wait 'till you're done generating all these blocks?

The latter makes more sense to me.