Stepwise scales

If we put together the tetrads joining 1 to 15/14, 16/15, 21/20, 25/24,
36/35, 49/48 or 50/49 stepwise, we get 30 tetrads. If we add to these
the complements under inversion (which in terms of the lattice
coordinates is x-->[-1,-1,-1]-x) we get 50 tetrads. These tetrads all
have the property that at least one of 1, 15/14, 14/15, 16/15,
15/16...is a chord element. Their notes comprise 55 notes to the
octave, giving the following scale:

[1, 50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9,
28/25, 9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7,
64/49, 21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24,
72/49, 3/2, 32/21, 49/32, 14/9, 25/16, 8/5, 80/49, 49/30, 5/3, 42/25,
12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18,
96/49, 49/25]

If we look at the intervals this gives us, the three smallest are
2401/2400, 225/224, and 1029/1024, making it a natural candidate for
miracle tempering. Tempering by miracle gives a 43 note scale, as follows:

[-26, -21, -20, -19, -18, -16, -15, -14, -13, -12, -11, -10, -9, -8,
-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15,16, 18, 19, 20, 21, 26]

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

> If we look at the intervals this gives us, the three smallest are
> 2401/2400, 225/224, and 1029/1024, making it a natural candidate for
> miracle tempering.

Hemiwuerschmidt is also a natural. If we take, not the smallest scale
steps, but the closest approximatations to 7-limit consonances, we get
as the five smallest commas, in order, 2401/2400, 6144/6125,
3136/3125, 225/224, 1029/1024. Taking the first three together gives
us hemiwuerschmidt.

In his Ancient Worlds record, Michael Harrison uses an (unspecified)
7-limit scale of 24 notes. It occured to me that if you wanted a
7-limit microtemperament for 24 notes, and if miracle was not quite
accurate enough for you, your best bet would probably be
Hemiweurschmidt[24]. Hemiwuerschmidt[25] is the DE, not 24, but I
don't think that matters much.

Here's the 55-note scale, reduced to 51 notes in hemiwuerschmidt:

[-32, -30, -28, -27, -25, -22, -21, -19, -18, -17, -16, -14, -13, -12,
-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 25, 27, 28, 30, 32]

>In his Ancient Worlds record, Michael Harrison uses an (unspecified)
>7-limit scale of 24 notes. It occured to me that if you wanted a
>7-limit microtemperament for 24 notes, and if miracle was not quite
>accurate enough for you, your best bet would probably be
>Hemiweurschmidt[24]. Hemiwuerschmidt[25] is the DE, not 24, but I
>don't think that matters much.

Might this be of interest to Aaron?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >In his Ancient Worlds record, Michael Harrison uses an (unspecified)
> >7-limit scale of 24 notes. It occured to me that if you wanted a
> >7-limit microtemperament for 24 notes, and if miracle was not quite
> >accurate enough for you, your best bet would probably be
> >Hemiweurschmidt[24]. Hemiwuerschmidt[25] is the DE, not 24, but I
> >don't think that matters much.
>
> Might this be of interest to Aaron?

I posted it. It might well be, it depends on how much tempering he
wants. If even miracle is a bit too much, then maybe. The 22 or 25
note marvelous dwarves are another possibility.