alabake

From: Gene Ward Smith (2003-06-13)
Subject: Baked Alaska

Responding to a question from Carl, I present Baked Alaska, an
alaska-style temperament with brats of exactly 3/2 and 2. This isn't
as nifty as it sounds, because since the octave is no longer pure,
what happens with close position no longer applies to wider spacings.
Anyway, it supplies one answer to what exact values we might use for
these things.

If we take a scale with steps a, b, b, repeated, then we have two
situations with major thirds and fifths--either we have a third equal
to a^2 b^2 and a fifth equal to a^3 b^4, or we have a third equal to
a b^3 and a fifth equal to a^2 b^5. If we set the brat for the first
equal to 3/2 and for the second equal to 2, and solve the resulting
system of two nonlinear equations in two unknowns via the resultant,
we get that a satisfies

320000*a^7-288000*a^4-1728*a^2+32400*a-151875 = 0

and b satisfies

1125*b^7+48*b^5+1800*b^4-4000 = 0

This now gives us the following alaska-style scale:

! alabake.scl
Baked alaska, with brats of 2 and 3/2
12
!
102.565223
201.130054
299.694886
402.260110
500.824942
599.389774
701.954998
800.519830
899.084662
1001.649885
1100.214717
1198.779549

From: Gene Ward Smith (2003-06-13)
Subject: Baked Alaska

Responding to a question from Carl, I present Baked Alaska, an
alaska-style temperament with brats of exactly 3/2 and 2. This isn't
as nifty as it sounds, because since the octave is no longer pure,
what happens with close position no longer applies to wider spacings.
Anyway, it supplies one answer to what exact values we might use for
these things.

If we take a scale with steps a, b, b, repeated, then we have two
situations with major thirds and fifths--either we have a third equal
to a^2 b^2 and a fifth equal to a^3 b^4, or we have a third equal to
a b^3 and a fifth equal to a^2 b^5. If we set the brat for the first
equal to 3/2 and for the second equal to 2, and solve the resulting
system of two nonlinear equations in two unknowns via the resultant,
we get that a satisfies

320000*a^7-288000*a^4-1728*a^2+32400*a-151875 = 0

and b satisfies

1125*b^7+48*b^5+1800*b^4-4000 = 0

This now gives us the following alaska-style scale:

! alabake.scl
Baked alaska, with brats of 2 and 3/2
12
!
102.565223
201.130054
299.694886
402.260110
500.824942
599.389774
701.954998
800.519830
899.084662
1001.649885
1100.214717
1198.779549

From: Carl Lumma (2003-06-14)
Subject: Re: [tuning-math] Baked Alaska

>Responding to a question from Carl, I present Baked Alaska,

Mmmm... baked...

Wait.  Maybe what we should be doing here is working on
the brat between the fifth and the octave.  The Alaska
temperaments all have only one size of octave and two sizes
of fifths.  How hard is this?

-Carl