The German sixth as a miracle chord

Working with Tonescape, it occurred to me that the German sixth is not
really intrinsically a meantone chord, but is a marvel chord. Since
225/224 plays a role in a number of important 7-limit
temperaments--meantone, miracle, magic, orwell, catakleismic, pajara,
negri, garibaldi--it's a pretty widely applicable business. The idea
is that, for example, the 16/15-4/3-8/5 major chord is pretty closely
related to 1/1, especially in minor, and (including in a minor key) so
is 15/8. Sticking them together as 16/15-4/3-8/5-15/8 gives a somewhat
out of tune otonal tetrad, whose tuning can be greatly improved, to a
very nice tetrad, on tempering out 225/224. Tempering out 81/80 does
not really play a role, and so the other comma aside from 225/224
defining the temperament doesn't really matter.

The same could be said about taking the e minor chord in C major,
5/4-3/2-15/8, and sticking a 16/15 below it. This isn't quite such a
natural addition to major as 15/8 is to minor, because the raised
seventh degree is used all the time in minor anyway, but I don't know
why 16/15-3/2-15/8-5/2 would be a big shock; it also is composed of
notes closely tied to 1/1 in 5-limit terms.

Dear Gene
divde the syntonic-comma in 4 different superparticular subfactors

81:80 = (324:323) * (323:322) * (322:321) * (321:320)
21.5062..C = 5.3848..C + 5.3681..C + 5.35..C + 5.34..C

not equal in (322.4961185... : 321.4961185...)^4 irrational
5.3765...C *4, avoid that!

start tempering from Eb:=81 cps(=Hz) using that proportions
to flat down the 5ths modulo octave, all values exact in cps

Eb:= 81,162,324(:323) *3> 'reduce fitst 5th by 324(:322)
Bb:= 969(:966,483) *3> 'same as 3times 323(:322) for 2nd 5th
F := 1449,2898(:2889,1444.5,722.25,361.125,180.5625)*3> '(322(:322))*9
C := 541.6875 'from that base on tune in pure 3ths

G := 81*5=405 'first pure 3th from Eb
D := (969,484.5,242.25,121.125)*5=605.625 '3th from Bb
A := (1449,724.5,362.25,181.125,90.5625)*5=452.8125 'choir pitch
E := (541.6875,270.84375,135.421875)*5=677.109375 'the C major 3th
B := 405*5=2025,1012.5,.... '3th from G
#F:= 605.625*5=3028.125,...'divide by as long as needed

#C:= 81*7=567 '7th from Eb instead A*5=566.015625 3th from A
#G:= (969,484.5,242.25,121.125)*7=847.875 '7th from Bb near E*5

then Eb:G:C# 'shorten that chord-frequencies by factor 81
and Bb:D:G# are
both 4:5:7 chords, fitting well to trumpets und horn
overtone-series in Eb and Bb, most common at that time.

that's imo why most german organ builders in baroque-meantone aera
choosed their normal pitch (Chor-Ton = choir-tone) in praxis at about
453cps near my theoretical value 452.8125cps

best
Andreas

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Working with Tonescape, it occurred to me that the German sixth is
not
> really intrinsically a meantone chord, but is a marvel chord. Since
> 225/224 plays a role in a number of important 7-limit
> temperaments--meantone, miracle, magic, orwell, catakleismic,
pajara,
> negri, garibaldi--it's a pretty widely applicable business. The idea
> is that, for example, the 16/15-4/3-8/5 major chord is pretty
closely
> related to 1/1, especially in minor, and (including in a minor key)
so
> is 15/8. Sticking them together as 16/15-4/3-8/5-15/8 gives a
somewhat
> out of tune otonal tetrad, whose tuning can be greatly improved, to
a
> very nice tetrad, on tempering out 225/224. Tempering out 81/80 does
> not really play a role, and so the other comma aside from 225/224
> defining the temperament doesn't really matter.
>
> The same could be said about taking the e minor chord in C major,
> 5/4-3/2-15/8, and sticking a 16/15 below it. This isn't quite such a
> natural addition to major as 15/8 is to minor, because the raised
> seventh degree is used all the time in minor anyway, but I don't
know
> why 16/15-3/2-15/8-5/2 would be a big shock; it also is composed of
> notes closely tied to 1/1 in 5-limit terms.