Septimal modified meantone

I have no idea how original this is, but it seems to work out very
nicely as an irregular 7-limit tuning for most repertoire which is
suited to ordinary meantone.

First the tuning, then the explanation. [NB: This will not be in
precise 'Scala' form. Tough. It is the logical order of tuning.]

C 1
F# 7/5
D 28/25
Bb 112/125
G# 98/125
G 3/2
E 5/4
B 15/16
F 75/112
A 375/448
C# 1875/1792
Eb 1875/1568

This combines the pure thirds of meantone with some pure fifths and
septimal intervals 4:7 and 5:7 in such a way that the 12 pitches can
be tuned directly by ear using pure intervals, and no fifth is much
out of tune except the 'wolf'.

If it is difficult to tune 5:7 by ear one may add Ab 8/5 and D# 45/32
in order to get F# and F respectively as 4:7.

The tuning is 'modified' from meantone in two ways. First C-G and E-B
are pure fifths, reflecting some possible historical tuning practice
and making the most used chord pure. That means the comma must be
distributed over G-D-A-E, which is explained later in more detail.

Second, E-G# and Eb-G are slightly wider than pure, which also may
reflect some historical practice in making G# a little less flat if
used as Ab, and Eb a little less sharp used as D#. This modification
is consistent with having the chords Bb-D-G# and Eb-A-C# pure (4:5:7
or 1/7:1/5:1/4 respectively). C#-G# and Eb-Bb are thus only slightly
flat (by 702464/703125, 1.6c).

The division of the syntonic comma in the pure thirds is by two
factors of 224/225 (7.7c, 'septimal kleisma'), and one factor
3125/3136 (6.1c), giving the three fifths G-D-A-E nearly 1/3 comma
each. The major third is divided into whole tones of 28/25 and
125/112. The nice feature of the tuning is the way that the sequences
F#-D-Bb and F-A-C# which start from septimal intervals slot neatly
between the pure G and E, as if tempered.

The intervals F-G# and Eb-C# are exactly 7.02464/6, 6.1c sharp of
septimal. Conversely Bb-C# is 7.0066../6 which is only 1.6c sharp.
(Compare the meantone wolf minor third of 7.0094../6.) The wolf fifth
G#-Eb is 28.9c sharp (cf. 35.7c in meantone).

Although pure septimal intervals were scarcely used in historical
practice, this tuning is fully usable for almost all music for which
meantone is suited, and is much simpler than the 49/32 method
explained by Ibo Ortgies for just, almost exact meantone.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> I have no idea how original this is, but it seems to work out very
> nicely as an irregular 7-limit tuning for most repertoire which is
> suited to ordinary meantone.

You might compare it to certain Fokker blocks, such as pipedum_12f,
from that point of view.

> First the tuning, then the explanation. [NB: This will not be in
> precise 'Scala' form. Tough. It is the logical order of tuning.]

Scala does not have an order to the tuning. For most purposes, sorting
everything into the octave 1 < q <= 2 is best, and I give a Scala file
for the scale below.

! dentirrmean.scl
Tom Dent's 7-limit irregular meantone
12
!
1875/1792
28/25
1875/1568
5/4
75/56
7/5
3/2
196/125
375/224
224/125
15/8
2