Dances with Wolves: Ibo Ortgies's Alternative Tuning Meantone (1/4 s.c.) with only Just Intervals

(sorry for cross-postings)

Hi,

several weeks ago I had finished reading several books by Martin Vogel
(prof. emeritus in musicology at the university of Bonn), especially his
books about "Schönberg und die Folgen [Schönberg and the consequences]"
(2 vols.), his "On the Relation of Tone" (Lehre von den Tonbeziehungen)
and the "Anleitung zur harmonischen Analyse und Intonation" [Manual for
harmical analysis and intonation].
All very interesting for many reasons and highly recommended.

Martin Vogel is known for proposing just intonation based on pure
fifths, major thirds and the just "natural" septima (7:4).

What is interesting here is, that somewhere in these books I happened
to notice the ratio 49/32 (i.e. 2 septimas 7/4, minus 1 octave). And
while falling asleep (I switched he lights out, when I awoke during the
night), I glanced at the Cent value of that ratio, which is 737.65 Cent.
That number gave a certain deja-vu feeling (now, who could possibly
speak against the use of Cents ...).

When I awoke next morning (31st Jan. 2006), the first thought
basically jumping to my mind was "Of course, that is the meantone wolf"
in standard quarter-comma meantone temperament, or the diminuished sixth
G#-Eb of 737.63 Cent.

And I immediately I thought, that it must be easily possible to base a
very simple tuning method on this interval: Namely, to tune all
intervals of quarter-comma meantone without actively tempering any of
intervals and avoiding the somewhat tempered fifth of that temperament
in the tuning process.

The basic idea was easily set:
If the wolf "fifth" easily can be tuned by pure intervals (and easily
audible like the septima is, once one has learned to listen to it, which
also isn't difficult), then the chain of pure major thirds on both side
of the wolf will already make half of the intervals being set, i.e.
C-E-G#/Eb-G-B
As you see, this chain includes not only the wolf and four pure major
thirds, but also 3 meantone tempered minor thirds (C-Eb, E-G, and G#-B),
and - most notably 2 meantone tempered fifths (C-G, E-B). The effect of
tempered intervals has thus been reached only by tuning justly, and only
in 6 tuning steps from th starting note!

The continuation to the remaining notes (to be tuned) is easy. One needs
to find a suitable other wolf! As two septimas constitute the wolf, you
can just proceed to from the last note of the already established chain
(C-E-G#/Eb-G-B) and build your "wolf" on it: B/Gb. If. like in this
case, the note you reach doesn't belong to the final temperament - with
12 notes per octave an F# would be by far the choice - then you proceed
to the next thirds until you have reached the alternative (Gb)-Bb-D-F# etc.

The advantages of such a method would be many:
- no tempering of any interval in the tuning process
- works from every starting note
- works on any pitch
- excellent control of purity of the intervals, because the
tuning is done (except of the octaves, of course) only with
major thirds (tenths) and septimas, which are more sensitive to
out-of tuneness
- easy to remember basic principle (you'll never again need to
write it down)
- since no tempered intervals need to be checked, you can (and
actually should) set immediately all octaves of any note
which you have just tuned
- needs tuning of maximum 5 notes more, than the whole keybaoard
has
- you can't set ordinary meantone temperament as precise by any
other means

Disadavantages
- requires that one is capable and able to listen to septimas (that)
should'nt be a major obstacle)
- works only on overtone/partial-rich instruments like
many harpsichords, regals, organ reed stops etc.

Below I'll provide the most easy way of setting this meantone
temperament, which actually is a just major-third/septima-tuning: All
intervals can be expressed in rather simple ratios of a limited number
of major thirds and septimas (even if the ratios themselves will easily
look awkward if calculated and written out).

------------

First the basic idea of the tuning process:

1) For an ordinary keyboard (12 notes per octave):

From C:

C
E
G#
(septimal F#)
(septimal E, which is meantone) Eb
G
B
(septimal A)
(septimal G, which is meantone Gb)
Bb
D
F#
(septimal E)
(septimal D, which is meantone Db)
F
A
C#

-------

2) If the instrument has split keys Ab and D# it is even more effective,
one can start with

C
Ab
(C-)E
G#
(septimal F#)
(septimal E, which is meantone) Eb
G
B
D#
(septimal C#)
(septimal B, which is meantone) Bb
D
F#
(septimal E)
(septimal D, which is meantone Db)
F
A
C#
-----------

To make the tuning process practical it must be userfriendly, easy and
comfortable.
Especially remembering, which notes in which octaves one has already
tuned, can be quite awkward.

But the process can be simplified greatly by

immediately tuning all octaves of a note, as soon as it has been set by
in the tuning process.
Then one doesn't need to remember which notes in which octaves one has
already tuned.

This is also useful because, one can after each tuned note (and its
octaves) freely choose the range in which an interval is most easily
heard (for example often in the tenor range - but that might differ a bit)

--------
Alternative 1

From any C
(Always tune all octaves immediately!)

1) Set C
2) C-E–G#
3) G#–(7F#)–(7E =) Eb
4) Eb–G–B
5) B–(7A)–(7G = Gb)–Bb
6) Bb–D–F#
7) F#–(7E)–(7D = Db)–F
(after tuning the octaves of f, retune the one
7E back to be pure C-E
8) F–A–C#

Amount of tuning steps: total number of keys + 5.

--------
Alternative 2

From any A
(Always tune all octaves immediately!)

Set A
A-F, A–C#
C#–(7B)-(7A = Ab)–C
C–E–G#
G#–(7F#)–(7E =) Eb
Eb–G–B
B–(7A)–(7G = Gb)–Bb
after tuning the octaves, retune the one 7A back to A
Bb–D–F#

Amount of tuning steps: total number of keys + 5.

--------
Alternative 3

From any F
(Always tune all octaves immediately!)

Set F
F-A, A–C#
...
continue like above

Amount of tuning steps: total number of keys + 5.

--------

kind regards

Ibo Ortgies

PS:
For the maths-interested

Some Ratios and Cents of Purely Tuned Meantone
related to the fundamental C

(Cent-figures rounded to maximum 2 decimals)

C = 0 Cent = 1/1
E = 386.31 Cent = 5/4
Double-septimal Abb (German: asas)/"Wolf"
= 737.65 Cent = 49/32 = (7/4)2
G# = 772.63 Cent = 50/32 = (5/4)^2
Eb = 310.28 Cent = 1225/1024 = (5/4)^2 * (7/4)^2 / (2/1)^2
Meantone G
= 696.59 Cent = 6125/4096 = (5/4)^3 * (7/4)^2 / (2/1)^2
Meantone minor semitone C#
= 76.03 Cent = 256/245 = (2/1)^2 / (5/4) / (7/4)^2

Compare the above figures with
(Theoretically Exact) Regular Meantone

C = 0 Cent
E = 386.31 Cent
Wolf/G#–Eb = 737.64 Cent
G# = 772.63 Cent
Eb = 310.26 Cent
G = 696.58 Cent
C# = 76.05 Cent