Yahoo Tuning Groups Ultimate Backup tuning Microtempering Partch

* Microtempering *

In earlier messages I gave the following simultaneous distribution of the
224:225 and 384:385 commas or "unison vectors". This particular
distribution minimises the maximum absolute error over all 11-limit
consonances.

Intvl Error (cents)
2:3 -1.36
4:5 -2.71
5:6 1.36
4:7 -0.43
5:7 2.28
6:7 0.93
4:9 -2.71
5:9 0.00
7:9 -2.28
4:11 -2.71
5:11 0.00
6:11 -1.36
7:11 -2.28
9:11 0.00

I earlier coined the term "wafso-just" for such tunings (Within A Fly
excrement Of just). I'd like to coin another: "microtemperament". The point
of microtempering an otherwise strictly-just scale is to increase the
number of available consonances while keeping the errors close to the limit
of tuning accuracy for non-electronic instruments and less than half those
of meantone (which is sometimes considered quasi-just).

Putting it another way. Those who consider themselves to be playing (for
example) guitar in a strictly-just tuning will typically have random tuning
errors of the order of those introduced by microtempering, and even if it
is perfectly tuned, they will sometimes deliberately use chords having
errors of the full 224:225 (7.7 c) or 384:385 (4.5 c). Why not nudge the
random errors in a useful direction and temper out these commas?

Carl Lumma complained that he could easily hear the difference between the
wafso-just and the just, (presumably in sustained dyads on an electronic
instrument). I don't doubt it in regard to the ratios of 11. I will
eventually get around to calculating the distribution that minimises the
maximum beats in the 4:5:6:7:9:11 chord. This will reduce the errors in the
ratios of 11 at the expense of the others, and should make the
microtempering even less obtrusive.

** The 11-limit diamond **

The 11-limit diamond itself cannot be improved by any such microtempering,
but harmonically and melodically it is crying out for two extra tones, one
of which simultaneously represents 15/8, 28/15 and 144/77 and the other,
their inversions, 16/15, 15/14 and 77/72.

The number of additional consonances created by distributing the 224:225
and 384:385 and adding these two tones is nothing short of astounding. Each
of the two new tones participates in 12 new dyads! (plus one new dyad
between the two tones). The two new tones complete 2 11-limit hexads, 8
7-limit tetrads (two having an 11 but no 9) and 2 5-limit triads!

I'd go so far as to say that the 11-limit diamond "wants" to become
31-notes and be microtempered in this way.

To see the lattice below, you will probably need to grow your window
sideways quite a bit. It shows all 29 tones of the microtempered 11-limit
diamond in a single 5-limit plane with the tones repeated to show their
ratio-of-7 and ratio-of-11 roles. The positions with letter names t T thru
z Z are places where the addition of a pair of tones will give a
significant increase in available consonances. But none anywhere near as
much as t and T, the aforementioned 15/8 and 16/15.

Y 14/11

/
z
x 16/11

20/11---- V ----- W
/ \
/ \ /

14/114/9/ \7/6/ 7/4
16/9 4/3 / 1/1 \
/ 3/2 \ / 9/8
z ----- x
----16/11---12/11---18/11
/ \ / \ / \ /
\ /
20/110/9/ V \5/3/ W \5/4/
\ t / 7/5
/10/7 \ / T \ / 8/5 \ /
6/5 \ / 9/5
5 u
----14/9-----7/6-----7/4-----
/ \ / \ / \ Y / \14/11/
/ 7 \ 16/116/912/114/318/111/1/
3/2 9/8
/ 11 \ / 8/7 \ /12/7 \ / 9/7 \ /
4-------6-------9 10/9-----5/3-----5/4----- t -----7/5
otonal hexad / \ / \20/11/ \ V / \ W /
legend 14/9\ /7/610/7/7/4\ T / \8/5/ 6/5
9/5
/11/9 \ /11/6 \ /11/8 \ / X \ / Z
16/9-----4/3-----1/1-----3/2-----9/8
z / \ x / \16/11/ \12/11/ \18/11/
10/9 5/3 /5/4\ / t \8/7/7/512/7/ \9/7/
utonal hexad
/ w \ / v \ /11/10\ / \ /
legend
10/7----- T -----8/5-----6/5-----9/5
1/9-----1/6-----1/4
/ \14/9 / \ 7/6 / \ 7/4 /
\ 1/11/
16/9 4/3 /1/111/9/3/211/6/9/811/8/
\1/7/
/11/7 \ / y \ / \ /
\ /
-----8/7----12/7-----9/7----- U
1/5
10/9 / \ 5/3 / \ 5/4 / \ t / \ 7/5 /
10/7 / T \ /8/5\ w /6/5\ v /9/511/10
/ \ / \ / \ / \ /
11/9----11/6----11/8----- X ----- Z
16/9 / \ 4/3 / \ 1/1 / 3/2 9/8
8/7 12/7\ /9/711/7/
/ \ / \ /
w ----- v ----11/10
/
11/8 X Z
11/7 y

t T 15/8 (28/15) 16/15 (15/14)
u U 28/27 (25/24) 27/14 (48/25)
v V 15/11 22/15
w W 45/44 (49/48, 56/55) 88/45 (96/49, 55/28)
x X 33/32 64/33
y Y 33/28 56/33
z Z 99/64 128/99

Regards,

-- Dave Keenan
http://dkeenan.com

In TD 483.18 I argued that the 11-limit diamond is "crying out for"
(a) the 15/8 and 16/15 to be added to it, for a total of 31 tones, and
(b) to be microtempered (i.e. temper out the 224:225 and 384:385).

The primary benefit is the huge number of additional consonances that
appear when these two things are done. 25 new dyads appear, completing
numerous hexads, pentads, tetrads and triads. Adding the 15/8 and 16/15
also breaks up some large melodic gaps.

Paul Erlich tells me that the 11-limit diamond plus 15/8 and 16/15 is the
31-tone tuning of Dean Drummond's "zoomoozophone." What's a zoomoozophone?
I wonder if Dean knows just how much his zoomoozophone would benefit from
distributing the 224:225 and 384:385 as described below.

The lattice I posted in TD483.18 was messed up by line wrapping, so rather
than risk it again I've split it in half vertically and offset the top half
to the left, duplicating the central chain of fifths. I also decided the
previous lattice was too cluttered with letters indicating other positions
where extra tones could be beneficially added. This time I've just left
holes in the lattice.

This is the microtempered 11-limit tonality diamond with all 29 tones shown
in a single triangulated 5-limit plane, with unlinked repetitions showing
their roles as ratios of 7 and 11. The 15/8 and 16/15 go in the only holes
that have 6 lines radiating from them. I hope you can see what I mean by
"crying out for them".

Sorry about the numbers that run into each other (and in some cases even
overlap!) but you can disambiguate them by knowing they are in the 11-limit
diamond, every fraction is between 1/1 and 2/1, no numerator is greater
than 20 and no denominator is greater than 11.

14/11
5 /
/ \ 16/11
/ 7 \ /
/ 11 \ 20/11---- -----
4-------6-------9 / \ / \ /
otonal hexad 14/114/9/ \7/6/ 7/4
legend 16/9 4/3 / 1/1 \ / 3/2 \ / 9/8
----- ----16/11---12/11---18/11
/ \ / \ / \ / \ /
20/110/9/ \5/3/ \5/4/ \ / 7/5
/10/7 \ / \ / 8/5 \ / 6/5 \ / 9/5
----14/9-----7/6-----7/4-----
/ \ / \ / \14/11/
16/116/912/114/318/111/1/ 3/2 9/8
/ 8/7 \ /12/7 \ / 9/7 \ /
10/9-----5/3-----5/4----- -----7/5
/ \ / \20/11/ \ / \ /
14/9\ /7/610/7/7/4\ / \8/5/ 6/5 9/5
/11/9 \ /11/6 \ /11/8 \ / \ /
16/9-----4/3-----1/1-----3/2-----9/8
|
`----> shift and join ------.
v
16/9-----4/3-----1/1-----3/2-----9/8
/ \ / \16/11/ \12/11/ \18/11/
10/9 5/3 /5/4\ / \8/7/7/512/7/ \9/7/
/ \ / \ /11/10\ / \ /
10/7----- -----8/5-----6/5-----9/5
/ \14/9 / \ 7/6 / \ 7/4 /
16/9 4/3 /1/111/9/3/211/6/9/811/8/
/11/7 \ / \ / \ /
-----8/7----12/7-----9/7-----
10/9 / \ 5/3 / \ 5/4 / \ / \ 7/5 /
10/7 / \ /8/5\ /6/5\ /9/511/10
/ \ / \ / \ / \ /
11/9----11/6----11/8----- -----
16/9 / \ 4/3 / \ 1/1 / 3/2 9/8 utonal hexad
8/7 12/7\ /9/711/7/ legend
/ \ / \ / 1/9-----1/6-----1/4
----- ----11/10 \ 1/11/
/ \1/7/
11/8 \ /
11/7 1/5

I worked out the tuning that gives the minimum maximum beat rate in the
4:5:6:7:9:11 chord. My philosophy is to maximise the consonance of the most
consonant chord. Here is a table of all the intervals in that chord, sorted
by interval width, showing the error in cents and the beat rate on a 130.8
Hz root.

Intvl Error Beat rate when the root is C below middle C (130.8 Hz)
(cents) (Hz)
6:7 0.31 1.0
5:6 2.13 4.8
9:11 0.47 3.5
4:5 -3.18 4.8
7:9 -1.35 6.4
5:7 2.44 6.4
4:6 -1.05 1.0
6:9 -1.05 1.4
7:11 -0.89 5.2
4:7 -0.74 1.6
5:9 1.08 3.7
6:11 -0.58 2.9
5:11 1.55 6.4
4:9 -2.10 5.7
4:11 -1.63 5.4

Note that equal worst beats occur in the 7:9, 5:7 and 5:11. The maximum
error has gone up to 3.2 c (in the 4:5).

Here's the scala file. I've included just enough alternative names in the
comments to allow you to see the two non-diamond hexads. Many other
alternative names would be needed to show *all* the non-diamond
consonances. Best just to look at the lattice above, with the 16/15 and
15/8 added.

!
! keenan6.scl
!
11-limit, 31 tones, 14 hexads within 3.2c of just, Dave Keenan 11-Jan-2000
31
! Microtempered 11-limit diamond plus 16/15 and 15/8. Min beats in
4:5:6:7:9:11.
115.9584761 ! 16/15
151.2176916 ! 12/11, 35/32
166.5545447 ! 11/10
181.3208835 ! 10/9
201.8137603 ! 9/8
231.9169522 ! 8/7
267.1761677 ! 7/6 , 64/55
317.7722363 ! 6/5
347.8754282 ! 11/9
383.1346438 ! 5/4
418.3938594 ! 14/11
433.7307124 ! 9/7 , 32/25
499.0931199 ! 4/3
549.6891885 ! 11/8
584.9484041 ! 7/5 , 45/32
615.0515959 ! 10/7 , 64/45
650.3108115 ! 16/11
700.9068801 ! 3/2
766.2692876 ! 14/9 , 25/16
781.6061406 ! 11/7
816.8653562 ! 8/5
852.1245718 ! 18/11
882.2277637 ! 5/3
932.8238323 ! 12/7 , 55/32
968.0830478 ! 7/4
998.1862397 ! 16/9
1018.679116 ! 9/5
1033.445455 ! 20/11
1048.782308 ! 11/6 , 64/35
1084.041524 ! 15/8
2/1

Would Partch have used it, if he'd known about it and had a way to tune it?

Regards,

-- Dave Keenan
http://dkeenan.com

Microtempering Partch

🔗David C Keenan <d.keenan@xx.xxx.xxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗David C Keenan <d.keenan@xx.xxx.xxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>