Naming intervals using Miracle

The Miracle temperament gives us a logical way of further extending the
Fokker extended-diatonic interval-names from 31-EDO to Miracle chains, and
hence to 41-EDO, 72-EDO and 11-limit JI.

Previously there was no obvious way of deciding which of a pair of nearby
intervals (such as the neutral seconds 10:11 and 11:12 or the minor
sevenths 5:9 and 9:16) should be called "wide" or "narrow", and which
should be unmodified. Now the answer is obvious. The unmodified one is the
one that is represented within a chain of +-15 Miracle generators. i.e. The
intervals available in Miracle-31 should be named the same as in 31-EDO,
without using "wide" or "narrow".

The table below shows how this scheme names the intervals of 72-EDO.

Legend for interval names:

1 unison
2 second
3 third
4 fourth
5 fifth
6 sixth
7 seventh
8 octave

m = minor
N = neutral
M = major

d = diminished
P = perfect
A = augmented

s = sub
S = super

n = narrow
W = wide

Legend for note names:

A,B,C,D,E,F,G,#,b as for 12-tET
] = quarter-tone up (+50 c)
> = sixth-tone up (+33 c)
^ = twelfth-tone up (+17 c)
v = twelfth-tone down (-17 c)
< = sixth-tone down (-33 c)
[ = quarter-tone down (-50 c)

No. Cents Intvl Note 11-limit
gens name frm C Ratio
---------------------------------
0 0 P1 C 1:1
31 17 W1 C^
-10 33 S1 C>
21 50 WS1 C]
-20 67 nsm2 C#<
11 83 sm2 C#v
-30 100 nm2 C#
1 117 m2 C#^
32 133 Wm2 C#>
-9 150 N2 D[ 11:12
22 167 WN2 D< 10:11
-19 183 nM2 Dv 9:10
12 200 M2 D 8:9
-29 217 nSM2 D^
2 233 SM2 D> 7:8
33 250 WSM2 D]
-8 267 sm3 Eb< 6:7
23 283 Wsm3 Ebv
-18 300 nm3 Eb
13 317 m3 Eb^ 5:6
-28 333 nN3 Eb>
3 350 N3 E[ 9:11
34 367 WN3 E<
-7 383 M3 E 4:5
24 400 WM3 E
-17 417 nSM3 E^ 11:14
14 433 SM3 E> 7:9
-27 450 ns4 F[
4 467 s4 F<
35 483 Ws4 Fv
-6 500 P4 F 3:4
25 517 WP4 F^
-16 533 nS4 F>
15 550 S4 F] 8:11
-26 567 nA4 F#<
5 583 A4 F#v 5:7
+-36 600 WA4/nd5 F#
-5 617 d5 F#^ 7:10
26 633 Wd5 F#>
-15 650 s5 G[ 11:16
16 667 Ws5 G<
-25 683 nP5 Gv
6 700 P5 G 2:3
-35 717 nS5 G^
-4 733 S5 G>
27 750 WS5 G]
-14 767 sm6 G#< 9:14
17 783 Wsm6 G#v 7:11
-24 800 nm6 G#
7 817 m6 G#^ 5:8
-34 833 nN6 G#>
-3 850 N6 A[ 11:18
28 867 WN6 A<
-13 883 M6 Av 3:5
18 900 WM6 A
-23 917 nSM6 A^
8 933 SM6 A> 7:12
-33 950 nsm7 A]
-2 967 sm7 Bb< 4:7
29 983 Wsm7 Bbv
-12 1000 m7 Bb 9:16
19 1017 Wm7 Bb^ 5:9
-22 1033 nN7 Bb> 11:20
9 1050 N7 B[ 6:11
-32 1067 nM7 B<
-1 1083 M7 Bv
30 1100 WM7 B
-11 1117 SM7 B^
20 1133 WSM7 B>
-21 1150 ns8 C[
10 1167 s8 C<
-31 1183 n8 Cv

Does anyone feel that any of these names are somehow wrong?

Does this conflict with any existing use of "wide" and "narrow"? e.g. Scala.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

On Wed, 04 Jul 2001 22:13:19 -0700, David C Keenan <D.KEENAN@UQ.NET.AU>
wrote:

>35 483 Ws4 Fv
>-6 500 P4 F 3:4
>25 517 WP4 F^

>-25 683 nP5 Gv
>6 700 P5 G 2:3
>-35 717 nS5 G^

I like this scheme in general, but I don't see any reason to avoid "narrow
perfect fourth" or "wide perfect fifth" (especially given that you have
"WP4" and "nP5". These are slightly closer to just than the 5-TET fourths
and fifths (which is about the limit of what I'd consider a good perfect
fourth or fifth).

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> On Wed, 04 Jul 2001 22:13:19 -0700, David C Keenan <D.KEENAN@U...>
> wrote:
>
> >35 483 Ws4 Fv
> >-6 500 P4 F 3:4
> >25 517 WP4 F^
>
> >-25 683 nP5 Gv
> >6 700 P5 G 2:3
> >-35 717 nS5 G^
>
> I like this scheme in general, but I don't see any reason to avoid
"narrow
> perfect fourth" or "wide perfect fifth" (especially given that you
have
> "WP4" and "nP5". These are slightly closer to just than the 5-TET
fourths
> and fifths (which is about the limit of what I'd consider a good
perfect
> fourth or fifth).

Good point. In 72-EDO, nP4 and Ws4 are indeed alternative names for
the same interval. Ws4 is +35 generators and nP4 is -37 generators.
The only time they might actually refer to different interval is on an
open Miracle chain with 38 notes or more, or a closed one with more
than 72 notes. So such distinctions are not really of any practical
interest.

There are 21 (=3*31-72) intervals with alternative names like this in
72-EDO. Then there are the alternative names allowed by the Fokker
31-EDO system itself (like sd5 and A4 for 5:7). These carry over to
the Miracle system as well.

Regards,
-- Dave Keenan

Dave Keenan wrote:
>Does anyone feel that any of these names are somehow wrong?
>Does this conflict with any existing use of "wide" and "narrow"? e.g.
Scala.

No direct conflicts, but I haven't seen the combinations "narrow neutral"
and "wide neutral" before, and find it a bit counterintuitive. I'd
prefer "large minor" and "small major".

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
>
> Dave Keenan wrote:
> >Does anyone feel that any of these names are somehow wrong?
> >Does this conflict with any existing use of "wide" and "narrow"?
e.g.
> Scala.
>
> No direct conflicts, but I haven't seen the combinations "narrow
neutral"
> and "wide neutral" before, and find it a bit counterintuitive. I'd
> prefer "large minor" and "small major".

Yes, wide minor and narrow major are certainly valid alternatives in
this system for the case of thirds and sixths.

But don't you agree that both 10:11 and 11:12 are neutral seconds and
so one of them must be a narrow or a wide neutral? And similarly for
sevenths.

-- Dave Keenan

Dave wrote:
>But don't you agree that both 10:11 and 11:12 are neutral seconds and
>so one of them must be a narrow or a wide neutral?

I don't know about calling 10:11 a neutral second, I'd have to take a
good listen. In the Scala list I called it a 4/5-tone. It's not that
far from the 10/9 whole tone.

Manuel

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
> I don't know about calling 10:11 a neutral second, I'd have to take
a
> good listen. In the Scala list I called it a 4/5-tone. It's not that
> far from the 10/9 whole tone.

It's closer to 11:12.

Ok. How about this argument. We agree that 11:12 is a (unmodified)
neutral second and that 8:9 is a (unmodified) major second. Although
you avoid the issue for major seconds by calling 9:10 a minor whole
tone and 8:9 a major whole tone (which is fine), your ninths make it
clear how they should be named as seconds. You have:
4:9 "major ninth"
9:20 "small ninth" (= narrow major ninth)

So doesn't it have to be:
11:12 neutral second
10:11 wide neutral second
9:10 narrow major second
8:9 major second?

If 10:11 is to be any kind of major second, then it would have to be a
"very narrow major second" or some such.

You call it Ptolemy's second (which is fine). Can you point me to any
scale it is used in, where one can say that it definitely functions as
a major second and not a neutral second.

Regards,
-- Dave Keenan

Dave, after consideration I agree that 10:11 is too small to
be called a major second and with calling it a neutral second.

Manuel