9-limit stepwise

Here are some 9-limit stepwise harmonizable scales, with the same
bound on size of steps--8/7 is the largest. In order to keep the
numbers down, I also enforced that the size of the largest step (in
cents) is less than four times that of the smallest step--the
logarithmic ratio is the fourth number listed.

In order I give scale type number on the list, scale steps,
multiplicities, largest/smallest, and number of steps in the scale.

As you can see, the largest scale listed has 41 steps, which is
getting up there. (64/63)/(81/80)=5120/5103 and (49/48)/(50/49) =
2401/2400; putting these together gives us hemififths, and hence
Hemififths[41] as a DE for this. Hemififths is into the
microtemperament range by most standards; it has octave-generator with
TOP values [1199.700, 351.365] and a mapping of
[<1 1 -5 -1|, <0 2 25 13|]. I've never tried to use it, and so far as
I know neither has anyone else, but this certainly gives a motivation.
Ets for hemififths are 41, 58, 99 and 140. We also get Schismic[29]
out of scale 4, Diaschismic[22] out of scale 5, Orwell[31] out of
scale 8 and Orwell[22] out of scale 24, Meantone[19] out of scales 10,
11, 13, or 19, Superkleismic[26] out of scale 12, Octacot[27] out of
scale 14, Semisixths[19] out of scale 29.

1 [81/80, 64/63, 50/49, 49/48] [10, 14, 5, 12] 1.659831 41
2 [81/80, 64/63, 50/49, 28/27] [10, 2, 5, 12] 2.927558 29
3 [81/80, 64/63, 49/48, 25/24] [10, 14, 7, 5] 3.286128 36
4 [81/80, 64/63, 28/27, 25/24] [10, 7, 7, 5] 3.286128 29
5 [81/80, 64/63, 25/24, 21/20] [3, 7, 5, 7] 3.927558 22
6 [81/80, 50/49, 36/35, 28/27] [8, 5, 2, 12] 2.927558 27
7 [81/80, 36/35, 28/27, 25/24] [3, 7, 7, 5] 3.286128 22
8 [64/63, 50/49, 49/48, 36/35] [4, 5, 12, 10] 1.788814 31
9 [64/63, 50/49, 49/48, 21/20] [4, 5, 2, 10] 3.098111 21
10 [64/63, 50/49, 28/27, 21/20] [2, 5, 2, 10] 3.098111 19
11 [64/63, 50/49, 25/24, 21/20] [4, 3, 2, 10] 3.098111 19
12 [64/63, 49/48, 36/35, 25/24] [4, 7, 10, 5] 2.592143 26
13 [64/63, 36/35, 25/24, 21/20] [4, 3, 5, 7] 3.098111 19
14 [50/49, 49/48, 36/35, 28/27] [5, 8, 10, 4] 1.800137 27
15 [50/49, 49/48, 36/35, 16/15] [5, 8, 6, 4] 3.194548 23
16 [50/49, 49/48, 28/27, 27/25] [5, 3, 4, 5] 3.809442 17
17 [50/49, 49/48, 21/20, 16/15] [5, 2, 6, 4] 3.194548 17
18 [50/49, 49/48, 16/15, 27/25] [5, 5, 4, 3] 3.809442 17
19 [50/49, 36/35, 28/27, 21/20] [5, 2, 4, 8] 2.415031 19
20 [50/49, 28/27, 25/24, 27/25] [2, 4, 3, 5] 3.809442 14
21 [50/49, 28/27, 21/20, 16/15] [5, 2, 8, 2] 3.194548 17
22 [50/49, 28/27, 21/20, 27/25] [5, 4, 6, 2] 3.809442 17
23 [50/49, 25/24, 21/20, 16/15] [3, 2, 6, 4] 3.194548 15
24 [49/48, 36/35, 28/27, 25/24] [3, 10, 4, 5] 1.979797 22
25 [49/48, 36/35, 28/27, 15/14] [3, 5, 4, 5] 3.346036 17
26 [49/48, 36/35, 25/24, 16/15] [3, 6, 5, 4] 3.130007 18
27 [49/48, 36/35, 16/15, 15/14] [3, 1, 4, 5] 3.346036 13
28 [49/48, 21/20, 16/15, 15/14] [2, 1, 4, 5] 3.346036 12
29 [36/35, 28/27, 25/24, 21/20] [7, 4, 5, 3] 1.731936 19
30 [36/35, 28/27, 25/24, 27/25] [4, 4, 5, 3] 2.731936 16
31 [36/35, 28/27, 25/24, 49/45] [7, 1, 5, 3] 3.022902 16
32 [36/35, 28/27, 25/24, 35/32] [7, 4, 2, 3] 3.181021 16
33 [36/35, 28/27, 21/20, 15/14] [2, 4, 3, 5] 2.449085 14
34 [36/35, 28/27, 15/14, 49/45] [2, 1, 5, 3] 3.022902 11
35 [36/35, 28/27, 15/14, 35/32] [5, 4, 2, 3] 3.181021 14
36 [36/35, 28/27, 35/32, 54/49] [3, 4, 3, 2] 3.449085 12
37 [36/35, 28/27, 35/32, 10/9] [5, 2, 3, 2] 3.740051 12
38 [36/35, 25/24, 21/20, 16/15] [3, 5, 3, 4] 2.290966 15
39 [36/35, 25/24, 21/20, 10/9] [3, 1, 3, 4] 3.740051 11
40 [36/35, 25/24, 16/15, 49/45] [6, 5, 1, 3] 3.022902 15
41 [36/35, 25/24, 16/15, 35/32] [3, 2, 4, 3] 3.181021 12
42 [36/35, 25/24, 49/45, 10/9] [6, 4, 3, 1] 3.740051 14
43 [36/35, 21/20, 15/14, 10/9] [2, 3, 1, 4] 3.740051 10
44 [36/35, 21/20, 35/32, 10/9] [3, 2, 1, 4] 3.740051 10
45 [36/35, 21/20, 54/49, 10/9] [1, 3, 1, 4] 3.740051 9
46 [36/35, 16/15, 15/14, 49/45] [1, 1, 5, 3] 3.022902 10
47 [36/35, 16/15, 15/14, 35/32] [1, 4, 2, 3] 3.181021 10
48 [36/35, 16/15, 35/32, 10/9] [3, 2, 3, 2] 3.740051 10
49 [36/35, 15/14, 49/45, 10/9] [2, 4, 3, 1] 3.740051 10
50 [36/35, 27/25, 35/32, 10/9] [1, 2, 1, 4] 3.740051 8
51 [36/35, 49/45, 35/32, 10/9] [4, 1, 2, 3] 3.740051 10
52 [28/27, 25/24, 15/14, 27/25] [4, 1, 4, 3] 2.116195 12
53 [28/27, 25/24, 15/14, 28/25] [1, 1, 4, 3] 3.116195 9
54 [28/27, 25/24, 27/25, 54/49] [4, 3, 3, 2] 2.671709 12
55 [28/27, 25/24, 27/25, 8/7] [2, 3, 3, 2] 3.671709 10
56 [28/27, 25/24, 54/49, 28/25] [1, 3, 2, 3] 3.116195 9
57 [28/27, 21/20, 16/15, 15/14] [2, 3, 2, 5] 1.897095 12
58 [28/27, 21/20, 15/14, 27/25] [4, 1, 5, 2] 2.116195 12
59 [28/27, 21/20, 15/14, 54/49] [4, 3, 3, 2] 2.671709 12
60 [28/27, 21/20, 15/14, 28/25] [2, 1, 5, 2] 3.116195 10
61 [28/27, 21/20, 15/14, 8/7] [2, 3, 3, 2] 3.671709 10
62 [28/27, 21/20, 54/49, 10/9] [1, 3, 2, 3] 2.897095 9
63 [28/27, 16/15, 15/14, 9/8] [2, 2, 2, 3] 3.238677 9
64 [28/27, 16/15, 35/32, 54/49] [1, 3, 3, 2] 2.671709 9
65 [28/27, 15/14, 27/25, 49/45] [3, 5, 2, 1] 2.341582 11
66 [28/27, 15/14, 27/25, 9/8] [4, 4, 2, 1] 3.238677 11
67 [28/27, 15/14, 49/45, 54/49] [1, 3, 3, 2] 2.671709 9
68 [28/27, 15/14, 49/45, 28/25] [1, 5, 1, 2] 3.116195 9
69 [28/27, 15/14, 28/25, 9/8] [2, 4, 2, 1] 3.238677 9
70 [28/27, 27/25, 35/32, 54/49] [4, 1, 2, 3] 2.671709 10
71 [28/27, 27/25, 35/32, 8/7] [1, 1, 2, 3] 3.671709 7
72 [28/27, 35/32, 54/49, 28/25] [3, 2, 3, 1] 3.116195 9
73 [25/24, 21/20, 16/15, 15/14] [2, 3, 4, 3] 1.690091 12
74 [25/24, 21/20, 16/15, 8/7] [2, 3, 1, 3] 3.271065 9
75 [25/24, 21/20, 10/9, 8/7] [1, 3, 1, 3] 3.271065 8
76 [25/24, 21/20, 28/25, 8/7] [2, 2, 1, 3] 3.271065 8
77 [25/24, 16/15, 15/14, 28/25] [2, 1, 3, 3] 2.776167 9
78 [25/24, 16/15, 49/45, 54/49] [2, 1, 3, 3] 2.380181 9
79 [25/24, 15/14, 10/9, 28/25] [1, 3, 1, 3] 2.776167 8
80 [25/24, 15/14, 28/25, 8/7] [2, 2, 3, 1] 3.271065 8
81 [25/24, 27/25, 28/25, 8/7] [3, 1, 2, 2] 3.271065 8
82 [25/24, 49/45, 54/49, 10/9] [1, 3, 3, 1] 2.580974 8
83 [25/24, 54/49, 28/25, 8/7] [3, 1, 3, 1] 3.271065 8
84 [25/24, 28/25, 9/8, 8/7] [2, 2, 1, 2] 3.271065 7
85 [21/20, 16/15, 15/14, 49/45] [1, 2, 5, 2] 1.745389 10
86 [21/20, 16/15, 15/14, 35/32] [1, 4, 3, 2] 1.836685 10
87 [21/20, 16/15, 15/14, 10/9] [3, 2, 3, 2] 2.159462 10
88 [21/20, 16/15, 35/32, 8/7] [1, 1, 2, 3] 2.736851 7
89 [21/20, 15/14, 27/25, 10/9] [1, 1, 2, 4] 2.159462 8
90 [21/20, 15/14, 49/45, 8/7] [1, 3, 2, 2] 2.736851 8
91 [21/20, 15/14, 10/9, 28/25] [1, 3, 2, 2] 2.322777 8
92 [21/20, 15/14, 10/9, 8/7] [3, 1, 2, 2] 2.736851 8
93 [21/20, 27/25, 54/49, 10/9] [2, 1, 1, 4] 2.159462 8
94 [21/20, 49/45, 54/49, 10/9] [2, 1, 2, 3] 2.159462 8
95 [21/20, 35/32, 10/9, 8/7] [2, 1, 1, 3] 2.736851 7
96 [21/20, 54/49, 10/9, 8/7] [3, 1, 3, 1] 2.736851 8
97 [21/20, 10/9, 9/8, 8/7] [2, 2, 1, 2] 2.736851 7
98 [16/15, 15/14, 49/45, 54/49] [1, 4, 3, 1] 1.505516 9
99 [16/15, 15/14, 49/45, 28/25] [1, 5, 2, 1] 1.755985 9
100 [16/15, 15/14, 49/45, 9/8] [2, 4, 2, 1] 1.825004 9
101 [16/15, 15/14, 35/32, 54/49] [4, 1, 3, 1] 1.505516 9
102 [16/15, 15/14, 35/32, 28/25] [3, 3, 2, 1] 1.755985 9
103 [16/15, 15/14, 35/32, 9/8] [4, 2, 2, 1] 1.825004 9
104 [16/15, 35/32, 54/49, 8/7] [3, 3, 1, 1] 2.069018 8
105 [16/15, 35/32, 9/8, 8/7] [2, 2, 1, 2] 2.069018 7
106 [15/14, 27/25, 49/45, 10/9] [2, 2, 1, 3] 1.527122 8
107 [15/14, 49/45, 54/49, 10/9] [2, 3, 2, 1] 1.527122 8
108 [15/14, 49/45, 54/49, 8/7] [3, 3, 1, 1] 1.935438 8
109 [15/14, 49/45, 10/9, 28/25] [4, 1, 1, 2] 1.642614 8
110 [15/14, 49/45, 28/25, 8/7] [4, 2, 1, 1] 1.935438 8
111 [15/14, 49/45, 9/8, 8/7] [2, 2, 1, 2] 1.935438 7
112 [15/14, 10/9, 28/25, 9/8] [2, 2, 2, 1] 1.707177 7
113 [27/25, 35/32, 10/9, 8/7] [2, 1, 3, 1] 1.735052 7
114 [49/45, 35/32, 9/8, 8/7] [1, 1, 1, 3] 1.568046 6
115 [49/45, 54/49, 10/9, 9/8] [2, 2, 2, 1] 1.383115 7
116 [49/45, 10/9, 9/8, 8/7] [1, 1, 2, 2] 1.568046 6
117 [10/9, 28/25, 9/8, 8/7] [2, 1, 2, 1] 1.267376 6

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

On the other end of the size scale we have these. Paul, have you ever
considered Pajara[6] as a possible melody scale?

> 88 [21/20, 16/15, 35/32, 8/7] [1, 1, 2, 3] 2.736851 7
Dom7[7]

> 111 [15/14, 49/45, 9/8, 8/7] [2, 2, 1, 2] 1.935438 7
Beatles[7]

> 112 [15/14, 10/9, 28/25, 9/8] [2, 2, 2, 1] 1.707177 7
Dicot[7]

> 114 [49/45, 35/32, 9/8, 8/7] [1, 1, 1, 3] 1.568046 6
Pajara[6]

> 115 [49/45, 54/49, 10/9, 9/8] [2, 2, 2, 1] 1.383115 7
Squares[7]

> 116 [49/45, 10/9, 9/8, 8/7] [1, 1, 2, 2] 1.568046 6
Pajara[6]

> 117 [10/9, 28/25, 9/8, 8/7] [2, 1, 2, 1] 1.267376 6
Tripletone[6]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> > 116 [49/45, 10/9, 9/8, 8/7] [1, 1, 2, 2] 1.568046 6
> Pajara[6]

It's six notes of pajara, but not Pajara[6]. The 49/45 and 10/9 are
seven fifths up and four octaves down--the apotome, 49/45 and 10/9 all
being the same in pajara. The 9/8 and 8/7 of course are the same, both
two fifths up and an octave down. So in terms of generators, it looks
like [2,2,7,2,2,7], which adds up to 22 fifths. In 22-equal, it would
be 443443, which is what you get from a generator of a half-octave
plus 8/22=4/11, which is a 9/7, and as a DE is associated to a
different temperament, with commas 50/49 and 245/243, which has come
up before but for which I don't have a name. So which temperament is
it? In practical terms, you probably take both, ending up with 22-et,
and a 6-note scale which is *not* Pajara[6] but which is related to
both pajara and the 9/7-generator temperament.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Here are some 9-limit stepwise harmonizable scales, with the same
> bound on size of steps--8/7 is the largest. In order to keep the
> numbers down, I also enforced that the size of the largest step (in
> cents) is less than four times that of the smallest step--the
> logarithmic ratio is the fourth number listed.
>
> In order I give scale type number on the list, scale steps,
> multiplicities, largest/smallest, and number of steps in the scale.
>
> As you can see, the largest scale listed has 41 steps, which is
> getting up there. (64/63)/(81/80)=5120/5103 and (49/48)/(50/49) =
> 2401/2400; putting these together gives us hemififths, and hence
> Hemififths[41] as a DE for this. Hemififths is into the
> microtemperament range by most standards; it has octave-generator
with
> TOP values [1199.700, 351.365] and a mapping of
> [<1 1 -5 -1|, <0 2 25 13|]. I've never tried to use it, and so far
as
> I know neither has anyone else, but this certainly gives a
motivation.
> Ets for hemififths are 41, 58, 99 and 140. We also get Schismic[29]
> out of scale 4, Diaschismic[22] out of scale 5,

I thought Diaschismic had no unique definition in the 7-limit.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> On the other end of the size scale we have these. Paul, have you
ever
> considered Pajara[6] as a possible melody scale?

Seems awfully improper, but descending it resembles a famous
Stravisky theme.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> On the other end of the size scale we have these. Paul, have you
ever
> considered Pajara[6] as a possible melody scale?

I'm confused -- I thought the largest step was supposed to be less
than 200 cents?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> > wrote:
> >
> > On the other end of the size scale we have these. Paul, have you
> ever
> > considered Pajara[6] as a possible melody scale?
>
> Seems awfully improper, but descending it resembles a famous
> Stravisky theme.

What I should have asked was if you've tried 443443 as a melody scale
in 22-et.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> > wrote:
> >
> > On the other end of the size scale we have these. Paul, have you
> ever
> > considered Pajara[6] as a possible melody scale?
>
> I'm confused -- I thought the largest step was supposed to be less
> than 200 cents?

I moved it up to 8/7. Carl thinks I could go even higher.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > > 116 [49/45, 10/9, 9/8, 8/7] [1, 1, 2, 2] 1.568046 6
> > Pajara[6]
>
> It's six notes of pajara, but not Pajara[6].

Oh!

> The 49/45 and 10/9 are
> seven fifths up and four octaves down--the apotome, 49/45 and 10/9
all
> being the same in pajara. The 9/8 and 8/7 of course are the same,
both
> two fifths up and an octave down. So in terms of generators, it
looks
> like [2,2,7,2,2,7], which adds up to 22 fifths. In 22-equal, it
would
> be 443443,

Ah. Well you can see on Manuel's list:

http://www.xs4all.nl/~huygensf/doc/modename.html

that I already gave its second mode a bland, technical name. Dave
Keenan shows it as a chord on his page:

http://www.uq.net.au/~zzdkeena/Music/ErlichDecChords.gif

and calls it "superaugmented subminor 7th 9th augmented 11th", which
is even worse.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > > wrote:
> > >
> > > On the other end of the size scale we have these. Paul, have
you
> > ever
> > > considered Pajara[6] as a possible melody scale?
> >
> > Seems awfully improper, but descending it resembles a famous
> > Stravisky theme.
>
> What I should have asked was if you've tried 443443 as a melody
scale
> in 22-et.

Right; I thought you were talking about 2 2 7 2 2 7, but then you
clarified. See /tuning-math/message/9886.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > > wrote:
> > > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > > > wrote:
> > > >
> > > > On the other end of the size scale we have these. Paul, have
> you
> > > ever
> > > > considered Pajara[6] as a possible melody scale?
> > >
> > > Seems awfully improper, but descending it resembles a famous
> > > Stravisky theme.
> >
> > What I should have asked was if you've tried 443443 as a melody
> scale
> > in 22-et.
>
> Right; I thought you were talking about 2 2 7 2 2 7, but then you
> clarified. See /tuning-
math/message/9886.

Oh yeah, the answer is yes.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> > wrote:
> >
> > > > 116 [49/45, 10/9, 9/8, 8/7] [1, 1, 2, 2] 1.568046 6
> > > Pajara[6]
> >
> > It's six notes of pajara, but not Pajara[6].
>
> Oh!
>
> > The 49/45 and 10/9 are
> > seven fifths up and four octaves down--the apotome, 49/45 and 10/9
> all
> > being the same in pajara. The 9/8 and 8/7 of course are the same,
> both
> > two fifths up and an octave down. So in terms of generators, it
> looks
> > like [2,2,7,2,2,7], which adds up to 22 fifths. In 22-equal, it
> would
> > be 443443,
>
> Ah. Well you can see on Manuel's list:
>
> http://www.xs4all.nl/~huygensf/doc/modename.html
>
> that I already gave its second mode a bland, technical name. Dave
> Keenan shows it as a chord on his page:
>
> http://www.uq.net.au/~zzdkeena/Music/ErlichDecChords.gif
>
> and calls it "superaugmented subminor 7th 9th augmented 11th", which
> is even worse.

Yow. I'll take 9-limit consonant whole-tone, thank you. Of course the
other whole tone scale counts also.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > > wrote:
> > >
> > > > > 116 [49/45, 10/9, 9/8, 8/7] [1, 1, 2, 2] 1.568046 6
> > > > Pajara[6]
> > >
> > > It's six notes of pajara, but not Pajara[6].
> >
> > Oh!
> >
> > > The 49/45 and 10/9 are
> > > seven fifths up and four octaves down--the apotome, 49/45 and
10/9
> > all
> > > being the same in pajara. The 9/8 and 8/7 of course are the
same,
> > both
> > > two fifths up and an octave down. So in terms of generators, it
> > looks
> > > like [2,2,7,2,2,7], which adds up to 22 fifths. In 22-equal, it
> > would
> > > be 443443,
> >
> > Ah. Well you can see on Manuel's list:
> >
> > http://www.xs4all.nl/~huygensf/doc/modename.html
> >
> > that I already gave its second mode a bland, technical name. Dave
> > Keenan shows it as a chord on his page:
> >
> > http://www.uq.net.au/~zzdkeena/Music/ErlichDecChords.gif
> >
> > and calls it "superaugmented subminor 7th 9th augmented 11th",
which
> > is even worse.
>
> Yow. I'll take 9-limit consonant whole-tone, thank you. Of course
the
> other whole tone scale counts also.

It doesn't, because 4+4+4=12 is not a 9-limit consonance in 22-equal.