Yahoo Tuning Groups Ultimate Backup tuning Amazing 11-limit Generator

Keis, thanks for making sense of those unison vectors.

Graham, yes, "6th root of 3/2" is a much better description. Thanks heaps
for that 7-limit plus neutral thirds lattice.

The fact that the blackjack scale can be seen as three 7-tone neutral third
MOS should be very useful melodically. In its tetrachordal mode the n3 MOS
goes mLmLmLm, where m is 150c and L is 200c.

Dan, because some 11-limit intervals are made up of a huge number of
generators (e.g. 22 for 5:11) then these scales are very sensitive to the
precise size of generator for keeping the errors in acceptable bounds. The
31-EDO and 41-EDO sizes of this generator are really the smallest and
largest acceptable. 116.1c and 117.1c. So so 51 and 52-EDO won't do, and
51, 52 and 103 are not 11-limit consistent. 113-EDO is ok but I think
72-EDO is better.
-------------------------

Here's a table showing, for all 13 generators, the denominators of
convergents and semi-convergents can be read in two ways. The low numbers
(say below 31) can be read as the numbers of tones required to acheive
Myhill's property, MOS, only two step sizes. The larger numbers (say 22 and
above) can be read as numbers of tones in EDOs that contain an
approximation to the generator.

In both cases the numbers in parenthesis (the semi-convergents) should be
considered second-class citizens. As MOS cardinalities they will give
improper scales (or maybe merely proper, not strictly proper). As EDO
cardinalities they will give poor approximations of the generator.

Notice that 31 is by far the most popular denominator, occurring in 9 out
of the 13 cases.

Gener- Hexad RMS MA Denominators of convergents and
ator width err err (semi-convergents) of the generator
(cents) (gens) (cents) as a fraction of an octave (< 100)
--------------------------------------------------------------------
78.1 21 5.2 8.5 15,(16),31,46,(77)
87.7 20 6.3 10.1 13,14,(27),41,(55,96)
116.7 22 2.3 3.4 10,(11,21),31,41,72
154.5 22 7.3 10.5 7,8,(15,23),31,(39,70)
232.1 20 7.9 10.8 5,(6,11,16,21),26,31,(39,70)
271.3 17 6.6 9.3 4,(5),9,(13),22,31,(53,84)
321.8 21 6.3 8.9 3,4,(7),11,15,(26),41,(56,97)
348.3 20 8.4 10.8 3,(4),7,(10,17,24),31,(38,69)
380.7 20 5.6 8.7 3,(4,7,10,13,16),19,22,41,(63)
387.3 20 6.0 9.5 3,(4,7,10,13,16,19,22,25,28),31,(34,65,96)
495.6 21 6.6 8.6 5,(7),12,17,(29),46,(63)
503.0 18 7.9 11.2 5,7,12,(19),31,(43,74)
580.3 22 6.2 10.4 2,(3,5,7,9,11,13,15,17,19,21,23,25,27),
29,31,(60),91

Here's the list of 11-limit consistent EDOs and the number of times they
occur above. Of course slightly different values for the generators could
easily change the occurrence of the larger EDOs (e.g 58 and above).

EDO 22 26 29 31 41 46 58 72 80 87 89 94
occur 3 2 2 9 4 2 0 1 0 0 0 0

Notice that some of the generators come in "families".
78.1c, 154.5c, 232.1c, 387.3c are all approximate multiples of 78.1c.
87.7c, 348.3c are both approximate multiples of 87.7c.
116.7c, 232.1c, 348.3c, 580.3c are all approximate multiples of 116.7c.
Some belong to more than one family.

Here's another useful table.

Gener- Hexad RMS MA Numbers of generators in each interval
ator width err err 3 5 6 7 7 7 9 9 9 11 11 11 11 11
(cents) (gens) (cents) 2 4 5 4 5 6 4 5 7 4 5 6 7 9
--------------------------------------------------------------------
78.1 21 5.2 8.5 9 5 4 -3 -8-12 18 13 21 7 2 -2 10-11
87.7 20 6.3 10.1 8 18-10 11 -7 3 16 -2 5 20 2 12 9 4
116.7 22 2.3 3.4 6 -7 13 -2 5 -8 12 19 14 15 22 9 17 3
154.5 22 7.3 10.5 -11-13 2-17 -4 -6-22 -9 -5-12 1 -1 5 10
232.1 20 7.9 10.8 3 12 -9 -1-13 -4 6 -6 7 -8-20-11 -7-14
271.3 17 6.6 9.3 7 -3 10 8 11 1 14 17 6 2 5 -5 -6-12
321.8 21 6.3 8.9 -9-10 1 3 13 12-18 -8-21 -2 8 7 -5 16
348.3 20 8.4 10.8 2 8 -6-11-19-13 4 -4 15 5 -3 3 16 1
380.7 20 5.6 8.7 5 1 4 12 11 7 10 9 -2 -8 -9-13-20-18
387.3 20 6.0 9.5 8 1 7 18 17 10 16 15 -2 20 19 12 2 4
495.6 21 6.6 8.6 -1-21 20-15 6-14 -2 19 13-11 10-10 4 -9
503.0 18 7.9 11.2 -1 -4 3-10 -6 -9 -2 2 8-18-14-17 -8-16
580.3 22 6.2 10.4 -5 11-16 12 1 17-10-21-22 3 -8 8 -9 13

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

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Paul Erlich asked how this generator compares as a 7-limit generator.

When I searched for 7-limit generators some time ago, I excluded any
generator where the tetrad spanned more than 9 generators. e.g. meantone
which spans 10.

A 116.6c generator has tetrads spanning 13 generators, so I suppose what
Paul wants to know is whether there are any other generators with a tetrad
width of 13 or less that have smaller errors than those generated by 116.6c.

We might as well see if there are any such with errors no worse than
meantone, since the only reason you'd go wider than meantone, is if you get
smaller errors than it (as 116.6c does).

I found are 6 (including meantone and 116.6c). All but one (310.2 c) were
also in the list of 11-limit generators I gave recently (some with slightly
different values).

Gener- Tetrad RMS MA
ator width err err
(cents) (gens) (cents)
-----------------------------
77.7 12 3.1 4.8
116.6 13 1.7 2.5 Erlich
232.2 13 3.6 5.4
271.3 11 2.6 4.3
310.2 10 3.7 5.5
503.4 10 3.7 5.4 meantone

Once again the 116.6c generator has the lowest errors by far.

Hoorah!

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

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗Graham Breed <graham@microtonal.co.uk>

Dave Keenan wrote :

> Gener- Hexad RMS MA Interval category
> ator width err err (31-tET based)
> (cents) (gens) (cents)
> --------------------------------------------------------------------
> 78.1 21 5.2 8.5 subminor second
> 87.7 20 6.3 10.1 subminor second
> 116.7 22 2.3 3.4 major second

He may already have corrected this, and it hasn't reached me yet, but
"major second" should be "minor second" or "diatonic semitone".

In a spirit of free exchange, note that I posted this

|-8 -1 0 0|
| 2 -1 2 -1| = 41
|-1 1 1 1|
| 5 0 0 -2|

wrongly before. It happens to be a way of defining 41-equal in the
11-limit using a schisma and three unison vectors it shares with 31-
and 72-equal. This is the first time I've seen three 11-limit scales
tied up consistently to a linear approximation. I know Dave
understands this (and you can read off some other families from one of
his charts) but I don't think it's been explained in full.

These are all 10+1 scales, in that they are made up of 10 large and 1
small intervals. I'll call these s and q respectively because they
are almost but not exactly entirely unlike a "semitone" and
"quartertone".

In 31=, s=3 steps and q=1 step. In 41=, s=4 and q=1. In 72=, s=7 and
q=2.

The interval s stands for 16:15 or 15:14, and q stands for 45:44,
55:54, 49:48, 56:55, 64:63 and it looks like even 100:99.

In the way I outline somewhere on my website, here's a definition of
the temperament:

(10 1)
H = (16 1)(s)
(23 3)(q)
(28 3)
(35 2)

Using this matrix relationship, you can approximate any 11-prime limit
interval in terms of s and q. If that's not intuitive enough, here's
an outline of the temperament. I'll talk of identity instead of
approximation for simplicity. Remember that just intervals are only
approximated.

The 3-limit is defined by 6s being a perfect fifth (3:2). Knowing s
represents 16:15 gives you the 5-limit, but it's simpler to see that
6s => 8:5. Again, s => 15:14 is enough for the 7-limit, but it
happens that 2s => 8:7 and 5s => 7:5. The 11-limit comes from 3s =>
11:9.

A 5:4 major third is 3s+q, and a 6:5 minor third is 3s-q. Hence q
stands for two kinds of 11-limit dieses. Also, 2q => 25:24.

A 9:8 tone is 2s-q, and a 10:9 minor tone is s+2q. That gives a
syntonic comma of s-3q. The difference between 9:8 and 8:7 also gives
the q => 64:63.

The 7:6 subminor third is 2s+q, giving q as the 49:48 comma with 8:7.

As 11:9 is 3s and 9:8 is 2s-q, that leaves 11:8 as 5s-q. 11:10 is
2s-2q and 12:ll is s+q.

If you were designing an 11-limit notation from scratch, this would be
the place to start. You set your ten nominals, and allow for
accidentals of q relative to them. You could even use the same ideas
for notating Paul Erlich's decatonic scales, as it's 10 nominals
again.

As I mentioned before, a 7-limit/neutral-third lattice already implies
two of the three common unison vectors. So you could already use that
as a generalized keyboard, or use a 10+1 layout for more intuitive
melody.

And, once you've set all this up, you don't need to use the the best
11-limit scales. There are borderline meantones that have value, I'm
sure the same will be true of these scales if you look at them closely
enough.

I think this part of the graph for the 21-note subset is common to the
whole family.

C-----Ev
/ \
/ \
D^----F#----A^
/ \/ / \ \/
/ /\ / \ /\
Db----Fv-/--Ab-/--Cv-\--Eb
/ \ / \/ / \/ \/ / \/
/ \ / /\ / /\ /\ / /\
C-----Ev----G-----Bv----D-----F^----A
/ \ / \/ / \/ \/ / \/ / \ /
/ \ / /\ / /\ /\ / /\ / \ /
D^-/--F#-\--A^-/--C#-\--E^-/--G#----B^
/ \/ / \/ \/ / \/ / \ /
/ /\ / /\ /\ / /\ / \ /
Db----Fv----Ab----Cv----Eb----Gv----Bb
\ / \/ / \/ \/ \/ \/ /
\ / /\ / /\ /\ /\ /\ /
G--\--Bv-/--D--\--F^-/--A
\/ \/ / \/ / \ /
/\ /\ / /\ / \ /
C#----E^----G#----B^
/ \ /
/ \ /
Gv----Bb

With suitable definition of "v" and "^", you could add the comma
shifts to make it work with 41 and 72=. There are some chords like
A^-Db-E^ which aren't spelt right, but hopefully it all comes out in
the wash. It would be much simpler using 10 nominals, but you also
have the investment in learning the new names.

Graham

🔗Graham Breed <graham@microtonal.co.uk>

Oh, and another thing ...

Going through the archives, it looks like there was a hunt for the
unison vectors of the 21-note periodicity block. These should do:

0 -3 0 0
-1 1 1 1
5 0 0 -2
-2 -2 1 0

Note that I'm using the kleisma now, don't know why I missed it
before. I think 243:242 and 225:224 are in the Book, can anybody
improve on 385:384?

You can also make the top line 4 2 0 0 to get a 10-note PB.

I also found that the message with which Paul started the "Dave
Keenan's Miracle Scale" thread gives through some simple intervals
like I did in my last message (not yet received:). To summarize that
bit:

6s => 3:2
-7s => 4:5
-2s => 7:8
12s => 9:4
15s => 11:4

Graham

🔗paul@stretch-music.com

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> Paul Erlich asked how this generator compares as a 7-limit
generator.
>
> When I searched for 7-limit generators some time ago, I excluded any
> generator where the tetrad spanned more than 9 generators. e.g.
meantone
> which spans 10.
>
> A 116.6c generator has tetrads spanning 13 generators, so I suppose
what
> Paul wants to know is whether there are any other generators with a
tetrad
> width of 13 or less that have smaller errors than those generated
by 116.6c.
>
> We might as well see if there are any such with errors no worse than
> meantone, since the only reason you'd go wider than meantone, is if
you get
> smaller errors than it (as 116.6c does).
>
> I found are 6 (including meantone and 116.6c). All but one (310.2
c) were
> also in the list of 11-limit generators I gave recently (some with
slightly
> different values).
>
> Gener- Tetrad RMS MA
> ator width err err
> (cents) (gens) (cents)
> -----------------------------
> 77.7 12 3.1 4.8
> 116.6 13 1.7 2.5 Erlich
> 232.2 13 3.6 5.4
> 271.3 11 2.6 4.3
> 310.2 10 3.7 5.5
> 503.4 10 3.7 5.4 meantone
>
> Once again the 116.6c generator has the lowest errors by far.
>
> Hoorah!

So the amazing 11-limit generator is also the amazing 7-limit
generator. Amazing! And look, it's within 0.1 cent of 7 steps in 72-
tET!

🔗paul@stretch-music.com

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> > Keis, thanks for making sense of those unison vectors.
> >
> > Graham, yes, "6th root of 3/2" is a much better description.
Thanks
> heaps
> > for that 7-limit plus neutral thirds lattice.
> >
> > The fact that the blackjack scale can be seen as three 7-tone
> neutral third
> > MOS should be very useful melodically. In its tetrachordal mode
the
> n3 MOS
> > goes mLmLmLm, where m is 150c and L is 200c.
>
> Yes! There are two other tetrachordal modes: LmLmmLm and mLmmLmL.

Huh? This must be some meaning of tetrachordal of which I am not
aware. Please explain. Are you really calling 350c+500c+350c a
"tetrachord". It thought that even to call something a "tetrachord"
(in scare-quotes) the outer intervals had to be a strong consonance.
Without the scare-quotes they have to be an approximate 3:4.

> The rest of your work is awesome. It should be published.

It has been. On the tuning list. But I understand you mean something
with wider circulation and a little more permanent than arrangements
of magnetic domains on a hard disk.

The presence of the 7-tone minor-thirds MOS was Graham's insight. Does
this scale occur in folk music from anywhere?

I think the paper will need to have at least 3 authors, and several
other tuning list acknowledgements. I think we should wait until the
dust settles a bit. I think we're still kicking it up. Who would you
suggest we submit the paper to?

This family of scales needs to be tested in real life. For that we
need some Scala scale files and keyboard mapping files. And then we
need to ask some of those composers out there to drop whatever they
are doing and see what they can make of the 7, 10, 11, 21 and 31 tone
subsets.

-- Dave Keenan

🔗paul@stretch-music.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., paul@s... wrote:

> > Yes! There are two other tetrachordal modes: LmLmmLm and mLmmLmL.
>
> Huh? This must be some meaning of tetrachordal of which I am not
> aware. Please explain.

Two identical 4:3 spans, either a 4:3 or a 3:2 apart.>
> > The rest of your work is awesome. It should be published.
>
> It has been. On the tuning list. But I understand you mean
something
> with wider circulation and a little more permanent than
arrangements
> of magnetic domains on a hard disk.
>
> The presence of the 7-tone minor-thirds MOS was Graham's insight.
Does
> this scale occur in folk music from anywhere?

It's a very rare scale in Arabic music.
>
> I think the paper will need to have at least 3 authors, and several
> other tuning list acknowledgements. I think we should wait until
the
> dust settles a bit. I think we're still kicking it up. Who would
you
> suggest we submit the paper to?

Perspectives of New Music, Journal of Music Theory, etc., etc.

🔗monz <joemonz@yahoo.com>

🔗Graham Breed <graham@microtonal.co.uk>

Dave Keenan wrote:

> --- In tuning@y..., paul@s... wrote:
> > Yes! There are two other tetrachordal modes: LmLmmLm and mLmmLmL.
>
> Huh? This must be some meaning of tetrachordal of which I am not
> aware. Please explain. Are you really calling 350c+500c+350c a
> "tetrachord". It thought that even to call something a "tetrachord"
> (in scare-quotes) the outer intervals had to be a strong consonance.

Paul has explained this. There's another variety of neutral third
scales that have equal tetrachords, and you'll find some of them in
the 31-note subset.

> > The rest of your work is awesome. It should be published.
>
> It has been. On the tuning list. But I understand you mean something
> with wider circulation and a little more permanent than arrangements
> of magnetic domains on a hard disk.

I was thinking publication in a recognized journal would be a good
idea. This is revolutionary stuff, and needs to be put where the
academics can see it.

> The presence of the 7-tone minor-thirds MOS was Graham's insight.
Does
> this scale occur in folk music from anywhere?

It was no great insight that an MOS with a generator that divides the
fifth into 6 equal parts will also divide it into 2 equal parts. By
the same insight I can say that the 31-note subset will contain 3
10-note neutral-thirds MOS scales (note the *neutral* ;) with a note
left over. Each of those will have all the properties I outlined
before at <http://x31eq.com/7plus3.htm>, including some
Rast-family modes.

With 31 notes, there'll be some diatonic scales. Has anybody
mentioned them?

I thought the extra simplification to the neutral-third lattice was
quite clever :)

> I think the paper will need to have at least 3 authors, and several
> other tuning list acknowledgements. I think we should wait until the
> dust settles a bit. I think we're still kicking it up. Who would you
> suggest we submit the paper to?

Did I sneak in at the eleventh hour to get co-authorship? :)

You and Paul seem to share the discovery, in that you found the
important properties between you. The list of "golden" generators
looks like it should be published independently. I'd like to see some
results for the 15-limit.

I've done some work converting existing scales into "decimal" notation
which I'll write up for the List sometime. I notice, by no surprise
at all, that the 41-note periodicity block behind Partch's 43-note
scale can be written with all notes ascending on the decimal staff.
But as there's no "q" from the 1/1 it can't be the 41-note MOS.

I've no idea where or how to submit these things.

> This family of scales needs to be tested in real life. For that we
> need some Scala scale files and keyboard mapping files. And then we
> need to ask some of those composers out there to drop whatever they
> are doing and see what they can make of the 7, 10, 11, 21 and 31
tone
> subsets.

Bleugh! How are we going to do the keyboard mapping? It's not really
something that fits the 7+5. I suppose we can have 5 black notes to
the octave where the last larger than average to be the s+q. Only you
have to ignore some of the other intervals being larger than average
as well.

C# D Eb E F F# G G# A Bb B C C#
0 0^ 1 1^ 2v 2 3v 3 4v 4 4^ 5v 5

C# D Eb E F F# G G# A Bb B C C#
5 6v 6 6^ 7v 7 8v 8 8^ 9 9^ 0v 0

Using a decimal notation where the number denotes generators from the
roots. Unfortunately, that scale isn't a single chain of generators,
so it'd have to be

C# D Eb E F F# G G# A Bb B C C#
0 1v 1 2w 2v 2 3v 3 4v 4 5w 5v 5

C# D Eb E F F# G G# A Bb B C C#
5 6v 6 7w 7v 7 8v 8 9v 9 0w 0v 0

C# D Eb E F F# G G# A Bb B C C#
0 0^ 1 1^ 1# 2 2^ 3 3^ 4 4^ 4# 5

C# D Eb E F F# G G# A Bb B C C#
5 5^ 6 6^ 6# 7 7^ 8 8^ 9 9^ 9# 0

where #==^^ looks better than M. That gives 24 notes, so enough for a
few Blackjacks.

Graham

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > --- In tuning@y..., paul@s... wrote:
>
> > > Yes! There are two other tetrachordal modes: LmLmmLm and
mLmmLmL.
> >
> > Huh? This must be some meaning of tetrachordal of which I am not
> > aware. Please explain.
>
> Two identical 4:3 spans, either a 4:3 or a 3:2 apart.

I should have remembered this from the discussion
about omnitetrachordality of the early Shrutar scale.

I just took a crash course in tetrachords from Monzo's
wonderful dictionary
http://www.ixpres.com/interval/dict/index.htm
which led me to his tetrachord tutorial and John Chalmer's website.

So now I know that I was mistakenly only considering scales with
_disjunct_ tetrachords as tetrachordal. But still, it's somewhat
redundant to list the conjunct modes for octave repeating scales since
the existence of a disjunct mode implies the existence of two conjunct
ones. All three have the same genus of tetrachord.

I also learnt that the tetrachord we're discussing (150+200+150 cents)
is in John Chalmers broad classification of "strictly-proper diatonic
genera". I assume it is can be described as one of the three
permutations of the neutral diatonic genus. The permutation that goes
200+150+150 (increasing pitch left to right) is the common Arabic (and
maybe early highland bagpipe) one. Paul said 150+200+150 is a rare
Arabic. Does anyone use the 150+150+200 permutation?

I wrote:
> > The presence of the 7-tone minor-thirds MOS was Graham's insight.

Brain bug again. Thanks for the correction Graham. I meant to write
_neutral_ thirds MOS. Alzheimer's setting in I guess. :-)

-- Dave Keenan

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., "Graham Breed" <graham@m...> wrote:
> Paul has explained this. There's another variety of neutral third
> scales that have equal tetrachords, and you'll find some of them in
> the 31-note subset.

You mean the common Arabic?

> I was thinking publication in a recognized journal would be a good
> idea. This is revolutionary stuff, and needs to be put where the
> academics can see it.

I'm convinced.

> With 31 notes, there'll be some diatonic scales. Has anybody
> mentioned them?

You mean Just ones with the broken D-A? No, but I was aware of it.

> I thought the extra simplification to the neutral-third lattice was
> quite clever :)

Indeed. Since it lets us do an 11-limit lattice with only 3 dimensions
instead of 5 or 6.

> Did I sneak in at the eleventh hour to get co-authorship? :)

That's what I had in mind. We'd certainly want to use some version of
that lattice, at least.

> The list of "golden" generators
> looks like it should be published independently.

What do you mean here by "golden" generators. The paper would be about
the 7/72 oct generator. We'd mention that computer searches found
nothing comparable.

>I'd like to see
some
> results for the 15-limit.

Nah. The 13 isn't good enough in 72-tET.

> I've done some work converting existing scales into "decimal"
notation
> which I'll write up for the List sometime.

Could you show it on the blackjack lattice?

> I notice, by no surprise
> at all, that the 41-note periodicity block behind Partch's 43-note
> scale can be written with all notes ascending on the decimal staff.
> But as there's no "q" from the 1/1 it can't be the 41-note MOS.

Ok.

> Bleugh! How are we going to do the keyboard mapping? It's not
really
> something that fits the 7+5.

Can we start by putting a 7-note neutral-thirds MOS on the white
notes? Can we then get a 10-note 7/72 oct MOS as well, using some
black notes. I guess not.

>I suppose we can have 5 black notes to
> the octave where the last larger than average to be the s+q. Only
you
> have to ignore some of the other intervals being larger than average
> as well.
>
> C# D Eb E F F# G G# A Bb B C C#
> 0 0^ 1 1^ 2v 2 3v 3 4v 4 4^ 5v 5
>
> C# D Eb E F F# G G# A Bb B C C#
> 5 6v 6 6^ 7v 7 8v 8 8^ 9 9^ 0v 0
>
> Using a decimal notation where the number denotes generators from
the
> roots. Unfortunately, that scale isn't a single chain of
generators,
> so it'd have to be
>
>
> C# D Eb E F F# G G# A Bb B C C#
> 0 1v 1 2w 2v 2 3v 3 4v 4 5w 5v 5
>
> C# D Eb E F F# G G# A Bb B C C#
> 5 6v 6 7w 7v 7 8v 8 9v 9 0w 0v 0
>
> or
>
> C# D Eb E F F# G G# A Bb B C C#
> 0 0^ 1 1^ 1# 2 2^ 3 3^ 4 4^ 4# 5
>
> C# D Eb E F F# G G# A Bb B C C#
> 5 5^ 6 6^ 6# 7 7^ 8 8^ 9 9^ 9# 0
>
> where #==^^ looks better than M.

I don't understand the above sentence. And I'll need some time to
digest the above.

-- Dave Keenan

🔗Graham Breed <graham@microtonal.co.uk>

Dave Keenan wrote:

> --- In tuning@y..., "Graham Breed" <graham@m...> wrote:
> > Paul has explained this. There's another variety of neutral third
> > scales that have equal tetrachords, and you'll find some of them
in
> > the 31-note subset.
>
> You mean the common Arabic?

There are all kinds of subtleties in Arabic tuning ... but yes, the
one your're thinking of.

> > The list of "golden" generators
> > looks like it should be published independently.
>
> What do you mean here by "golden" generators. The paper would be
about
> the 7/72 oct generator. We'd mention that computer searches found
> nothing comparable.

I meant the optimal, irrational generators you came up with. They
should at least be listed on a website somewhere. Some look familiar,
like the meantone fifth and neutral third. There's also a positive
fifth (negative fourth) that crosses schismic and diaschismic.

> >I'd like to see
> some
> > results for the 15-limit.
>
> Nah. The 13 isn't good enough in 72-tET.

I meant re-running the program to throw out some 15-limit generators.

> > I've done some work converting existing scales into "decimal"
> notation
> > which I'll write up for the List sometime.
>
> Could you show it on the blackjack lattice?

I worked out 31 notes on paper last night, but deliberately left it at
home so I couldn't be destracted from my work and start typing it in.

> > Bleugh! How are we going to do the keyboard mapping? It's not
> really
> > something that fits the 7+5.
>
> Can we start by putting a 7-note neutral-thirds MOS on the white
> notes? Can we then get a 10-note 7/72 oct MOS as well, using some
> black notes. I guess not.

Sure, you can start with neutral thirds on the white notes, and you'll
end up with a neutral third scale. The semitone-generated 10 note MOS
would only have 3 or 4 notes in each chain of neutral thirds.

> > where #==^^ looks better than M.
>
> I don't understand the above sentence. And I'll need some time to
> digest the above.

Raising by two "q"s would mean ^^. You could write that as M (like I
write vv as w) for space but I chose #. It's the same identity as in
31-equal notation, and the same idea as ##("double sharp")==x.

It may be in general that # should refer to ^^^, I'm undecided.

Graham