Yahoo Tuning Groups Ultimate Backup tuning Re: Chain-of-minor-thirds tuning

Here's a periodic table of ETs (EDOs) that have reasonable 5:6 minor
thirds. Errors are between -15.6c and +17.7c.

0
4
8 11
12 15 18
16 19 22
20 23 26 29
24 27 30 33 36
28 31 34 37 40
32 35 38 41 44 47
36 39 42 45 48 51 54
40 43 46 49 52 55 ... (+17.7c)
44 47 50 53 56 ...
48 51 54 57 ...
52 55 58 ...
56 59 ... (0.15c)
60 ...

(-15.6c)

The formula is N = 4Y + 3x, x and Y are natural numbers where 3x <= 2Y.
Ideally Y would be 5^3/3^3*2^2 = 125/108 (253.1c) and x would be
3^4*2^3/5^4 = 648/625 (62.6c). So the ideal Y/x is very close to 4 and so
ETs which are multiples of 19 have very good 5:6 minor thirds.

The formula suggests a 7 tone scale with two step sizes, but in no ET does
this have more than 2 triads (in 19-tET). By going to 11 tones we get 10
triads while still having only two step sizes. Here's the scale in 19-tET.

C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B Cb(C)
* * * * * * * * * * * (*)

Here's a 5-limit lattice of it

Cb (note Cb = B# in 19-tET)
G# D#
B F#
D A
F C
Ab Eb
Cb

Another view to show how it loops around.

D A
F C
Ab Eb
Cb
G# D#
B F#
D A

If you can tolerate its approximations to ratios of 7, where the augmented
6th is the 4:7, then this 11 of 19-tET scale has 10 tetrads!

Cb
/F \
G#--- D#
\ F /Ab\
B ----F#
\ Ab/Cb\
D ----A
\ Cb/D#\
F ----C
\ D#/F#\
Ab----Eb
\ F#/
Cb

Sorry about the lazy 7-limit lattice drawing. Hope you can get it.

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

🔗D. Stearns <stearns@xxxxxxx.xxxx>

[David C Keenan:]
>Here's a periodic table of ETs (EDOs) that have reasonable 5:6 minor
thirds.

Hi Dave,
File this one under FWIW... When I first stated posting, I posted quite a
bit on my own attempts to use number sequences to generate sales... This
was one of them which puts the 'interior set' at 1/28th (>1/4th and
<2/7ths) by setting the minor third boarders at 4&7 (1/4th and 2/7ths):

s+1=L s+2=L s+3=L s+4=L s+5=L s+6=L s+7=L
(4) 1 5 2 6 3 (7)
11 (8) (12) 9 13 10 (14)
18 15 19 (16) (20) 17 (21)
25 22 26 23 27 (24) (28)
...

Taking this sequence where 7=0 would create an +L+s+L+s+L+L+s scale
template, which expressed in simplest terms (i.e., L=s+1) would be 0, 2, 5,
6, 8, 10, 11 (in 11e).

This is another with a 5/99ths (>2/9ths and <3/11ths) 'interior set'
created by setting the minor third boarders at 9&11 (2/9th and 3/11ths):

+1=lL +2=L +3=L +4=L +5=L +6=L +7=L +8=L +9=L +10=L +11=L
(9) 7 5 3 1 10 8 6 4 2 (11)
20 (18) 16 14 12 21 19 17 15 13 (22)
31 29 (27) 25 23 32 30 28 26 24 (33)
42 40 38 (36) 34 43 41 39 37 35 (44)
53 51 49 47 (45) (54) 52 50 48 46 (55)
64 62 60 58 56 65 (63) 61 59 57 (66)
75 73 71 69 67 76 74 (72) 70 68 (77)
86 84 82 80 78 87 85 83 (81) 79 (88)
97 95 93 91 89 98 96 94 92 (90) (99)
...

Taking this sequence where 11=0 would create an +L+L+L+L+s+L+L+L+L+L+s
scale template, which expressed in simplest terms would be 0, 2, 4, 6, 8,
9, 11, 13, 15, 17, 19, 20 (in 20e).

Of course both of these scales are using generators (by that I only mean
numbers that represent equal divisions of the octave) that fall outside the
>4/7ths and <3/5ths fifth parameters of the 5&7 periodic table, and as such
these particular types of scales will probably be palatable to few... but
perhaps some of the similarities these methods share with the current
thread of post (periodic tables, chain-of-thirds tunings, etc.) might be
relevant, or interesting...

Dan

🔗D. Stearns <stearns@xxxxxxx.xxxx>

🔗David C Keenan <d.keenan@xx.xxx.xxx>

Here's a chain-of-minor-thirds tuning with a near-optimum size of minor
third for 7-limit harmony, when:
a fifth is 6 minor thirds,
a major third is 5 minor thirds,
a 4:7 augmented sixth is 3 minor thirds,
a 5:7 augmented fourth is -2 minor thirds,
a 6:7 augmented 2nd is -3 minor thirds.

It has a minor third of 317c and the errors in cents are:
2:3 4:5 5:6 4:7 5:7 6:7
0.04 -1.3 1.4 -17.8 -16.5 -17.9

RMS error of 12.3c
Max Abs error 17.9c
(only slightly worse than 22-tET)

With 11 tones it has only two step sizes (4 at 181c and 7 at 68c, max even).

It is essentially Just at the 5-limit. The 11 tones give 10 triads. If you
can tolerate the error in the 7's then this becomes 10 complete 7-limit
tetrads. I gotta tell ya those 7's are pretty bad. But it's interesting
enough for its 5-limit properties alone.

Unfortunately it's not 'proper'.

A scala file
-----------------------
! keenan3.scl
!
Chain-of-minor-thirds, 10 tetrads, Dave Keenan, 30-Jun-99, TD235

11
!
68.0
136.0
317.0
385.0
453.0
634.0
702.0
770.0
951.0
1019.0 ! 1087.0 ! 12th note for keyboard mapping
2/1
-----------------------

Keyboard mapping with offsets in cents from 12-tET:

Note Where
name on kbd Offset
--------------------
Cb C -9
C C# -41
D D +40
D# D# +8
Eb E -24
F F +57
F# F# +25
Gb G -7
G# G# +6
Ab A -26
A A# -58
B B +23

The Gb (on the G key) is not part of the 11-tone scale but allows it to be
modulated up by one minor third. Here's the lattice done properly.

Cb
/|\
/ | \
/ F \
/,' \`.\
G#-----\--D#
\`. ,\/|\
\ F /\| \
\ |\/ Ab \
\|/,' \`.\
B------\--F#
\`.\ ,\/|\
\ Ab /\| \
\ |\/ Cb \
\|/,' \`.\
D------\--A
\`.\ ,\/|\
\ Cb /\| \
\ |\/ D# \
\|/,' \`.\
F------\--C
\`.\ ,\/|\
\ D# /\| \
\ |\/ F# \
\|/,' `.\
Ab--------Eb
\`.\ ,'/
\ F# /
\ | /
\|/
Cb

I'm sure this could be redrawn in other enlightening ways. Note that many
notes appear 3 times above. Cb appears 4 times.

Wot fun.

I expect I'm duplicating Erv Wilson's work here (or someone's?). Can anyone
point me to where he examined that 11 of 19 scale (a minor-third MOS)?

Graham, In Wilson's term "Moment Of Symmetry", what symmetry is it
referring to?

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

🔗Graham Breed <g.breed@xxx.xx.xxx>

Dave Keenan wrote:

> I expect I'm duplicating Erv Wilson's work here (or
> someone's?). Can anyone
> point me to where he examined that 11 of 19 scale (a minor-third MOS)?

This page:

http://www.anaphoria.com/scaletree.html

makes more sense to me than last time I saw it. Now, go to this page:

http://www.anaphoria.com/ST03.html

5/19 is the scale. It looks a bit like 5/15 but it must be 5/19. It means
the generating interval is 5 steps from 19, so this is a minor third.
There's no commentary yet, so we don't know what he'll have to say about it!

Beyond that, I don't think he's ever dealt with an MOS where the generating
interval isn't taken to be a fourth. We may be on virgin territory!

> Graham, In Wilson's term "Moment Of Symmetry", what symmetry is it
> referring to?

Why ask me? I'm not completely sure of the definition. But I think the
idea is that the spiral of fourths (or whatever) is symmetric. Or that the
middle note (D for meantone, GA for schismic) is the moment of symmetry.
That's how I understand it, anyway.

This paper:

http://weber.u.washington.edu/~pnm/CLAMPITT.pdf

says that the MOS is equivalent to other things most of the time anyway. It
also covers that neutral third-based scale in a diatonic sort of way.

Incidentally, I have more to say about that article. However, I don't have
the time and inclination to argue it through properly. So here are some
random comments:

The connection of 5+2 and 3+4 isn't so surprising as there are only so many
7-note MOSs. Well, three: 5+2, 3+4 and 6+1.

Their derivation of the neutral scale from major (Ionian) and minor
(Aeolian) could also be used to get the tetrachordal scale Paul Erlich was
talking about but a digest ago:

>Aside from my
>paper, the most recently discussed example was 7-out-of-24, where the
>maximally even scale, which has three modes with two identical tetrachords,
>is far less common than the usual Arabic scale, which has five modes with
>two identical tetrachords.

Averaging Ionian and Dorian gives Paul's preferred pattern. It all confirms
my view that the maximally even and tetrachordal scales go together, in a
crazy sort of way. Between the two, you get 8 modes with identical
tetrachords, by Paul's counting. Although I don't see where the 8th comes
from, as each tetrachordal neutral scale is the anti-scale of a tetrachordal
diatonic scale. The process is as follows:

1) First, catch your tetrachords!

2) The two tetrachords are divided by a whole tone.
Keep it as a major tone.

3) Each tetrachord has three seconds in it.
Swap every major second for a neutral second,
and every minor second for a major second.

By this process, the maximally even scale is the average of Ionian and
Phrygian, or is anti-Dorian.

Right, that's all for today.

Graham
http://www.cix.co.uk/~gbreed/tuning.htm

🔗Daniel Wolf <DJWOLF_MATERIAL@compuserve.com>

Nachricht geschrieben von INTERNET:tuning@onelist.com
>

I expect I'm duplicating Erv Wilson's work here (or someone's?). Can anyone
point me to where he examined that 11 of 19 scale (a minor-third MOS)?

Graham, In Wilson's term "Moment Of Symmetry", what symmetry is it
referring to?

Regards,
-- Dave Keenan
http://dkeenan.com<

Good questions.

First, look at the articles by Larry Hansen in Xenharmonikon. In these
articles, Erv was essentially publishing through Larry, who had himself
overlooked the minor third basis of his keyboard design. Second, the 11 of
19 MOS is easily located on Wilson's scale tree, on Kraig Grady's site.
(Although most of us work with MOS generated by intervals in the fourth or
fifth range, the tree eventually includes all possible generators). Third,
the symmetry referred to is that among the intervals in the complete scale
subtended by cycle create by the generating interval and one atypical
closing interval. Example: for the 5 out of 12 MOS generated by 7/12 octave
the moment of symmetry is:

Generating Series: F C G D A F
Subtended Melodic Series: G A D F A C F G C D

(The axis of symmetry is the middle of the atypical interval A - F).

🔗Kraig Grady <kraiggrady@anaphoria.com>

Graham Breed wrote:

> From: Graham Breed <g.breed@tpg.co.uk>
>
> Dave Keenan wrote:
>
> > I expect I'm duplicating Erv Wilson's work here (or
> > someone's?). Can anyone
> > point me to where he examined that 11 of 19 scale (a minor-third MOS)?
>
> This page:
>
> http://www.anaphoria.com/scaletree.html
>
> makes more sense to me than last time I saw it. Now, go to this page:
>
> http://www.anaphoria.com/ST03.html
>
> 5/19 is the scale. It looks a bit like 5/15 but it must be 5/19. It means
> the generating interval is 5 steps from 19, so this is a minor third.
> There's no commentary yet, so we don't know what he'll have to say about it!

The scale tree is meant to be used both acoustically and/or logrithmically.

>
> Beyond that, I don't think he's ever dealt with an MOS where the generating
> interval isn't taken to be a fourth. We may be on virgin territory!

Unfortunately, Erv is wide open to the idea of any interval being a generator.
Since there is an infinite of branches off the tree though it is full of virgin
territory. Thats it use!

>
>
> > Graham, In Wilson's term "Moment Of Symmetry", what symmetry is it
> > referring to?

The resulting two interval patterns that a generator produces.

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

🔗David C Keenan <d.keenan@uq.net.au>

In a previous post I gave a tuning of an improper 11-tone minor-third MOS
with 10 tetrads, that minimised the maximum error over all 7-limit ratios.
I've since tried minimising the maximum beat rate in the 4:5:6:7 chord
instead. This is equivalent to weighting the 6:7 error at 1.5 times the 4:7
error and then equalising them. In this scale the 6:7 and 7:8 are
represented by the same interval.

It turns out that the size of minor-third that gives minimum beat rates for
this scale is the 19-tET one, or at least completely indistinguishable from
it at 315.797 cents (versus 315.790 for 19-tET).

Of course they are not actually audible as beats, since they are at about
0.34 of the frequency of the root of the chord. But even so, this version
gives better sounding 4:5:6:7's to my ear (despite the 21.4c error in the
4:7). We lose the near-Just 2:3's and 4:5's, but in the previous version
the perfect triads just seemed to accentuate how bad the tetrads were.

Putting it in 19-tET also cuts down the number of different intervals from
20 to 18.

The diesis-pump chord progression G#,B,D,F,Ab,G# is interesting, but
maddening to play on a standard keyboard, even with the keys labelled
(according to the mapping I gave earlier).

In addition to the consonant intervals shown on the earlier lattice, there
are four 5:9's (7c flat), G#:F#, B:A, D:C, F:Eb, and three 7:9's (7c sharp)
B:Eb, Eb:G#, G#:C. Two pairs of these combine to give two 5:6:7:9
half-diminished 7th chords, D:F:G#:C, F:Ab:B:Eb.

It's a pity it's so far from proper.

Paul E., are there any modes of this scale that fulfil your generalised
diatonic criteria if modified to allow minor-thirds to take over the role
of fifths for chord-progression purposes? Here it is again in 19-tET with a
tetrad shown against it.

C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B Cb(C)
* * * * * * * * * * * (*)
4 5 6 7

I see that only the mode starting on Cb has anything approaching your 1(b)
tetrachordality, but Cb isn't the root of any tetrad. Does it fit your 1(a)
max evenness? I'm not sure how to interpret it.

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

🔗David C Keenan <d.keenan@xx.xxx.xxx>

Regarding the 11 of 19-tET minor-third-MOS (3+1+1+3+1+3+1+1+3+1+1).

I found a much better keyboard mapping. Unfortunately it requires me to
change the particular 11 that I've been using from 19-tET (or change the
note names, depending how you look at it). They shift up by 4 divisions.
What used to be Cb is now D etc. This new mapping makes the white keys a
7-tone 4+1+4+1+4+1+4 minor-third MOS. It also makes it so only 4 of the 11
keys are not named consistently with 12-tET but are instead sharpened or
flattened. So chord patterns are more consistent. Unfortunately it blows
any chance of using a +63,-64c tuning table, and doesn't allow a 12th note
to be added to the chain of minor-thirds.

19-tET note: Cb C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B (Cb)
Key colour : b b b b b
w w w w w w w (w)
Key name : C C# D D# E F F# G A A# B (C)
Unused key : G#

Here's the lattice with the new names.

D
/|\
/ | \
/ G# \
/,' \`.\
Bb-----\--F
\`. ,\/|\
\ G# /\| \
\ |\/ B \
\|/,' \`.\
Db-----\--Ab
\`.\ ,\/|\
\ B /\| \
\ |\/ D \
\|/,' \`.\
(Fb) E#-----\--Cb (B#)
\`.\ ,\/|\
\ D /\| \
\ |\/ F \
\|/,' \`.\
G#-----\--D#
\`.\ ,\/|\
\ F /\| \
\ |\/ Ab \
\|/,' `.\
B---------F#
\`.\ ,'/
\ Ab /
\ | /
\|/
D

Here are most of the available chords shown on the renamed chain of minor
thirds. The names of the chords are based on the names of the notes, not
the keys they are played on.

* Otonal triads and tetrads (one per row)

Bb Db E# G# B D F Ab Cb D# F#
7 5 6 9 half-dim 7th
7 5 6 9 half-dim 7th
7 5 6 dim
4 7 5 6 aug 6th (or flat 7th?)
4 7 5 6 aug 6th (or flat 7th?)
4 7 5 6 aug 6th (or flat 7th?)
4 7 5 6 aug 6th (or flat 7th?)
4 7 5 6 aug 6th (or flat 7th?)
4 7 5 aug 6th (or flat 7th?) no 5th
9 4 7 (this one only happens in 19-tET)

* Utonal triads and tetrads

Flip the above end-for-end. e.g.
4 7 5 6
becomes
1 1 1 1
- - - - minor aug 6ths (or minor flat 7ths?)
6 5 7 4

1/9:1/7:1/6:1/5 are supermajor 7ths

* Other saturated chords:

Bb Db E# G# B D F Ab Cb D# F#
3.3 4 5 6 minor 7ths
3.3 4 5 6
3.3 4 5 6
3.3 4 5 6

Bb Db E# G# B D F Ab Cb D# F#
7 6 10.5 9 subminor 7ths
7 6 10.5 9

There are no major 7th chords, alas.

* Inconsistent (necessarily tempered) chords:

Bb Db E# G# B D F Ab Cb D# F#
4 5 6 7.2 dominant 7ths
4 5 6 7.2
4 5 6 7.2
4 5 6 7.2

Bb Db E# G# B D F Ab Cb D# F#
7 8.4 5 6 diminished 7ths
7 8.4 5 6
7 8.4 5 6
7 8.4 5 6
7 8.4 5 6
7 8.4 5 6
7 8.4 5 6
7 8.4 5 6

Bb Db E# G# B D F Ab Cb D# F#
6.4 4 5 augmented (actually doubly-augmented in
this notation) This one only occurs in 19-tET.

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

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

Dave Keenan wrote,

>Paul E., are there any modes of this scale that fulfil your generalised
>diatonic criteria if modified to allow minor-thirds to take over the role
>of fifths for chord-progression purposes? Here it is again in 19-tET with a
>tetrad shown against it.

>C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B Cb(C)
>* * * * * * * * * * * (*)
> 4 5 6 7
>
>I see that only the mode starting on Cb has anything approaching your 1(b)
>tetrachordality, but Cb isn't the root of any tetrad.

1(b) does not distinguish among modes. You must be reading it wrong.

>Does it fit your 1(a)
>max evenness? I'm not sure how to interpret it.

Since 1(a) is really distributional evenness, let's interpret it that way.
We can easily check by listing the specific intervals in each class:

"second": 1, 3
"third": 2, 4
"fourth": 5, 7
"fifth": 6, 8
"sixth": 7, 9

Looks like there are no more than two specific sizes for each generic size,
so 1(a) is satisfied.

For 1(b), since the minor third is structural, we should maybe consider it
rather than the fourth as the framing interval for "tetrachords". My
intuition tells me that under an appropriate redefinition, this scale would
be OK with 1(b).

Moving on to the other rules, your outline of a 4:5:6:7 tetrad above
suggests 1,5,8,10 as the scalar template. This template will produce major
tetrads but no minor tetrads, and since I think you said there are 10
consonant tetrads, I'm assuming 5 were major and 5 out of 11 is not a
majority. So we fail rule (2). So it looks like the answer to your question
is "no".

If we relax rule (2) but still require the number of complete consonant
chords to be more than half the number of notes we have a lot of scales to
consider such as 19-out-of-31 and a scale generated by major thirds,
22-out-of-41. Should we look at some of your recent scales with some sort of
relaxed set of rules?

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

🔗David C Keenan <d.keenan@xx.xxx.xxx>

🔗Graham Breed <g.breed@xxx.xx.xxx>

Paul Erlich (digest 237.10) wrote:
> Some modes have more than one pair of identical tetrachords.
> Maybe that's
> why there are more distinct tetrachordal modes with four
> neutral thirds[sic] and
> three major seconds than there are modes of the diatonic scale?

Indeed it is! I worked all this out in my head last night. Another way of
thinking about it is that you need 8 notes to get both types of neutral
scale at the same time. And, sure enough, a different tetrachordal mode
starts on each of those 8 notes.

In both cases (fifth and neutral third based) there are 3 different
tetrachords, each of which can be used in 3 ways, which gives a maximum of 9
scales. For 5+2, G and A mode can both be taken two ways. (Would it make
sense to distinguish them mixolydian/hypoionan and lydian/hypodorian?) For
3+4, this mode:

T t t T t t T

(T>t) could be either anti-locrian (anti-hypophrygian):

T t t T t t T

or anti-lydian (anti-hyperionian??????):

T t t T t t T

So that leaves us with 8 distinct modes. While I'm at it, there's only one
more family of 7-note tetrachordal scales : 1+6. Here are the examples in
22=:

3 3 3 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 3 3 3

Graham
http://www.cix.co.uk/~gbreed/

🔗David C Keenan <d.keenan@uq.net.au>

I've made a start at writing up the 11-note minor-third MOS. Of most
interest is the 9-limit lattice. I found a good projection of it, but not
in ASCII this time. Colour has been used to good effect (I hope). See
http://dkeenan.com/Music/ChainOfMinor3rds.htm

Here is a keyboard chord chart for this scale. The keys are shown in two rows:
black
white

Remember that the white keys A,C,E,G are in fact tuned as Ab,Cb,E#,G#, all
others are as named below. The "x" indicates that the G#/Ab key is unused.

*Otonal chords
1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 1 3 1 3 (19-tET steps)
Db D# F# x Bb Db D# F# x Bb Db D#
D E F G A B C D E F G A B C D
5--------6-----7--------------9
+4-----------5--------6-----7
4--------------------------7
5--------6--------------------9
5--------6-----7
4-----------5--------6-----7
*4--------------------------7--------9
5--------6--------------------9
4-----------5--------6-----7
4-----------5--------6-----7
5--------6-----7--------------9
4-----------5--------6-----7
4-----------5--------------7
Db D# F# x Bb Db D# F# x Bb Db D#
D E F G A B C D E F G A B C D

+ The only otonal tetrad on all white keys
* This improper 7:9 only occurs in 11-of-19-tET

*Utonal chords
Db D# F# x Bb Db D# F# x Bb Db D#
D E F G A B C D E F G A B C D
9--------------7-----6--------5
+6--------5-----------4-----7
7--------------------------4
9--------------------6--------5
7----6--------5
*9--------7--------------------------4
9--------------------6--------5
6--------5-----------4-----7
6--------5-----------4-----7
9--------------7-----6--------5
6--------5-----------4-----7
7--------------5-----------4
Db D# F# x Bb Db D# F# x Bb Db D#
D E F G A B C D E F G A B C D

+ The only utonal tetrad on all white keys
* This improper 7:9 only occurs in 11-of-19-tET

* Other saturated chords:
Db D# F# x Bb Db D# F# x Bb Db D#
D E F G A B C D E F G A B C D
3.3------4-----------5--------6
3.3------4-----------5--------6
3.3------4-----------5--------6
3.3------4-----------5--------6
6-----7--------------9----10.5
6-----7--------------9----10.5
Db D# F# x Bb Db D# F# x Bb Db D#
D E F G A B C D E F G A B C D

* Inconsistent (necessarily tempered) chords:
Db D# F# x Bb Db D# F# x Bb Db D#
D E# F G# Ab B Cb D E# F G# Ab B Cb D
5--------6--------7.2
5--------6--------7.2
5--------6--------7.2
5--------6--------7.2
5--------6--------7.2
5--------6--------7.2
5--------6--------7.2
5--------6--------7.2
5--------6--------7.2
Db D# F# x Bb Db D# F# x Bb Db D#
D E F G A B C D E F G A B C D
4-----------5--------6--------7.2
4-----------5--------6--------7.2
4-----------5--------6--------7.2
4-----------5--------6--------7.2
Db D# F# x Bb Db D# F# x Bb Db D#
D E F G A B C D E F G A B C D
5--------6-----7--------8.4
5--------6-----7--------8.4
5--------6-----7--------8.4
5--------6-----7--------8.4
5--------6-----7--------8.4
5--------6-----7--------8.4
5--------6-----7--------8.4
5--------6-----7--------8.4
Db D# F# x Bb Db D# F# x Bb Db D#
D E F G A B C D E F G A B C D
4-----------5--------6.4

The above augmented chord only has a 7:9 F#:Bb in 19-tET.

-- Dave Keenan
http://dkeenan.com

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

🔗David C Keenan <d.keenan@uq.net.au>

[Paul H. Erlich TD245.4]
>Dave, that is truly wonderful. If you could construct similar diagrams for
>the diatonic scale and my scales, you'd be a hero.

Eventually.

>The connectivity
>calculations are great too -- they remind be of a post I made a long time
>ago about monkeys randomly playing intervals on keyboards (I gave the
>probability of consonance for various scales), but I would suggest a
>_weighted_ connectivity measure where lower-limit intervals are weighted
>more.

Excellent idea!

>Your page says, "Of course I'm assuming something about what are maximum
>acceptable errors here. Namely about 18 cents in any interval." But the 7:4
>is 21.5 cents off, right?

Only in with the 19-tET minor third as the generator. With the sixth root
of 3 as the generator (just fifths) we get less than 18c error in all
intervals. 19-tET minimises the maximim otonal beat rate, i.e. the max beat
rate of any interval in the 4:5:6:7:9. This was in an earlier post in this
thread (but only mentioning 7-limit). Oops maybe 9-limit doesn't work
except in the 19-tET version. I'll check when I have time.

-- Dave Keenan
http://dkeenan.com

🔗David C Keenan <d.keenan@uq.net.au>

🔗David C Keenan <d.keenan@xx.xxx.xxx>

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

🔗Carl Lumma <clumma@xxx.xxxx>

Sent to Dave Keenan on July 20...

I checked out the new version of your 11-tone MOS article. Forgive me if I
haven't been responsive to some of the great work you've been doing lately.
I can't respond to everything on the list, and it didn't seem that there
was much for me to say... until now! :)

I'm wondering: why are you using diatonic (that is, based on the 7 fifths
semitone) nomenclature for this scale and the chords in your diagram?
Would it be better to use scale degree numbers up to 11, with 4, 8, or 11
nominal symbols? With 4 nominals there may be more accidentals than is
convenient, but with 8 or 11 there are weird effects too, since there will
be step sizes between nominals smaller than the size of the accidentals.

Speaking of the 8-tone subsets of this scale... they are proper, and due to
Rothenberg and Miller's results, might they be heard even when the 11-tone
superset is desired? What are the harmonic characteristics of the 8-tone
subset? Scala file is attached.

In the legend of your lattice diagram, there seems to be an extra blue line
between 4 and 9?

How can the error go from 18 cents to 21.5 cents when the generator goes
from 6/5 to 315.789? In the 7:4 you say? The difference is only 3 * 0.1485.

>That is, it has only two step sizes and has reflective symmetry (about D
in >the notation below). This is also called Myhill's property.

There is a conflict between this and my understanding of Myhill's property.
I thought Myhill's property was: the scale has no more than two sizes of
interval in each interval class. It seems the above quote talks about
interior intervals of the scale. Also, I was not aware that Myhill's
required reflective symmetry. Having only two interior intervals with
reflective symmetry may be enough to guarantee what I thought was Myhill's
property --- I'll take a look at it later (must go fishing now) --- but I
doubt the converse is true?

Thanks for the soapbox! Do you think we should take this on-list?

-C.

🔗David C Keenan <d.keenan@xx.xxx.xxx>

[Carl Lumma]
>I'm wondering: why are you using diatonic (that is, based on the 7 fifths
>semitone) nomenclature for this scale and the chords in your diagram?
>Would it be better to use scale degree numbers up to 11, with 4, 8, or 11
>nominal symbols? With 4 nominals there may be more accidentals than is
>convenient, but with 8 or 11 there are weird effects too, since there will
>be step sizes between nominals smaller than the size of the accidentals.

You haven't exactly made a case for why we should use 4, 8 or 11 nominals.
Perhaps you can spell out what you have in mind. I used diatonic notation
because it's familiar. Since there are many diatonic chords, why shouldn't
they have diatonic notation? Although, in effect, it _is_ a 7 nominal + 4
accidentals notation. Ab, Cb, E#, G# may be considered nominals since there
is no A, C, E, or G (or their other flats or sharps).

In general I don't like using degree numbers because they change when you
change the number of tones in the chain.

>Speaking of the 8-tone subsets of this scale... they are proper, and due
>to Rothenberg and Miller's results, might they be heard even when the 11-
>tone superset is desired?

I'm not familiar with these results. Can you give me a URL?

That's interesting that 8 tones is proper. I hadn't imagined there were any
proper chains between 4 and 15 tones exclusive. 4 and 15 are strictly
proper and have only two step sizes. I've checked them all with Scala now.
8 tones is proper but it has 3 step sizes (and so does not have Myhill's
property and is not a MOS). In that range, only 7 and 11 have 2 step sizes.
Any length chain (of anything) will have symmetry, so since 2, 3, 4, 7, 11,
15 and 19 (minor 3rds) have 2 step sizes they are also MOS's. They also
have Myhill's property.

Consideration of meantone chains leads me to think that having 2 step sizes
is more important than propriety, since 4, 5, 6, 8 and 12 of meantone are
all proper, but of these, only 5, 7 and 12 have only 2 step sizes (+
Myhill's + MOS). Although they are also the only ones that are strictly
proper.

>What are the harmonic characteristics of the 8-tone
>subset? Scala file is attached.

Well, for every tone you drop, you lose a utonal and otonal tetrad, so
going from 11 down to 8 tones you go from 10 down to 4 tetrads. If you're
only interested in 5-limit it's the same; from 10 down to 4 triads. But
it's better than 7 tones with only two (one major and one minor).

>In the legend of your lattice diagram, there seems to be an extra blue
>line between 4 and 9?

Yes, there are no 4:9's in an 11 tone chain of m3's. One would have to go
to 13 or more tones. But you will see it used in the 9-limit lattice for
diatonic, at the end of the article.

In this case there are no 4:9's because there is no chain of two fifths,
but in general for temperaments we may have consecutive fifths with
acceptable errors but they may add up to an unacceptable error in the 4:9.

>How can the error go from 18 cents to 21.5 cents when the generator goes
>from 6/5 to 315.789? In the 7:4 you say? The difference is only 3 *
>0.1485.

I'll have to make this clearer. Thanks. It goes from 18c to 21.5c when the
generator goes from 316.99 (1/6-kleisma, the one with the lowest highest
absolute error) to 315.79 (19-tET).

>>That is, it has only two step sizes and has reflective symmetry (about D
>in >the notation below). This is also called Myhill's property.
>
>There is a conflict between this and my understanding of Myhill's property.
> I thought Myhill's property was: the scale has no more than two sizes of
>interval in each interval class. It seems the above quote talks about
>interior intervals of the scale.

Don't you mean exterior? Better clue me into the definition of interior and
exterior, I'm not too good on this melodic stuff. Isn't an inner interval
just what I'm calling a step.

>Also, I was not aware that Myhill's
>required reflective symmetry. Having only two interior intervals with
>reflective symmetry may be enough to guarantee what I thought was Myhill's
>property

I should have said "is equivalent to Myhill's property". I got this from
the Scala Help for the "MOS" command. The Scala Help agrees elsewhere (FIT
/MODE) with your understanding of Myhill's. So Manuel, should this be MOS
=> Myhill's, not MOS <=> Myhill's? And BTW what is pseudo-Myhill's property?

Regards,

-- Dave Keenan
http://dkeenan.com

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

🔗Kraig Grady <kraiggrady@xxxxxxxxx.xxxx>

🔗Carl Lumma <clumma@xxx.xxxx>

>You haven't exactly made a case for why we should use 4, 8 or 11 nominals.
>Perhaps you can spell out what you have in mind. I used diatonic notation
>because it's familiar. Since there are many diatonic chords, why shouldn't
>they have diatonic notation? Although, in effect, it _is_ a 7 nominal + 4
>accidentals notation. Ab, Cb, E#, G# may be considered nominals since there
>is no A, C, E, or G (or their other flats or sharps).

No I didn't make a case, I asked. However. I find this 'familiar' stuff
to be way over-rated. Composing in this scale will be the challenge.
Finding structures --- took 500 years to get where we are with the diatonic
scale. That's a huge challenge. To get the chords under your fingers on
your instrument. That's a huge challenge. To learn a new notation is
absolutely trivial. The skills of sight reading are not revoked by
changing the number of nominal symbols in the notation. It's a 2-hour
matter of flash cards to learn the notes names on the staff. Observe what
happened when Easley Blackwood made retention of diatonic notation a
priority. He wound up with the most obtuse and hard-to-read notation I can
imagine. His decatonic scale is a piece of cake, for example, with ten
nominals and one octave to a conventional 5-line staff. The chords pop
out, the key signatures are a sinch, and melodic motion is intuitive.

The 7-fifths semitone exists in your ear, but not in this scale. By using
7 nominals, you reinforce it in the compositional framwork of the scale. I
suggested 4, 8, and 11 because they represent places where the chain gets
close to closing against the octave, just as 5 and 7 fifths almost close
around the octave.

>In general I don't like using degree numbers because they change when you
>change the number of tones in the chain.

You're using them, tho?

>>Speaking of the 8-tone subsets of this scale... they are proper, and due
>>to Rothenberg and Miller's results, might they be heard even when the 11-
>>tone superset is desired?
>
>I'm not familiar with these results. Can you give me a URL?

Miller wrote a paper on short-term memory which has become a classic of
cognitive psychology, and has been discussed before on this list...

http://www.well.com/user/smalin/miller.html

As far as Rothenberg goes, I just meant that the 11-tone chain is not
proper, while the 8-tone chain is.

>Consideration of meantone chains leads me to think that having 2 step sizes
>is more important than propriety, since 4, 5, 6, 8 and 12 of meantone are
>all proper, but of these, only 5, 7 and 12 have only 2 step sizes (+
>Myhill's + MOS). Although they are also the only ones that are strictly
>proper.

Rothenberg was able to explain the selection of 5 and 7 using only two
measures: stability and efficiency. Stability is essentially the
percentage of the scale's intervals which are "ambiguous", that is, which
appear in more than one interval class. For a definition of efficiency,
see TD #52 and 55. What I like about Rothenberg's stuff is that he came up
with it first, actually for a speech recognition model, and then applied it
to music and got results that jived.

>Well, for every tone you drop, you lose a utonal and otonal tetrad, so
>going from 11 down to 8 tones you go from 10 down to 4 tetrads. If you're
>only interested in 5-limit it's the same; from 10 down to 4 triads. But
>it's better than 7 tones with only two (one major and one minor).

Interesting.

>>In the legend of your lattice diagram, there seems to be an extra blue
>>line between 4 and 9?
>
>Yes, there are no 4:9's in an 11 tone chain of m3's. One would have to go
>to 13 or more tones. But you will see it used in the 9-limit lattice for
>diatonic, at the end of the article.
>
>In this case there are no 4:9's because there is no chain of two fifths,
>but in general for temperaments we may have consecutive fifths with
>acceptable errors but they may add up to an unacceptable error in the 4:9.

Oh wait- there's not an extra line, you've just bent the 4:9 so it's
distinct from 4:6 and the 6:9?

>>How can the error go from 18 cents to 21.5 cents when the generator goes
>>from 6/5 to 315.789? In the 7:4 you say? The difference is only 3 *
>>0.1485.
>
>I'll have to make this clearer. Thanks. It goes from 18c to 21.5c when the
>generator goes from 316.99 (1/6-kleisma, the one with the lowest highest
>absolute error) to 315.79 (19-tET).

Ah-ha.

>>>That is, it has only two step sizes and has reflective symmetry (about D
>>>in the notation below). This is also called Myhill's property.
>>
>>There is a conflict between this and my understanding of Myhill's property.
>>I thought Myhill's property was: the scale has no more than two sizes of
>>interval in each interval class. It seems the above quote talks about
>>interior intervals of the scale.
>
>Don't you mean exterior? Better clue me into the definition of interior and
>exterior, I'm not too good on this melodic stuff. Isn't an inner interval
>just what I'm calling a step.

A B C D

According to me, A:B, B:C, C:D, and D:A is a complete list of this scale's
interior intervals. Basically, all the 2nds. I understood Myhill's
property to be: no more than two kinds of 2nd, no more than two kinds of
3rd, and so on...

>I should have said "is equivalent to Myhill's property". I got this from
>the Scala Help for the "MOS" command. The Scala Help agrees elsewhere (FIT
>/MODE) with your understanding of Myhill's. So Manuel, should this be MOS
>=> Myhill's, not MOS <=> Myhill's? And BTW what is pseudo-Myhill's property?

I also noticed "pseudo-Myhill's property" and wondered what it was.

-C.

🔗David C Keenan <d.keenan@xx.xxx.xxx>

[Paul H. Erlich TD256.5]
>"Double diminished" is not a good name for the septimal diminished chords
>since "doubly diminished" means lowered by two half-steps in conventional
>theory.

In the notation used, the "fifths" of these chords _are_ lowered by two
chromatic semitones (it's just that they are rather small ones). An
augmented fourth is the same as a double-diminished fifth here. So I don't
feel comfortable calling them any kind of (singly) diminished chord either.
Any other suggestions?

>The page says,
...
>A kleisma is the difference between a just fifth (2:3) and
>an octave reduced chain of 6 just minor thirds (5:6), i.e. 56/(35.26) or
>about 8.11c.

That was 5^6/(3^5*2^6), not 56/(35.26), in case anyone hasn't read the HTML
and is puzzled.

>Is this the accepted definition of a kleisma?

See http://www.ixpres.com/interval/dict/kleisma.htm
It might have been defined as the difference between 6 just major thirds
and 5 just fifths (octave reduced), but this is equivalent to my simpler
description. Can John Chalmers (or anyone) tell us how and why Shoh� Tanaka
defined it.

>If so, Dave's page probably
>represents the most detailed study of kleismatic temperament to date! Dave
>you're awesome!

I find it very difficult to believe that no-one else has discovered the
harmonic riches of chains of 1/6-kleisma (or nearby) minor-thirds before.
Maybe Shoh� Tanaka?

[Paul H. Erlich TD256.7]
>Dave Keenan wrote,
>
>>http://dkeenan.com/Music/ChainOfMinor3rds.htm
>
>>I added a colour 9-limit lattice of the diatonic scale with chord
>>templates, for comparison.
>
>You say G, D, and A are 100% connected. How do you figure that?

Only at the 9-limit. There is an approximate 9-limit consonance between any
one of these notes and every other note (in some octave/inversion).

Here are the relationships for D shown on a chain of fifths. Just slide it
up one and down one for A and G.
. F C G D A E B .
4 5
6 5
4 9
9 5
2 3
2 3
9 5
4 9
6 5
4 5

How might I reword this to avoid whatever misunderstanding we had?

It is obvious that dyadic connectedness (whether 9 or 5 limit) doesn't
determine tonics, since we know A and C are the strong favourites.

[Paul H. Erlich TD256.8, Subject: Question for Neil]
>With Dave Keenan delving deep into 11-out-of-19, I distinctly remember Neil
>Haverstick mentioning an 11-out-of-19 scale on this list several years ago
>(when it was on the Mills server). Someone he knew made it up (was it
>Richard Krantz?). I guessed it was the result of a one-to-one mapping of the
>keyboard to 19-tET and playing 12 consecutive white keys, but Neil didn't
>say. Neil, are you out there?

Any scale that could have been generated in that manner will not be a chain
of minor thirds since its steps could only be 1/19 and 2/19 of an octave.
Chains of minor thirds in 19-tET have steps of at least 3/19oct unless you
have 15 or more tones. The 11-tone chain has 4 such steps. It is not a
maximally even 11 of 19-tET but it is maximally distributed for that pair
of step sizes, 1/19oct and 3/19oct. Did I get that right?

I've made corrections and added more stuff to the page.
http://dkeenan.com/Music/ChainOfMinor3rds.htm

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

🔗David C Keenan <d.keenan@xx.xxx.xxx>

[Carl Lumma TD257.7]
>I find this 'familiar' stuff
>to be way over-rated. Composing in this scale will be the challenge. ...
>To get the chords under your fingers on
>your instrument. That's a huge challenge. To learn a new notation is
>absolutely trivial....
>Observe what
>happened when Easley Blackwood made retention of diatonic notation a
>priority...
>I suggested 4, 8, and 11 because they represent places where the chain
>gets close to closing against the octave, just as 5 and 7 fifths almost
>close around the octave.

You've made a good case. I've added 4-natural and 8-natural notations to
the article. URL at end of this message.

>>In general I don't like using degree numbers because they change when
>>you change the number of tones in the chain.
>
>You're using them, tho?

I don't follow. How, or in what sense, am I using degree numbers?

Thanks for the Miller URL. Of course this talks about 7 +- 2 notes _total_
in short term memory at any given time, not 7 +- 2 notes _per_octave_ in a
scale. But I guess it _is_ desirable for a whole octave to fit in STM.

>Rothenberg was able to explain the selection of 5 and 7 using only two
>measures: stability and efficiency. Stability is essentially the
>percentage of the scale's intervals which are "ambiguous", that is, which
>appear in more than one interval class. For a definition of efficiency,
>see TD #52 and 55.

Thanks for this explanation and pointer too. But I'm not sure I understand
them well enough to apply them to 4, 7, 8 and 11 of m3rds. Could you, please?

There may also be useful scales that have gaps in the chain of m3rds in the
same way as that 12 of meantone with 6 tetrads. I haven't looked for these
yet.

I wrote:
>>... 4, 5, 6, 8 and 12 of meantone are all proper ...

That should have been
... 4, 5, 6, 7 and 12 of meantone are all proper ...

>What I like about Rothenberg's stuff is that he came up
>with it first, actually for a speech recognition model, and then applied
>it to music and got results that jived.

Sounds great. I really want to learn this stuff, but I don't know when I'll
make time to go and find the papers.

The Scala Help actually agrees with Myhill's being _exactly_ two sizes in
each interval class. Could pseudo-Myhill's mean an _average_ of two sizes
per interval class?

See what you think of it now.
http://dkeenan.com/Music/ChainOfMinor3rds.htm

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