Yahoo Tuning Groups Ultimate Backup tuning 10 triads, 10 notes

A while back Dave Keenan discussed the 10-tone 7/22 MOS...

0 1 6 7 8 13 14 15 16 21 22
1 5 1 1 5 1 1 1 5 1

...It has ten triads (5 major, 5 minor)! We can improve
the tuning without loosing any chords by taking it from
41-tET...

0 2 11 13 15 24 26 28 30 39 41
2 9 2 2 9 2 2 2 9 2

...Both versions are wildly un-proper, but the 41-tET
version is slightly more stable, at least in log-freq.
space.

Recently, Dave posted this...

>Single chain: No. gene- Notes in m/n
> No. rators in smallest Gen. is
> triads Min Min interval proper approx
>Generator in 8 7-limit 7-limit 2 4 5 MOS with m steps
>(+-1c) notes RMS err MA err. 3 5 6 >4 notes of n-tET
>----------------------------------------------------------------------
>380c M3 6 4.6c 6.0c 5 1 4 16 or 19 6/19, 13/41

...which is 8-out-of the scale above. Its chord/note ratio is lower,
and the chords no longer fall on a regular pattern of scale steps.

But by omitting 1 note, we have another MOS...

0 11 13 24 26 37 39 41
11 2 11 2 11 2 2

...with 4 triads, falling on a regular pattern of scale steps. Like the
10-note version, the pattern is different for the major and minor chords,
which means we can not consider these scales generalized diatonics, at
least by my book. But they are intersting.

Also like the 10-note and 8-note versions, this 7-note one is wildly un-
proper. But, as I told Dave, I believe chord coverage can at least
partially replace propriety in many kinds of music. The 10-tone version
is obviously covered, with its 1:1 chord/note ratio. I wrote...

>I haven't checked yet, but with six triads in eight notes, the 380c scale
>has probably got the coverage thing down.

Indeed it does, although, as my rules state, for coverage to work in
place of propriety, the triads must fall on a regular scale-step pattern,
which they do not in the 8-note scale. They do in the 7-note, but only
6 of its 7 tones are covered by its 4 triads.

In short... On paper, I don't like the 8-note version, but I do like the
10-note one, and the 7-note one is somewhere in between. Now to play
with 'em...

-Carl

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Carl Lumma wrote:

>A while back Dave Keenan discussed the 10-tone 7/22 MOS...
>
>0 1 6 7 8 13 14 15 16 21 22
>1 5 1 1 5 1 1 1 5 1

I don't remember describing it in 22-tET. But merely as a chain of major
thirds in general. Possibly I tuned it in 19-tET.

>...It has ten triads (5 major, 5 minor)! We can improve
>the tuning without loosing any chords by taking it from
>41-tET...
>
>0 2 11 13 15 24 26 28 30 39 41
>2 9 2 2 9 2 2 2 9 2
>
>...Both versions are wildly un-proper, but the 41-tET
>version is slightly more stable, at least in log-freq.
>space.

Good. And of course the 19-tET version (1411411141) is slightly more stable
again, but tuned worse.

>Recently, Dave posted this...
>
>>Single chain: No. gene- Notes in m/n
>> No. rators in smallest Gen. is
>> triads Min Min interval proper approx
>>Generator in 8 7-limit 7-limit 2 4 5 MOS with m steps
>>(+-1c) notes RMS err MA err. 3 5 6 >4 notes of n-tET
>>----------------------------------------------------------------------
>>380c M3 6 4.6c 6.0c 5 1 4 16 or 19 6/19, 13/41
>
>...which is 8-out-of the scale above. Its chord/note ratio is lower,
>and the chords no longer fall on a regular pattern of scale steps.
...
>In short... On paper, I don't like the 8-note version, but I do like the
>10-note one, and the 7-note one is somewhere in between. Now to play
>with 'em...

You misunderstood my purpose in giving the number of triads in 8 notes. I'm
sorry I did not make it clear. I'll spell it out in gruesome detail below
so that others may benefit as well.

The quote above only shows one line of a longer table, but of course there
were other generators. I merely wanted an easily understood way of
comparing the triads-per-note for the given generators. There was no
intention to suggest that chains of 8 notes was in any way a "scale" for
any of them. I chose 8 notes because that was the minimum number that gave
more than zero triads for the worst generators listed.

This was exactly the same reason I gave "number of tetrads in 10 notes"
when listing 7-limit generators some months back. Not because I thought 10
notes was a particularly good scale for any of them (although it did happen
to be so for some).

The notes are the fence-posts while the generators are the gaps in between.
There is always one more post than there are gaps. So 8 notes means a chain
of 7 generators.

In general, when considering a single chain of some generator, the complete
x-limit chord [having (x+1)/2 notes] has a certain pattern on the chain.
The pattern has a certain width, given by the maximum of the number of
generators in a dyad over all dyads. As you can read from the table, this
is 5 in the above chain-of-major-thirds example (a chain of 5 generators is
needed to make a 2:3). Of course octave-equivalence is assumed in this
discussion.

The complete utonality always has the same width on the chain as the
otonality. It is simply the otonal pattern rotated 180 degrees. So there
will always be as many otonal as utonal (i.e. as many major as minor in the
5-limit case).

If our scale consists of a chain which is only as wide as our chord (e.g. 6
notes in our case) then we will have only 2 chords (one major one minor).
For every note we add, we get to slide the chord patterns one more position
along the chain, and so we get 2 more chords for every note we add.

Or we lose two chords for every note we remove (assuming we remove it from
an end of the chain, so we still have a single chain)*.

This goes for any odd-limit and for any generator in a single chain. Given
the width "w" of the chord, in generators. A chain of n notes will give you
max(0, 2*(n-w)) chords.

This doesn't apply for N chains spaced 1/N of an octave apart (where N is
typically 2, 3 or 4).

The actual choice of the number of notes for a scale should be based also
on melodic criteria and playability criteria as Carl has done. Myhills
property (or MOS) (only two sizes of scale step) seems to be an almost
essential requirement for good behaviour in these regards.

------------------------------------------------------------------
* Removing a note from the interior of the chain _nearly_ always breaks
more chords than removing one from an end. A notable exception is this
(well known?) interrupted chain of meantone-fifths at the 7-limit.

Db Ab . . F C G D A E B F# . . D# A#
1 3 5 7 otonal pattern
7 5 3 1 utonal pattern

This has 6 tetrads whereas a single chain of 12 notes would only have 4. If
you look at the holes left when the otonal and utonal chord patterns
overlap on this chain you'll see why this one works. It is proper too, but
really has too many notes to be a "scale".

In general it can be very useful to draw these chain diagrams for different
generators, with the chord patterns, so one can _see_ how they work.
-------------------------------------------------------------------

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

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

🔗Carl Lumma <CLUMMA@NNI.COM>

>>A while back Dave Keenan discussed the 10-tone 7/22 MOS...
>>
>>0 1 6 7 8 13 14 15 16 21 22
>>1 5 1 1 5 1 1 1 5 1
>
>I don't remember describing it in 22-tET. But merely as a chain of
>major thirds in general. Possibly I tuned it in 19-tET.

You suggested 19, 22, and 41 in the first post on the subject I can
find.

>You misunderstood my purpose in giving the number of triads in 8 notes.
...
>The quote above only shows one line of a longer table, but of course there
>were other generators. I merely wanted an easily understood way of
>comparing the triads-per-note for the given generators. There was no
>intention to suggest that chains of 8 notes was in any way a "scale" for
>any of them.

Oh, I knew that! Didn't mean to imply otherwise.

>I chose 8 notes because that was the minimum number that gave
>more than zero triads for the worst generators listed.

Nice.

>The complete utonality always has the same width on the chain as the
>otonality. It is simply the otonal pattern rotated 180 degrees. So there
>will always be as many otonal as utonal (i.e. as many major as minor in the
>5-limit case).

A very valuable observation!

>Given the width "w" of the chord, in generators. A chain of n notes will
>give you max(0, 2*(n-w)) chords.*

Sweet.

-Carl

*As you say, for single chains only.

🔗Carl Lumma <CLUMMA@NNI.COM>

>The actual choice of the number of notes for a scale should be
>based also on melodic criteria and playability criteria as Carl
>has done. Myhills property (or MOS) (only two sizes of scale step)
>seems to be an almost essential requirement for good behaviour
>in these regards.

I disagree, and I see you do now, too.

>But N chains spaced 1/N of an octave apart will result in an non-
>Myhill, non-MOS scale, though it still may have only two step sizes.
>There's an "official" name for scales with two step sizes -- I forget
>what it is.
>
>Let's keep forgetting it.

:) Good call. But by "two step sizes", do you mean two sizes of 2nd??

>I think two-step-size is way more important than Myhill's/MOS.

I don't think any of it is important, per se. They do simplify things,
greatly, but I don't believe in any _sudden_ change in goodness from
3 step sizes to 2 (as opposed to 4 to 3). If I had to guess, I'd say
the best measure for this business was Rothenberg's mean variety.

-Carl

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Carl Lumma wrote:

>You suggested 19, 22, and 41 in the first post on the subject I can
>find.

Oh good.

>>There was no
>>intention to suggest that chains of 8 notes was in any way a "scale" for
>>any of them.
>
>Oh, I knew that! Didn't mean to imply otherwise.

Ok. Sorry.

So why did you even consider a chain of 8 (380c major thirds)?

>:) Good call. But by "two step sizes", do you mean two sizes of 2nd??

Er. Yes. What else could "step" mean in this context?

>>I think two-step-size is way more important than Myhill's/MOS.
>
>I don't think any of it is important, per se. They do simplify things,
>greatly, but I don't believe in any _sudden_ change in goodness from
>3 step sizes to 2 (as opposed to 4 to 3).

Ok. I can go along with that. Particularly since one may have two step
sizes that differ by only a few cents. But one tends to map these things
onto the nearest N-tET where N <= 41 before talking about numbers of
different step sizes.

>If I had to guess, I'd say
>the best measure for this business was Rothenberg's mean variety.

I don't recall that one. Is it easy to explain?

-- Dave Keenan
http://dkeenan.com

🔗Carl Lumma <CLUMMA@NNI.COM>

>>:) Good call. But by "two step sizes", do you mean two sizes of 2nd??
>
>Er. Yes. What else could "step" mean in this context?

Well, in our recent discussion of Myhill's property, we used "step sizes"
to mean all generic intervals (not just 2nds).

>>>I think two-step-size is way more important than Myhill's/MOS.
>>
>>I don't think any of it is important, per se. They do simplify things,
>>greatly, but I don't believe in any _sudden_ change in goodness from
>>3 step sizes to 2 (as opposed to 4 to 3).
>
>Ok. I can go along with that. Particularly since one may have two step
>sizes that differ by only a few cents.

Sure, but I think it goes deeper. Basically, I think it's one of those
false positives... just like prime-limit for chords. Many times, two
types of seconds will result in a simple matrix entirely, and in these
cases I believe it's the fact the _entire_ matrix is simple that counts;
nothing special about 2nds. To check this, try a two-2nds-sizes scale
where the 2nds are un-evenly distributed (thus, causing havoc on the
rest of the matrix), and see what you think of the scale. I've tried
this, and concluded that stuff like maximal evenness is suspiciously
tack-on.

>>If I had to guess, I'd say the best measure for this business was
>>Rothenberg's mean variety.
>
>I don't recall that one. Is it easy to explain?

You'll be delighted. It's the average number of sizes per generic
interval, excluding the interval of equivalence. So for Myhill scales,
the mean variety is always 2.

-Carl