chain-of-minor-third scale revisited

Earlier this year, Graham Breed performed a computer search for
the "best" linear temperaments. "Best" is roughly defined as meaning
the greatest number of complete, consonant chords, tuned as closely
to JI as possible, for a given number of notes in a melodically
coherent scale (MOS, with interval of repetition equal to the octave,
or one-half octave, or one-third octave, etc). His search was
performed by "cross-Breeding" a selection of consistent ETs in
various ways.

One might have expected the #1 rank in the 5-limit to go to meantone
temperament, with its richly harmonizing diatonic and chromatic
scales. But meantone is actually only #2!

#1 is the Stearns/Breed/Keenan chain-of-minor-thirds temperament. The
generator is a minor third of 317 cents. Thus it is approximated
quite well 19-tone equal temperament. There is a 7-tone MOS, which in
19-tET has steps 1 4 1 4 4 1 4, or degrees 0 1 5 6 10 14 15 (19). A
bit of music in this scale can be heard here:

http://www.angelfire.com/mo/oljare/images/skopnytt.mid

A lattice reveals the major and minor triads:

15
\
\
\
\
1
/ \
/ \
/ \
/ \
14---------6
\ /
\ /
\ /
\ /
0
\
\
\
\
5

There are only two complete triads here. Why does this beat meantone,
where a 7-tone scale has six complete triads? Because the maximum
error from JI here is 1.4 cents -- almost four times less than the
maximum error in meantone. More triads can be found in the next-
higher MOS, an 11-tone one, which gives four additional major and
four additional minor triads. The scale in 19-tET has steps of 1 3 1
1 3 1 1 3 1 1 3, or scale degrees 0 1 4 5 6 9 10 11 14 15 16 (19).
The lattice:

10
/ \
/ \
/ \
/ \
4--------15
\ / \
\ / \
\ / \
\ / \
9---------1
\ / \
\ / \
\ / \
\ / \
14---------6
\ / \
\ / \
\ / \
\ / \
0--------11
\ / \
\ / \
\ / \
\ / \
5--------16
\ /
\ /
\ /
\ /
10

It's latticed and explored in greater detail, including 9-limit
considerations, at
<http://www.uq.net.au/~zzdkeena/Music/ChainOfMinor3rds.htm>). This
scale, though, is highly improper, with many of the "thirds" smaller
than many of the "seconds" . . .

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> I'm not sure if I had anything to do with the origins of this scale
or
> not... I forget (Dave Keenan has a beautiful page on it).
>
> However, I did post on the [7,4] 11-tone quite a bit, and another
way
> to possibly think of this in a 5-limit context is as a chain of 5/3s
> that when converted from 1D to 2D becomes somewhat analogous to the
> 7-tone 4:5:6, only here it's an 11-tone 3:4:5.
>
> So tempering out the 15625/15552 theoretically gives five consonant
> (or alike) 3:4:5s and four

five?

> alike 1/(5:4:3)s.
>
> Here's the 2D 7-tone MOS,
>
> (*)--*---*---*
> \ / \
> *---*---*---*
>
> and here's the 2D 11-tone MOS,
>
> (*)--*---*---*---*---*
> \ / \ / \ / \ / \ / \
> *---*---*---*---*---*
>
>
> --Dan Stearns

Isn't this exactly what was in my post? What new insight have you
added -- I'm missing it.
>
> (btw, I liked the 22-tet microfest mp3s John Starrett posted earlier
> today, especially the '60s-ish pop ballad one)

Which one is that??

Sorry, but MINOR or MAJOR third's have never be one
interval referance or one just intonation intervals
!!!
It's one completly wrong way to follow !
The third is and has everytime one non consonant
interval ! Remember it, pls :)
And stop to go to...nothing!

Dimitrov :)

--- Paul Erlich <paul@stretch-music.com> a �crit�: >
--- In tuning@y..., "D.Stearns" <STEARNS@C...>
> wrote:
> > Paul,
> >
> > I'm not sure if I had anything to do with the
> origins of this scale
> or
> > not... I forget (Dave Keenan has a beautiful page
> on it).
> >
> > However, I did post on the [7,4] 11-tone quite a
> bit, and another
> way
> > to possibly think of this in a 5-limit context is
> as a chain of 5/3s
> > that when converted from 1D to 2D becomes somewhat
> analogous to the
> > 7-tone 4:5:6, only here it's an 11-tone 3:4:5.
> >
> > So tempering out the 15625/15552 theoretically
> gives five consonant
> > (or alike) 3:4:5s and four
>
> five?
>
> > alike 1/(5:4:3)s.
> >
> > Here's the 2D 7-tone MOS,
> >
> > (*)--*---*---*
> > \ / \
> > *---*---*---*
> >
> > and here's the 2D 11-tone MOS,
> >
> > (*)--*---*---*---*---*
> > \ / \ / \ / \ / \ / \
> > *---*---*---*---*---*
> >
> >
> > --Dan Stearns
>
> Isn't this exactly what was in my post? What new
> insight have you
> added -- I'm missing it.
> >
> > (btw, I liked the 22-tet microfest mp3s John
> Starrett posted earlier
> > today, especially the '60s-ish pop ballad one)
>
> Which one is that??
>
>

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--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> <<Isn't this exactly what was in my post? What new insight have you
> added -- I'm missing it.>>
>
> Well my point for whatever it's worth was that you could base it on
> the 3:4:5

Don't know what that means (as opposed to basing it on the 4:5:6),
but I'm confident that your way of seeing it is as good as any. If
you'd care to elaborate in as simple terms as you can, I'd welcome it.

> so that it's a PB where the UVs are 15625/15552 and
> 16/15--or did you already mention that!

I didn't mention that, and it's very important -- thanks! In
particular, the kleisma or 15625/15552 is tempered out (so it's the
commatic unison vector), while 16/15 is not (so it's the chromatic
unison vector).

> >
> > Which one is that??
>
>
> One of them.
>
Sorry -- what was the context?

--- In tuning@y..., Latchezar Dimitrov <latchezar_d@y...> wrote:
> Sorry, but MINOR or MAJOR third's have never be one
> interval referance or one just intonation intervals
> !!!

What do you mean?

> It's one completly wrong way to follow !
> The third is and has everytime one non consonant
> interval !

The third is non consonant?

> Remember it, pls :)
> And stop to go to...nothing!

Thanks for your open-mindedness.