Optimal Just tunings for 12 notes

Hey all,

I think I remember seeing something on monz's site,with a Paul Erlich
reference? Or was it Carl Lumma?

Anyway, the idea was that there was a maximum number of consonances possible
with a certain JI scale....can anyone point me to that reference...I tried to
backtrack, but can't find it.....

Best,
Aaron.

--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

hi Aaron,

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
>
>
> Hey all,
>
> I think I remember seeing something on monz's site,
> with a Paul Erlich reference? Or was it Carl Lumma?
>
> Anyway, the idea was that there was a maximum number
> of consonances possible with a certain JI scale....
> can anyone point me to that reference...I tried to
> backtrack, but can't find it.....

i thought you had in mind a Dictionary page about ASSes
(anomalous saturated suspensions), but i don't have that
defined yet!

i did find this:
http://sonic-arts.org/td/lumma/carl-cps.htm

i was about to search my site for some other possibilities,
but i'm having strange internet troubles right now.
something probably got messed up in the fire. anyway,
hope that one helps.

if you search the sonic-arts.org/td directory i think
that's where you'll find the page you want. it's an old
tuning-list post from Carl.

- monz

>Hey all,
>
>I think I remember seeing something on monz's site,with a Paul Erlich
>reference? Or was it Carl Lumma?
>
>Anyway, the idea was that there was a maximum number of consonances
>possible with a certain JI scale....can anyone point me to that
>reference...I tried to backtrack, but can't find it.....

Hard to pin down. This general goal is lost in antiquity. I famously
asked about 7-limit consonances in a 12-tone scale, around 1998/99.
About two years later, Joseph Pehrson famously asked about (any limit?)
consonances in a 19-tone scale. The answer to both questions involved
tempering out or ignoring the 225:224 interval. Joseph wound up with a
21-tone scale in the Miracle temperament called Blackjack, and has been
working with it ever since. I can post the scales I wound up with if
you like.

If you refuse to temper and don't care about having pairs of pitches
in your tuning that are only a comma apart, the answer is generally
any convex, compact sphericalish thing on the lattice. This includes
things like Partch's tonality diamond and Wilson's CPS and stellated-
CPS scales (which Kraig Grady and others have worked with).

If you don't mind limited and/or a more modal style of modulation, and
you like high-limit sonorities, you can have 100% of your intervals
just by using harmonic series segments. Denny Genovese has made some
very compelling (to me) music with them.

-Carl

hi Aaron,

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Aaron,
>
> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
> wrote:
> >
> >
> > Hey all,
> >
> > I think I remember seeing something on monz's site,
> > with a Paul Erlich reference? Or was it Carl Lumma?
> >
> > Anyway, the idea was that there was a maximum number
> > of consonances possible with a certain JI scale....
> > can anyone point me to that reference...I tried to
> > backtrack, but can't find it.....
>
>
>
> i thought you had in mind a Dictionary page about ASSes
> (anomalous saturated suspensions), but i don't have that
> defined yet!
>
>
> i did find this:
> http://sonic-arts.org/td/lumma/carl-cps.htm
>
>
> i was about to search my site for some other possibilities,
> but i'm having strange internet troubles right now.
> something probably got messed up in the fire. anyway,
> hope that one helps.

strange ... now my internet is working OK again.

> if you search the sonic-arts.org/td directory i think
> that's where you'll find the page you want. it's an old
> tuning-list post from Carl.

look at "Topic No. 2" on this old Tuning List Digest:

http://sonic-arts.org/td/1599.htm

Carl presents two 12-tone 7-limit scales that have 30 and
31 consonances.

then in "Topic No. 9" Paul Hahn illustrates one of 24
12-tone scales which have 32 consonances.

-monz

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> If you don't mind limited and/or a more modal style of
> modulation, and you like high-limit sonorities, you can
> have 100% of your intervals just by using harmonic series
> segments. Denny Genovese has made some very compelling
> (to me) music with them.

i write pieces like this when i compose on my Rational Guitar,
which has staggered frets giving up to the 27th harmonic.
examples:

Kuh-meah [712 Kb]
http://sonic-arts.org/monzo/worklist/kuh-meah.mp3

Kraigian [548 Kb]
http://sonic-arts.org/monzo/worklist/kraigian.mp3

(this is one of the main things i have in common
with Denny.)

i particularly delight in creating analogues of typical
5-limit JI / meantone chords, but with higher-prime
intervals substituted for the familiar 5-limit ones.

for instance: the skewed "major-7th" triad at 0:18
in "Kuh-meah", D< : A< : C> in my 72edo-HEWM notation,
with the proportions 14:21:26 (resolving into the
"tonic" chord E : G#- : B = 8:20:24).

the lower interval is simply a 2:3 "perfect-5th".

the upper interval is a 21:26 = ~369.7467544 cents,
a narrow 7,13-al "major-3rd".

the bounding interval, which is the one that gets me hot,
is 7:13 = ~1071.701755 cents, a narrow 7,13-al "major-7th".

i arrive at such interesting cofigurations simply because
all of the notes on the guitar are harmonics of the deepest E.

the only other choice available for a "D major-7th" chord
was to use "C#" 27:16 and make the "major-7th" the 3,7-al
14:27 = ~1137.039096 cents, which was much bigger than what
i wanted.

-monz

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
>
>
> Hey all,
>
> I think I remember seeing something on monz's site,

yeah, probably go to sonic-arts.org, and click on "tuning digest
archives" -- that's the very end of the mills archive (right before
the archives you can find here on yahoo) and contains the discussion
about maximizing the number of just 7-limit consonances in a 12-tone
7-limit just scale.

>with a Paul Erlich
> reference? Or was it Carl Lumma?
>
> Anyway, the idea was that there was a maximum number of consonances
possible
> with a certain JI scale....can anyone point me to that
reference...I tried to
> backtrack, but can't find it.....

you probably mean this; paul hahn gave the final answer:

http://www.sonic-arts.org/td/1599.htm#1599-9

but you'll want to read the surrounding archives, including the
beginning of the yahoo ones, to understand the full discussion.

>> If you don't mind limited and/or a more modal style of
>> modulation, and you like high-limit sonorities, you can
>> have 100% of your intervals just by using harmonic series
>> segments. Denny Genovese has made some very compelling
>> (to me) music with them.
>
>i write pieces like this when i compose on my Rational Guitar,
>which has staggered frets giving up to the 27th harmonic.
>examples:
>
>Kuh-meah [712 Kb]
>http://sonic-arts.org/monzo/worklist/kuh-meah.mp3
>
>Kraigian [548 Kb]
>http://sonic-arts.org/monzo/worklist/kraigian.mp3
>
>
>(this is one of the main things i have in common
>with Denny.)

Awesome examples!!!

-CARL

hi Carl,

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> If you don't mind limited and/or a more modal style of
> >> modulation, and you like high-limit sonorities, you can
> >> have 100% of your intervals just by using harmonic series
> >> segments. Denny Genovese has made some very compelling
> >> (to me) music with them.
> >
> >i write pieces like this when i compose on my Rational Guitar,
> >which has staggered frets giving up to the 27th harmonic.
> >examples:
> >
> >Kuh-meah [712 Kb]
> >http://sonic-arts.org/monzo/worklist/kuh-meah.mp3
> >
> >Kraigian [548 Kb]
> >http://sonic-arts.org/monzo/worklist/kraigian.mp3
> >
> >
> >(this is one of the main things i have in common
> >with Denny.)
>
> Awesome examples!!!

thanks, glad you liked them. actually, i expected you
to like them, since you like some of my other stuff a lot,
and these two really are "harmonic-series".

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

/tuning/topicId_48249.html#48256

>
> i write pieces like this when i compose on my Rational Guitar,
> which has staggered frets giving up to the 27th harmonic.
> examples:
>
> Kuh-meah [712 Kb]
> http://sonic-arts.org/monzo/worklist/kuh-meah.mp3
>
> Kraigian [548 Kb]
> http://sonic-arts.org/monzo/worklist/kraigian.mp3
>

***These really do have a great sound, Monz!

jP