n-note MOS of (2n-1)-EDO

Today I was having a closer look at one of the pages of the italian site of Armodue (http://www.armodue.com/risorse.htm) and learned about something the Armodue makers call "semi-equalized Armodue", and which is a temperament generated by two steps of 31EDO (77.4 cents). It is close to 16EDO (one step of which is 75 cents) but, on one hand, with better approximations of the important consonant interval ratios and, on the other hand, not equal but actually a 16-note MOS (of 31EDO, of course). You can see on http://xenharmonic.wikispaces.com/Armodue+theory#Semi-equalized%20Armodue

A little point is that the xenwiki page of 31edo mentions this temperament as "Valentine temperament" - so there is a terminology issue here that might be kept in memory...

But another point was that I felt reminded on Ozan Yarman's 79-tone (or 80-tone, respectively) MOS of 159EDO. I wonder whether there is a more general pattern here - n-note MOS of (2n-1)-EDO - that would be worth being explored?
--
Hans Straub

--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
>
> Today I was having a closer look at one of the pages of the italian site of Armodue (http://www.armodue.com/risorse.htm) and learned about something the Armodue makers call "semi-equalized Armodue", and which is a temperament generated by two steps of 31EDO (77.4 cents).

I edited that some more to reflect the fact that we are talking about a MOS and one which could use other tunings, such as 3\46, 5\77 etc.

> A little point is that the xenwiki page of 31edo mentions this temperament as "Valentine temperament" - so there is a terminology issue here that might be kept in memory...

A 16-note MOS is a scale, not a temperament.

On Wed, Jul 6, 2011 at 1:57 PM, hstraub64 <straub@...> wrote:
>
> Today I was having a closer look at one of the pages of the italian site of Armodue (http://www.armodue.com/risorse.htm) and learned about something the Armodue makers call "semi-equalized Armodue", and which is a temperament generated by two steps of 31EDO (77.4 cents). It is close to 16EDO (one step of which is 75 cents) but, on one hand, with better approximations of the important consonant interval ratios and, on the other hand, not equal but actually a 16-note MOS (of 31EDO, of course). You can see on http://xenharmonic.wikispaces.com/Armodue+theory#Semi-equalized%20Armodue
>
> A little point is that the xenwiki page of 31edo mentions this temperament as "Valentine temperament" - so there is a terminology issue here that might be kept in memory...
>
> But another point was that I felt reminded on Ozan Yarman's 79-tone (or 80-tone, respectively) MOS of 159EDO. I wonder whether there is a more general pattern here - n-note MOS of (2n-1)-EDO - that would be worth being explored?

Yeah, I've been thinking about that for a while. So you end up with:

5&6 - machine temperament, 11-EDO primary chromatic EDO, 17-EDO
primary enharmonic EDO
6&7 - either tetracot or 2.5.9 meantone temperament, 13-EDO primary
chromatic EDO, 19-EDO primary enharmonic (meantone) 20-EDO primary
enharmonic (tetracot)
7&8 - porcupine temperament, 15-EDO primary chromatic EDO, 22-EDO
primary enharmonic EDO
8&9 - progression temperament, 17-EDO primary chromatic EDO, 26-EDO
primary enharmonic EDO
9&10 - negri temperament, 19-EDO primary chromatic EDO, 28-EDO and
29-EDO are both enharmonic EDOs
10&11 - miracle temperament, 21-EDO primary chromatic EDO, 31-EDO
primary enharmonic EDO

This is a bit arbitrary because there are a few options for 6&7, and I
think there are a few for 10&11 as well, so I've listed what I feel is
the most important temperament in each case.

This is related to the work we did on MODMOS a while ago. Temperaments
like these have two diatonic-sized or "albitonic" scales, with which
the smaller one might be said to be a "haplotonic" scale, analogous to
meantone's pentatonic scale.

At any rate the "primary chromatic" EDO may not be the one with the
best tuning (definitely not in the case of porcupine or miracle), but
it does serve the purpose of being the simplest EDO that enables full
chromatic modulation around the MOS in question. In that sense 15-EDO
is cognitively simpler than 22-EDO for porcupine, but the tradeoff is
that 22-EDO is more accurate. This is analogous to how 12-EDO is
simpler than 19-EDO, but 19-EDO is more accurate. Simplicity may offer
additional benefits in that the extra tempering can lead to simpler
and more useful comma pumps, once you get over the higher tuning
error.

I've been exploring a new idea these days, which is that the
categorical perception of intervals is really what matters, with
purity of tuning being the icing on the cake. It might be a good idea
to first "establish" a tuning in the listener's mind by playing a
composition in a "purer" tuning (so start with quarter-comma meantone
or TOP-RMS porcupine or TOP-RMS semaphore), and then translate that to
the simpler EDO once the understanding of the temperament's "logic" is
set up, which likely means the categorical perception of the
temperament's intervals (then dumb it down to 12-EDO or 15-EDO or
14-EDO).

-Mike

"hstraub64" <straub@...> wrote:
> Today I was having a closer look at one of the pages of
> the italian site of Armodue
> (http://www.armodue.com/risorse.htm) and learned about
> something the Armodue makers call "semi-equalized
> Armodue", and which is a temperament generated by two
<snip>

> A little point is that the xenwiki page of 31edo mentions
> this temperament as "Valentine temperament" - so there is
> a terminology issue here that might be kept in memory...

The Wayback Machine only has that page from 2004. Whether
what they describe is Valentine temperament or not, I think
Valentine was Valentine by 2004.

> But another point was that I felt reminded on Ozan
> Yarman's 79-tone (or 80-tone, respectively) MOS of
> 159EDO. I wonder whether there is a more general pattern
> here - n-note MOS of (2n-1)-EDO - that would be worth
> being explored?

MOS of an MOS is an idea that's out there. Maybe it goes
back to Erv Wilson.

Graham