Proportional commas

Recently we briefly discussed proportions between tempered commas and step sizes. Here is a music example which happens to take advantage of simple proportions between commas;

http://soundcloud.com/cameron-bobro/eveninghorizon-cbobro

Namely, the diesis, minor chroma, and septimal minor chroma are all tempered to the same size, which in turn is twice the tempered syntonic comma and half the minor diatonic step. At the same time, the intervals used in the tuning remain very close to Just intonations. The ability to "temper back in" commas as step sizes is a great feature of 41 equal divisions of the octave, along with the good approximations to rational intervals.

Very nice little appetizer--I would love an entrée of the same flavors!

AKJ

On Tue, May 24, 2011 at 8:22 AM, lobawad <lobawad@...> wrote:

> Recently we briefly discussed proportions between tempered commas and step
> sizes. Here is a music example which happens to take advantage of simple
> proportions between commas;
>
> http://soundcloud.com/cameron-bobro/eveninghorizon-cbobro
>
> Namely, the diesis, minor chroma, and septimal minor chroma are all
> tempered to the same size, which in turn is twice the tempered syntonic
> comma and half the minor diatonic step. At the same time, the intervals
> used in the tuning remain very close to Just intonations. The ability to
> "temper back in" commas as step sizes is a great feature of 41 equal
> divisions of the octave, along with the good approximations to rational
> intervals.
>
>
>
> ------------------------------------
>
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--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

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

> Namely, the diesis, minor chroma, and septimal minor chroma are all tempered to the same size, which in turn is twice the tempered syntonic comma and half the minor diatonic step.

The ratio of the minor chroma to the diesis is (25/24)/(128/125) = 3125/3072, and presuming by "septimal minor chroma" you mean 28/27 (seems like a good name, anyone have a better one?) then (25/24)/(28/27) = 225/224 and this is magic temperament. It would be nice to use your Soundcloud stuff as examples of magic and wuerschmidt temperaments on the Xenwiki.

The "Gallery of Just Intervals" here:

http://xenharmonic.wikispaces.com/Gallery+of+Just+Intervals

calls 21/20 "minor semitone, (large) (septimal) chromatic semitone", by the way.

On Tue, May 24, 2011 at 12:34 PM, genewardsmith
<genewardsmith@...t> wrote:
>
> --- In tuning@yahoogroups.com, "lobawad" <lobawad@...> wrote:
>
> > Namely, the diesis, minor chroma, and septimal minor chroma are all tempered to the same size, which in turn is twice the tempered syntonic comma and half the minor diatonic step.
>
> The ratio of the minor chroma to the diesis is (25/24)/(128/125) = 3125/3072, and presuming by "septimal minor chroma" you mean 28/27 (seems like a good name, anyone have a better one?) then (25/24)/(28/27) = 225/224 and this is magic temperament. It would be nice to use your Soundcloud stuff as examples of magic and wuerschmidt temperaments on the Xenwiki.
>
> The "Gallery of Just Intervals" here:
>
> http://xenharmonic.wikispaces.com/Gallery+of+Just+Intervals
>
> calls 21/20 "minor semitone, (large) (septimal) chromatic semitone", by the way.

I'm not actually here right now, as I'm extremely busy.

Yeah, I was going to say magic temperament, but I didn't know what
"minor diatonic step" meant. Is "minor diatonic step" 16/15, and
"major diatonic step" 27/25 or something? Either way, the fact that
magic equates 128/125 and 25/24 is definitely one of its strongest
features, even more so in my opinion than the more usual and cliche
setup whereby 5 5/4's equals 3/1.

Since we're now exploring this paradigm of what small intervals
temperaments equate with one another - rather than the more usual
paradigm of what the generator for a temperament is and how many
generators it takes you to get to 3/1 or 5/1 or some other such
milestone - it might be useful for me to come up with a list showing,
for any 5-limit regular temperament, what intervals in the set {10/9,
16/15, 25/24, 27/25, 81/80, 128/125} are equated with one another for
a particular temperament. That might really be helpful for folks to
really intuitively understand the logic behind each 5-limit
temperament. For example, take negri - what does negri equivocate with
25/24? What about 16/15? etc.

Lastly in my stream of consciousness post, the fact that 19-tet
supports both magic and negri only bolsters its status in my mind as
The Best Meantone Ever.

Again, notably absent from this list am I, for I have such an enormous
amount of work to do.

-Mike

By "septimal minor chroma" I was thinking 36/35, but now that you mention it, probably you're right that the term better suggests 28/27.
Maybe "quintaseptimal chroma" would be better for 36/35, as it separates both the subminor septimal third or 7/6 from the quintal minor third of 6/5, and the septimal supramajor third of 9/7 from the quintal major third of 5/4.

In the case of septimal "magic" temperament tuned to 41 equal divisions of the octave, which is how this bit is done, the 875/864 comma between 36/35 and 25/24 (the interval between 6/5 and 5/4, for those following along) is tempered out*, equating the two intervals and that's an essential thing here: the septimal interals are easily reached by the same chromatic step size taken between the quintal thirds. Notice that the very first intervals appearing in the lower pentachord in the "magic" tuning scheme are the subminor third, quintal minor third and quintal major third, seperated from each other by this uniform tempered minor chroma.

I'll get those links into the wiki- there should also be a page on proportions betweens commas and between commas and stepsizes, as that is a very natural usage of the "regular temperament paradigm", and probably more easily grasped than comma pumps.

*not something I'd intuitively expect, as the comma is nearly the size of a 41-edo step size, and it's impressive how such a large comma is spread out such that the rational approximations remain so audibly decent.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "lobawad" <lobawad@> wrote:
>
> > Namely, the diesis, minor chroma, and septimal minor chroma are all tempered to the same size, which in turn is twice the tempered syntonic comma and half the minor diatonic step.
>
> The ratio of the minor chroma to the diesis is (25/24)/(128/125) = 3125/3072, and presuming by "septimal minor chroma" you mean 28/27 (seems like a good name, anyone have a better one?) then (25/24)/(28/27) = 225/224 and this is magic temperament. It would be nice to use your Soundcloud stuff as examples of magic and wuerschmidt temperaments on the Xenwiki.
>
> The "Gallery of Just Intervals" here:
>
> http://xenharmonic.wikispaces.com/Gallery+of+Just+Intervals
>
> calls 21/20 "minor semitone, (large) (septimal) chromatic semitone", by the way.
>

Thanks- I'll see what I can do about that. There already more, but one part I'm dreading logistically, because it turns out that what I cooked up over my morning beer requires a very different subset of 41-edo, because it's in what's probably "miracle" rather than "magic" temperament.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@...> wrote:
>
> Very nice little appetizer--I would love an entrée of the same flavors!
>
> AKJ
>
> On Tue, May 24, 2011 at 8:22 AM, lobawad <lobawad@...> wrote:
>
> > Recently we briefly discussed proportions between tempered commas and step
> > sizes. Here is a music example which happens to take advantage of simple
> > proportions between commas;
> >
> > http://soundcloud.com/cameron-bobro/eveninghorizon-cbobro
> >
> > Namely, the diesis, minor chroma, and septimal minor chroma are all
> > tempered to the same size, which in turn is twice the tempered syntonic
> > comma and half the minor diatonic step. At the same time, the intervals
> > used in the tuning remain very close to Just intonations. The ability to
> > "temper back in" commas as step sizes is a great feature of 41 equal
> > divisions of the octave, along with the good approximations to rational
> > intervals.
> >
> >
> >
> > ------------------------------------
> >
> > You can configure your subscription by sending an empty email to one
> > of these addresses (from the address at which you receive the list):
> > tuning-subscribe@yahoogroups.com - join the tuning group.
> > tuning-unsubscribe@yahoogroups.com - leave the group.
> > tuning-nomail@yahoogroups.com - turn off mail from the group.
> > tuning-digest@yahoogroups.com - set group to send daily digests.
> > tuning-normal@yahoogroups.com - set group to send individual emails.
> > tuning-help@yahoogroups.com - receive general help information.
> > Yahoo! Groups Links
> >
> >
> >
> >
>
>
> --
> Aaron Krister Johnson
> http://www.akjmusic.com
> http://www.untwelve.org
>

--- In tuning@yahoogroups.com, "lobawad" <lobawad@...> wrote:
>
> By "septimal minor chroma" I was thinking 36/35, but now that you mention it, probably you're right that the term better suggests 28/27.
> Maybe "quintaseptimal chroma" would be better for 36/35, as it separates both the subminor septimal third or 7/6 from the quintal minor third of 6/5, and the septimal supramajor third of 9/7 from the quintal major third of 5/4.

I'd suggest "septimal quarter tone" for 36/35:

Still wondering about the temperament examples.

"Septimal quartertone" is good, though it doesn't link to the "chroma" nature of 36/35 (major to minor a chroma, minor to subminor a chroma, makes sense to me).

What were you wondering about the temperament examples? I'll put them in the wiki, and the last one was indeed "magic" temperament, if that was what you were wondering.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "lobawad" <lobawad@> wrote:
> >
> > By "septimal minor chroma" I was thinking 36/35, but now that you mention it, probably you're right that the term better suggests 28/27.
> > Maybe "quintaseptimal chroma" would be better for 36/35, as it separates both the subminor septimal third or 7/6 from the quintal minor third of 6/5, and the septimal supramajor third of 9/7 from the quintal major third of 5/4.
>
> I'd suggest "septimal quarter tone" for 36/35:
>
> Still wondering about the temperament examples.
>

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

> What were you wondering about the temperament examples? I'll put them in the wiki, and the last one was indeed "magic" temperament, if that was what you were wondering.

I was asking for permission to add them to the wiki.

Sure- the Wuerschmidt example has a good explanation, I think, I can put something about the temperament into the text for the "magic" example.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "lobawad" <lobawad@> wrote:
>
> > What were you wondering about the temperament examples? I'll put them in the wiki, and the last one was indeed "magic" temperament, if that was what you were wondering.
>
> I was asking for permission to add them to the wiki.
>