Term temperament

If you take the dent comma (225/224)^3/(81/80) and optimize a tuning,
and then look for commas which it shrinks a lot, 32805/32768 is the
standout. Taking both commas together gives the 159&171 temperament, a
strong microtomperament which has shown up a number of times on my
lists on tuning-math, but which never got a name because of its high
complexity. I'm proposting "term", from "tertia marvel", but if I knew
what the heck the "dec" in Tom's "tertiadec" meant, that sounds like a
temperament name.

Anyway, it's compatible with 12-et and has a fifth as a generator and
400 cents as a period, which might be helpful from an adaptive point of
view. 171 is fine as an edo, though it can be improved on if you wish
(eg, 1209.)

Wedgie: <<3 -24 -54 -45 -94 -58||
Generator mapping: [<3 5 5 4|, <0 -1 8 18|]

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> If you take the dent comma (225/224)^3/(81/80) and optimize a tuning,
> and then look for commas which it shrinks a lot, 32805/32768 is the
> standout. Taking both commas together gives the 159&171 temperament, a
> strong microtomperament

Aha! I get my name in after all

> which has shown up a number of times on my
> lists on tuning-math, but which never got a name because of its high
> complexity. I'm proposting "term", from "tertia marvel", but if I knew
> what the heck the "dec" in Tom's "tertiadec" meant, that sounds like a
> temperament name.

Gene, it looks like you invented this name yourself:

http://bahamas.eshockhost.com/~xenharmo/commalist.htm

- one of only two Google matches for 'tertiadec'. I plead not guilty.

As for the rest, complexity is the thing!! I'm looking for the
opposite - the simplest notation for specifying nice chords well enough.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

I guess my reply to this didn't get posted.

> Gene, it looks like you invented this name yourself:
>
> http://bahamas.eshockhost.com/~xenharmo/commalist.htm
>
> - one of only two Google matches for 'tertiadec'.

I'd forgotten that. The name comes from the fact that two other
very strong microtemperaments temper it out--tetiaseptal, the 140%171
temperament, and enneadecal, the 152&171 temperament.

171 = 9*19, and so you can look at it as a stack of 19 9-et cycles (the
ennealimma point of view) or a stack of 9 19-et cycles (the enneadecal
point of view.) The period of enneadecal is 1/19 octave, and the
generator can be taken as 225/224, which in 171-et is one step. You can
do something like George's 217 thing with 171; a stack of 19-et cyckes
(notated in meantone) with adjustments up and down by a 225/224, three
ofwhich make an 81/80.

In 171, though not in enneadecal or tertiaseptal, there's also the
landscape comma around, which says two 126/125 give a 64/63. Moreover,
the tertiadec comma not only is equal to (225/224)^3/(81/80) it is also
(126/125)/(225/224)^2, so that two of the basic comma unit gives a
126/125. Term temperament can in fact be defined as the temperament
which tempers out both landscape and tertiadec, so that all of the
above is true. And, as I said, you can even push the accuracy a little
higher than 171-et if you want.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> Gene, it looks like you invented this name yourself:
>
> http://bahamas.eshockhost.com/~xenharmo/commalist.htm
>
> - one of only two Google matches for 'tertiadec'. I plead not
guilty.

Ah! You are right. I was naming commas after the conjunction of two
temperaments, in this case two other very strong tempeaments which
temper out the comma--tertiaseptal, the 140&171 temperament, and
enneadecal, the 152&171 temperament.

> As for the rest, complexity is the thing!! I'm looking for the
> opposite - the simplest notation for specifying nice chords well
enough.

That would be yet another strong temperament which tempers out the
tertiadec comma--meantone. Actually, 171 factors as 171 = 9*19, and so
can be thought of as 19 9-ets stacked (the ennealimmal point of view)
or 9 19-ets stacked (the enneadecal point of view.) The latter has
the advantage that you already have a notation for 19-et, and hence
adjustments up or down can be taken from there.

> 171 = 9*19, and so you can look at it as a stack of 19 9-et
> cycles (the ennealimma point of view) or a stack of 9 19-et
> cycles (the enneadecal point of view.) The period of
> enneadecal is 1/19 octave, and the generator can be taken
> as 225/224, which in 171-et is one step.

None of these are term (<<3 -24 -54 -45 -94 -58||)?

-Carl

> Generator mapping: [<3 5 5 4|, <0 -1 8 18|]

Uh, wouldn't this be

< 3 0 0 4 |
< 0 -1 8 18 |

?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > 171 = 9*19, and so you can look at it as a stack of 19 9-et
> > cycles (the ennealimma point of view) or a stack of 9 19-et
> > cycles (the enneadecal point of view.) The period of
> > enneadecal is 1/19 octave, and the generator can be taken
> > as 225/224, which in 171-et is one step.
>
> None of these are term (<<3 -24 -54 -45 -94 -58||)?

A dominating equal temperament, which 171 certainly is in the 7 limit,
will be at the confluence of a lot of notable systems. In this case we
have ennealimmal, sesquiqartififths, tertiaseptal, enneadecal, pontiac,
term, neptune, 58&171, 22&171, 46&171, 94&171.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Generator mapping: [<3 5 5 4|, <0 -1 8 18|]
>
> Uh, wouldn't this be
>
> < 3 0 0 4 |
> < 0 -1 8 18 |

If you wanted to temper out 32805 and 11250000/343, which personally I
never do. :)

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> >
> > > Generator mapping: [<3 5 5 4|, <0 -1 8 18|]
> >
> > Uh, wouldn't this be
> >
> > < 3 6 0 4 |
> > < 0 -1 8 18 |
>
> If you wanted to temper out 32805 and 11250000/343, which
> personally I never do. :)

Oh sorry, I forget this isn't octave equivalent. But
surely then we must have

< 3 6 -3 -14 |
< 0 -1 8 18 |

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> Oh sorry, I forget this isn't octave equivalent. But
> surely then we must have
>
> < 3 6 -3 -14 |
> < 0 -1 8 18 |

We must have that if the generator is fourth, but not otherwise. Take
400 cents away from the fourth, and you have a semitone generator of 98
cents.

> > Oh sorry, I forget this isn't octave equivalent. But
> > surely then we must have
> >
> > < 3 6 -3 -14 |
> > < 0 -1 8 18 |
>
> We must have that if the generator is fourth, but not otherwise.
> Take 400 cents away from the fourth, and you have a semitone
> generator of 98 cents.

Aha! - C .