Associated chord in a linear temperament

In a chord in a linear temperament, there is always a unique chord
interval which is widest in terms of generator steps between chord
elements. We can define a chord associated to the given chord by
inverting in a transposition such that this widest element is preserved.

Has anyone given a name to this? An example would be C major and a
minor in meantone. Note that I mean linear in the strict sense; the
definition does not apply to all rank-two temperaments.

>In a chord in a linear temperament, there is always a unique chord
>interval which is widest in terms of generator steps between chord
>elements. We can define a chord associated to the given chord by
>inverting in a transposition such that this widest element is preserved.

Why not use the narrowest element?

>Has anyone given a name to this? An example would be C major and a
>minor in meantone.

Um, isn't that keeping the narrowest element (the fifth)?

>Note that I mean linear in the strict sense; the
>definition does not apply to all rank-two temperaments.

So linear temperaments are rank 2 these days? Refresh my memory.

-Carl

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

> >In a chord in a linear temperament, there is always a unique chord
> >interval which is widest in terms of generator steps between chord
> >elements. We can define a chord associated to the given chord by
> >inverting in a transposition such that this widest element is
preserved.
>
> Why not use the narrowest element?

Because the widest element seems the most interesting, and in any case
is unique. If c is a chord and ~c is the associated chord, then ~c is
well-defined and ~~c = c. In any MOS, if you have a chain of n chords
of type c, you have a corresponding chain of n chords of type ~c, and
you can play a sort of major to relative minor type of game. The
nature of ~c, if c is a p-limit n-ad, is determined by the most
complex element of the corresponding diamond. In 5-limit meantone, the
most complex element is the major third, and hence that is preserved,
leading to the diatonic scale of three major triads defining the major
key, and three minor triads the minor key. In heemiwuerschidt, where
3/2 is more complex than 5/4 and 7/4, the corresponding tetrad to a
otonal tetrad on a given root will be the utonal tetrad on the same
root, and hemiwuerschmidt leads to scale where both the major and
minor tetrads exist for a given root if one or the oher exists. Hence,
this seems to define an interesting property of any linear temperament.

> >Has anyone given a name to this? An example would be C major and a
> >minor in meantone.
>
> Um, isn't that keeping the narrowest element (the fifth)?

No, C major is CEG, and a minor is ACE, so the CE major third is the
preserved element for ~C major = a minor. Of course, for
CEGA# it's entirely different, as C-A# is now the preserved element,
and so ~CEGA# = D#F#A#C.

> >Note that I mean linear in the strict sense; the
> >definition does not apply to all rank-two temperaments.
>
> So linear temperaments are rank 2 these days? Refresh my memory.

I thought we decided that linear temperaments are rank 2 with a period
of an octave. Paul wants it that way on historical grounds, and I sort
of like it since it allows us to use the two different terms to mean
different things. The problem is that we've been using it the other
way for so long, and I'm not sure we've really gotten a vote in favor
from Graham or Dave.

>> >In a chord in a linear temperament, there is always a unique chord
>> >interval which is widest in terms of generator steps between chord
>> >elements. We can define a chord associated to the given chord by
>> >inverting in a transposition such that this widest element is
>> >preserved.
>>
>> Why not use the narrowest element?
>
>Because the widest element seems the most interesting, and in any case
>is unique.

Would you say it's allowed in linear temperament for the generator
to exactly divide the period? Or is this degenerate in some way?

>If c is a chord and ~c is the associated chord, then ~c is
>well-defined and ~~c = c. In any MOS, if you have a chain of n chords
>of type c, you have a corresponding chain of n chords of type ~c, and
>you can play a sort of major to relative minor type of game. The
>nature of ~c, if c is a p-limit n-ad, is determined by the most
>complex element of the corresponding diamond. In 5-limit meantone, the
>most complex element is the major third, and hence that is preserved,
>leading to the diatonic scale of three major triads defining the major
>key, and three minor triads the minor key. In heemiwuerschidt, where
>3/2 is more complex than 5/4 and 7/4, the corresponding tetrad to a
>otonal tetrad on a given root will be the utonal tetrad on the same
>root, and hemiwuerschmidt leads to scale where both the major and
>minor tetrads exist for a given root if one or the oher exists. Hence,
>this seems to define an interesting property of any linear temperament.

For sure; cool.

>> >Has anyone given a name to this? An example would be C major and a
>> >minor in meantone.
>>
>> Um, isn't that keeping the narrowest element (the fifth)?
>
>No, C major is CEG, and a minor is ACE,

Oh, I missed the "a".

>> >Note that I mean linear in the strict sense; the
>> >definition does not apply to all rank-two temperaments.
>>
>> So linear temperaments are rank 2 these days? Refresh my memory.
>
>I thought we decided that linear temperaments are rank 2 with a period
>of an octave.

I see this is indeed what "we" decided.

>Paul wants it that way on historical grounds, and I sort of like it
>since it allows us to use the two different terms to mean different
>things. The problem is that we've been using it the other way for so
>long, and I'm not sure we've really gotten a vote in favor from
>Graham or Dave.

Just checking.

-Carl

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

> Would you say it's allowed in linear temperament for the generator
> to exactly divide the period? Or is this degenerate in some way?

If the generator divides the period you no longer have a rank two
temperament. However, one step of n equal can be a tuning for a
temperament, in which case you have the usual business of tuning what
starts out as a rank two temperament in rank one form; meantone in 31
equal, for example.