tetrachordality

Paul,

My current model works like this:

pentachordal
(0 109 218 382 491 600 709 873 982 1091)
(1193 102 211 375 484 593 702 811 920 1084)
7 7 7 7 7 7 7 62 62 7

symmetrical
(0 109 218 382 491 600 709 818 982 1091)
(1193 102 211 320 484 593 702 811 920 1084)
7 7 7 62 7 7 7 7 62 7

So obviously, these two scales will come out
the same. But you've view -- and I remember
doing some listening experiments that back you
up (the low efficiency of the symmetrical
version was the other theory there) -- is that
the symmetrical version is not tetrachordal.

So what's going on here? Where's the error
in tetrachordality = similarity at transposition
by a 3:2?

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> Paul,
>
> My current model works like this:
>
> pentachordal
> (0 109 218 382 491 600 709 873 982 1091)
> (1193 102 211 375 484 593 702 811 920 1084)
> 7 7 7 7 7 7 7 62 62 7
>
> symmetrical
> (0 109 218 382 491 600 709 818 982 1091)
> (1193 102 211 320 484 593 702 811 920 1084)
> 7 7 7 62 7 7 7 7 62 7
>
> So obviously, these two scales will come out
> the same. But you've view -- and I remember
> doing some listening experiments that back you
> up (the low efficiency of the symmetrical
> version was the other theory there) -- is that
> the symmetrical version is not tetrachordal.
>
> So what's going on here? Where's the error
> in tetrachordality = similarity at transposition
> by a 3:2?

An octave species is homotetrachordal if it has identical melodic
structure within two 4:3 spans, separated by either a 4:3 or a 3:2.
In the pentachordal scale, _all_ of the octave species are
homotetrachordal (some in more than one way). In the symmetrical
scale, _none_ of the octave species are homotetrachordal.

>>So obviously, these two scales will come out
>>the same. But you've view -- and I remember
>>doing some listening experiments that back you
>>up (the low efficiency of the symmetrical
>>version was the other theory there) -- is that
>>the symmetrical version is not tetrachordal.
>>
>>So what's going on here? Where's the error
>>in tetrachordality = similarity at transposition
>>by a 3:2?
>
>An octave species is homotetrachordal if it has identical melodic
>structure within two 4:3 spans, separated by either a 4:3 or a 3:2.
>In the pentachordal scale, _all_ of the octave species are
>homotetrachordal (some in more than one way). In the symmetrical
>scale, _none_ of the octave species are homotetrachordal.

That's the def. in your paper. But:

() I never understood how it reflects symmetry at the 3:2.

() "homotetrachordal" is a new term on me. Are there precise
defs. of homo- vs. omni- around? How did you choose these
prefixes?

() We agreed a bit ago that 'the number of notes that change
when a scale is transposed by 3:2 index its omnitetrachordality',
right? My current approach is just a re-scaling of this. So
do we want to revise this agreement?

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>So obviously, these two scales will come out
> >>the same. But you've view -- and I remember
> >>doing some listening experiments that back you
> >>up (the low efficiency of the symmetrical
> >>version was the other theory there) -- is that
> >>the symmetrical version is not tetrachordal.
> >>
> >>So what's going on here? Where's the error
> >>in tetrachordality = similarity at transposition
> >>by a 3:2?
> >
> >An octave species is homotetrachordal if it has identical melodic
> >structure within two 4:3 spans, separated by either a 4:3 or a 3:2.
> >In the pentachordal scale, _all_ of the octave species are
> >homotetrachordal (some in more than one way). In the symmetrical
> >scale, _none_ of the octave species are homotetrachordal.
>
> That's the def. in your paper. But:
>
> () I never understood how it reflects symmetry at the 3:2.

4:3 more clearly than 3:2. However, you could look at 3:2 spans if
you wished, and still see a large gulf between the pentachordal and
symmetrical decatonic scales.

> () "homotetrachordal" is a new term on me. Are there precise
> defs. of homo- vs. omni- around?

Were those not precise enough for you?

> How did you choose these
> prefixes?

Homo = same -- two 4:3 spans that are the same
Omni = all -- all octave species are homotetrachordal.

> () We agreed a bit ago that 'the number of notes that change
> when a scale is transposed by 3:2 index its omnitetrachordality',
> right?

We did? I don't see transposition as coming into this -- rather, it's
a property of the _untransposed_ scale, heard in its full,
unmodulating glory.

> My current approach is just a re-scaling of this. So
> do we want to revise this agreement?

I guess so!!

>>That's the def. in your paper. But:
>>
>>() I never understood how it reflects symmetry at the 3:2.
>
>4:3 more clearly than 3:2. However, you could look at 3:2 spans
>if you wished, and still see a large gulf between the pentachordal
>and symmetrical decatonic scales.

I accept symmetry at 4:3 being tied to symmetry 3:2 in an octave-
equivalent universe. Just trying to see why we're getting
different results.

>> () "homotetrachordal" is a new term on me. Are there precise
>> defs. of homo- vs. omni- around?
>
> Were those not precise enough for you?

Never saw them!

>> How did you choose these prefixes?
>
> Homo = same -- two 4:3 spans that are the same
> Omni = all -- all octave species are homotetrachordal.

Aha! Now I've seen them!

>> () We agreed a bit ago that 'the number of notes that change
>> when a scale is transposed by 3:2 index its omnitetrachordality',
>> right?
>
>We did? I don't see transposition as coming into this -- rather,
>it's a property of the _untransposed_ scale, heard in its full,
>unmodulating glory.

You said something like "aren't we done? we just count the number
of notes that change...".

I always thought the basis for tetrachordality was that for any
note heard, it's 3:2 transposition was floating in the listener's
ear. Thus, if _that_ note is then played, it sounds natural (or,
you could say, it makes it easier to sing melodies). Also, there
was the anecdotal stories about folks at parties singing tunes
a 3:2 off. So that's an absolute pitch thing.

The interval pattern stuff (the L-L-s of conventional theory) is
a relative pitch thing... ?

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> The interval pattern stuff (the L-L-s of conventional theory) is
> a relative pitch thing... ?

Not sure what you mean.

>>The interval pattern stuff (the L-L-s of conventional theory) is
>>a relative pitch thing... ?
>
>Not sure what you mean.

I'm trying to understand the psychoacoustical basis for the
version in your paper, and recent posts about ethnic scales
on the main list (x+x+y, etc.). And I'm trying to understand
the lack, if any, of a psychoacoustical basis for my stuff
(absolute pitches being transposed by a 3:2).

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>The interval pattern stuff (the L-L-s of conventional theory) is
> >>a relative pitch thing... ?
> >
> >Not sure what you mean.
>
> I'm trying to understand the psychoacoustical basis for the
> version in your paper, and recent posts about ethnic scales
> on the main list (x+x+y, etc.).

You mean when I talk about tetrachordal (now homotetrachordal)
octave species of ethnic scales?

> And I'm trying to understand
> the lack, if any, of a psychoacoustical basis for my stuff
> (absolute pitches being transposed by a 3:2).

Psychoacoustic basis? Well, when you hear the step pattern repeated
at a 4:3 or a 3:2, it spells a more coherent tonal organization than
when you hear it repeated at some other interval. The patterns sound
more similar to one another in the first case, so the information
can be "compressed" more efficiently, if that makes any sense . . .