notational consistency

[Paul Erlich, TD 165]
> Joe Monzo wrote,
>
>> The point, as far as this posting goes, is that altho I think
>> Johnston's notation has its merits, it also has a whole series
>> of inconsistencies (in the Erlichean sense)
>
> Not really; it's a different kind of inconsistency. My kind of
> inconsistency is one of approximations to JI by ETs and can never
> arise in Just Intonation itself.

I apologize for extrapolating the concept into an area where you
didn't intend it to be used.

What I meant by 'in the Erlichean sense' was that in Johnston's
notation, adding two intervals with the same 'type of spelling'
sometimes results in an interval that requires a different
'spelling', in the same way that adding the best approximations
of two ET intervals sometimes results in an interval whose best
approximation in that ET uses a different 'spelling' than the one
that results from the addition.

Is there any name or formal [mathematical] description for this
type of inconsistency?

> Joe Monzo wrote [re: Schoenberg's _Harmonielehre_],
>> The inconsistency that Erlich would point out would have called
>> these notes Bb, F#, and C#.
>
> Never. That's yet another kind of consistency concept that
> couldn't be further divorced from my own. Using the Erlich
> consistency that one gets with stretched partials, I would call
> these notes B, F#, and C#.
>

Hmmm... I never knew you were using stretched partials in your
consistency definition.

Again, is there a more formal description of this kind of
inconsistency? And could you explain *why* this is a
'kind of consistency concept that couldn't be further divorced
from my own'?

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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Joe Monzo wrote,

>What I meant by 'in the Erlichean sense' was that in Johnston's
>notation, adding two intervals with the same 'type of spelling'
>sometimes results in an interval that requires a different
>'spelling', in the same way that adding the best approximations
>of two ET intervals sometimes results in an interval whose best
>approximation in that ET uses a different 'spelling' than the one
>that results from the addition.

Since each interval is spelled depending on where
it appears in pitch, it makes no sense to speak of
adding two intervals with the same type of
spelling in Johnston's notation.

I wrote,

>> Never. That's yet another kind of consistency concept that
>> couldn't be further divorced from my own. Using the Erlich
>> consistency that one gets with stretched partials, I would call
>> these notes B, F#, and C#.

Joe Monzo wrote,

>Hmmm... I never knew you were using stretched >partials in your
>consistency definition.

This refers to the statement I made earlier in
that post to the effect that 12-tET is consistent
through the 11th partial if one uses a
stretched partial template as Terhardt suggests.

>Again, is there a more formal description of this kind of
>inconsistency?� And could you explain *why* this is a
>'kind of consistency concept that couldn't be further divorced
>from my own'?

Becuase you derived your "consistent" answer by choosing the closest approximations of the
ratios that each pitch makes with C, totally ignoring the ratios that each pitch makes
with all other non-C pitches. This is the same procedure you used in your transcription of Partch's system into 144-tET. In both cases,
you not only violate the most basic principle behind Erlich consistency, which is that
the consonant ratios and only the consonant ratios must be approximated as well as possible, but you also introduce another kind of inconsistency
by approximating the same ratio in different ways in different parts of the scale.