additional meaning for "diaschisma"

reading Albert Seay's translation of the _expositio manus_
of Johannes Tinctoris (_Journal of Music Theory_ vol 9 #2,
Winter 1965), i noticed that Tinctoris used the word
"diaschismata" to refer to two successive instances of
some "diaschisma", but referring to a diaschisma different
in size from the one we mean when we use the term today.

see the new #2 definition at
/tuning/files/dict/diaschisma.htm

Tinctoris was referring to the same interval which
Arixtoxenos called the "enharmonic diesis". see
/tuning/files/monzo/aristoxenus/318tet.htm#enh-di
esis

-monz
"all roads lead to n^0"

monz,

what did aristoxenos call the interval that today we call
the "pythagorean comma"?

thanks . . .

--- In tuning@y..., "monz" <monz@a...> wrote:

> Tinctoris was referring to the same interval which
> Arixtoxenos called the "enharmonic diesis". see
>
/tuning/files/monzo/aristoxenus/318tet.htm#
enh-di
> esis
>
>
>
>
> -monz
> "all roads lead to n^0"

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
/tuning/topicId_39780.html#39782
>
> monz,
>
> what did aristoxenos call the interval that today we call
> the "pythagorean comma"?

he didn't. he never once in his entire treatise refers to
any rational measurements, prefering to speak in terms of
string *tension* (_syntonon_ = "tense", _malakon_ = "relaxed",
which most others have unfortunately translated as "soft")
instead of string *length*.

he gives the method of "tuning by concords" (which i referred
to in a link in my last post), meaning that from an origin pitch
one tunes by "perfect 4ths and 5ths". of course, these would be
*close* to the 4:3 and 3:2 ratios, but after tuning 12 different
notes by this method, Aristoxenus ends up with a "5th" which is
supposed to be "consonant", rather than the quite discordant
262144:177147 (~678 cents) which would result if those "4ths"
and "5ths" actually *were* tuned as 4:3 and 5:3, which means
that some tempering has to take place to make that last "5th"
sound "consonant", like all the others.

he never says anything about tempering or about the discrepancy
of the Pythagorean comma, insisting that musicians know what
"perfect 5ths and 4ths" sound like and know how to tune them
by ear. in essence, without giving any of the math, he was
describing a closed 12edo system. considering that the "4ths"
and "5ths" of 12edo really don't sound all that different from
the Pythagorean ratios, his theory seems plausible. i can speak
from my own experience of tuning pianos that it is quite possible
to tune something very close to 12edo entirely by ear using
Aristoxenus's "tuning by concords" method.

anyway, he speaks in terms of "1/4-tones" and "1/3-tones",
and the measurement of one of his genera compares these, which
requires a division into "1/12-tones", and he uses the term
"diesis" to mean several different small intervals, but there's
never any mention of any kind of "comma", which infers that
commatic differences between Pythagorean pitches was irrelevant
to his theory.

-monz
"all roads lead to n^0"