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At a quick glance, that tuning sounds interesting. It is clearly very
uneven as a melodic scale, which isn't necessarily good or bad, but an
important property.

But for whatever it's worth, I don't think that 9 would be considered a
limit on the grounds that it's not prime.

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🔗Matt Nathan <mattn@...>

PAULE wrote:
>
> Gary wrote,
>
> >But for whatever it's worth, I don't think that 9 would be considered a
> >limit on the grounds that it's not prime.
>
> I, for one, completely disagree. I would define the limit as the largest odd
> number that occurs in ratios that are considered consonant. I differ with
> many JI advocates in that I do not think dissonances should be represented
> as ratios

Maybe I misunderstand your point, but how else would you represent them?
I notice you use ratios as you continue:

> -- in 5-limit diatonic music, a major second is a major second
> whether you call it 9/8 or 10/9.

You've just used ratios to distinguish two _different_ major seconds. I've
read these described as the large just major second and the small just
major second, and other names. These would be distinguished from the septimal
major second. These various and imprecise English-y names can be dispensed with
by the use of ratios.

> (Not to deny that one could create a
> 9-limit diatonic style in which these would be two distinct consonances --
> this is hinted at by the use of chords such as C-D-E-G as consonances; i.e.,
> 8:9:10:12.)

They are distinct consonances (or dissonances, depending on how you define the
consonance/dissonance threshold). Assuming this typical diatonic major scale,
C-D-E-F-G-A-B-C 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1,
compare the chord C-D-E-G or 8:9:10:12,
with the chord G-A-B-D which is not 8:9:10:12.

In another example, assuming a pan-modal approach (drawing chords from multiple
diatonic modes on the same tonic), compare the D in the chord C-D-E-G 8:9:10:12
with the D in the chord Bb-C-D-F 8:9:10:12, keeping C the same. These are two
different D's. The best way that I know of to distinguish them and at the same
time describe their ideal tuning is to use ratios. The first D is 9/8 in the key
of C, and the second is 10/9 in the key of C.

> The simplest equal temperament that clearly distinguishes all 9-limit
> intervals is 41-equal.

Each equal temperament may provide a maximum of one rational interval only.
41-equal contains no 9 limit intervals save 2/1. To say that it distinguishes
9/8 from 10/9 (and the rest of the 9-limit intervals) is to say that it contains
intervals which may be used to misrepresent these rational intervals by
approximation using a different number of 41-equal degrees for each.

By my hearing, most equal temperaments work in proportion to the extent that
they get close to rational intervals, thanks or curses to the ear's natural
attraction to rational intervals and its propensity for interpretting
random intervals as perversions of rational intervals--which leads me to think
it best use the rational intervals directly. The rest of the equal temperaments
work to the extent that they get away from rational intervals and give us a
way to hear which lets us break away from the tyranny of the gestalt of rational
intervals. Even this though is a reactionary way of hearing which bases itself
on the gestalt of the rational intervals as something to get away from.

If you are suggesting that we attach et-41 names to 9-limit intervals
for the sake of ease of handling and communication, while understanding
that the intervals are to be tuned justly, that's fine, until you want to
include other rational intervals. Then you run into the problem of finding
the next best ET whose names can be mapped over the intervals we require,
and renaming all the intervals we've already come to recognize using et-41
names. I think using ratios is the most explicit, and most extensible notation
available for handling rational intervals in discussions like this.

Matt Nathan

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🔗Daniel Wolf <106232.3266@...>

Matt Nathan wrote:

'' I think using ratios is the most explicit, and most extensible notation
available for handling rational intervals in discussions like this.''

To which I agree, but also recognize that it is often useful to mapped onto
ETs in order to design keyboards, notations, or to collect my pitches into
reasonable scales.

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Ooops...

🔗Gary Morrison <71670.2576@...>

🔗Gary Morrison <71670.2576@...>

🔗Matt Nathan <mattn@...>

🔗Daniel Wolf <106232.3266@...>