Yahoo Tuning Groups Ultimate Backup tuning What is a good name for the meantone double diminished second?

In 12-EDO, the following meantone equivalences are made:

aug2 = m3
aug3 = P4
aug4 = dim5
aug6 = m7

...and so on, you all know the drill. From a meantone standpoint, this
is because 12-EDO tempers out the diesis, which is a special name for
the "diminished second."

In 19-EDO, we have a much more complex (and, IMO, more beautiful) set
of enharmonic equivalences, which I'm currently trying to internalize
fully. There are lots of nice equivalences there I hadn't even thought
of, all of which are related to the other regular temperaments and
MOS's in 19-EDO. For instance, you have

aug2 = dim3
aug3 = dim4
aug4 = double dim 5
aug6 = dim7

All of these are really related to the same core enharmonic
equivalence, which is that the "double diminished second" is tempered
out (as Keenan pointed out). What is the name for this interval, other
than the "double diminished second?"

Put another way:
Minor second = diatonic semitone, if you temper this out you get 5-EDO
Augmented unison = chromatic semitone, if you temper this out you get 7-EDO
Diminished second = diesis, if you temper this out you get 12-EDO
Double diminished second = ?????????, if you temper this out you get 19-EDO

Does anyone know of the proper name for this interval? It's the
difference between the diesis and the chromatic semitone. (And "magic
comma" isn't what I'm after here, this is specifically an interval in
the meantone tuning system.)

-Mike

🔗Mike Battaglia <battaglia01@...>

On Mon, Oct 22, 2012 at 6:08 AM, Mike Battaglia <battaglia01@...> wrote:
>
> Does anyone know of the proper name for this interval? It's the
> difference between the diesis and the chromatic semitone. (And "magic
> comma" isn't what I'm after here, this is specifically an interval in
> the meantone tuning system.)

And for the tuning theorist type, let's say that the meantone tmonzo
|1 0> represents the meantone octave, and the tmonzo |0 1> represents
the meantone fifth. Then the tval <12 7| maps the octave to 12 steps
and the fifth to 7 steps; this is 12-EDO from a meantone perspective.
The dual of this is the tmonzo |7 -12>, which is what's being tempered
out here. The meantone name for this interval is "diesis."

The tval <7 4| then represents 7-EDO from a meantone perspective. The
dual of this is the tmonzo |4 -7>, which is what you have to temper
out from meantone to get 7-EDO. The meantone name for the interval |4
-7> is "chromatic semitone."

The tval <19 11| then maps meantone onto 19-EDO. The dual is |11 -19>,
which is the meantone interval you have to temper out to get to
19-EDO. The meantone name for the interval |11 -19> is what??

If there's no established meantone name for this, there should
definitely be one. Are there prominent historic authors who have
referred to this interval as something?

-Mike

🔗Charles Lucy <lucy@...>

🔗genewardsmith <genewardsmith@...>

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > Put another way:
> > Minor second = diatonic semitone, if you temper this out you get 5-EDO
> > Augmented unison = chromatic semitone, if you temper this out you get 7-EDO
> > Diminished second = diesis, if you temper this out you get 12-EDO
> > Double diminished second = ?????????, if you temper this out you get 19-EDO
>
> Except the positive interval is a doubly augmented seventh, down an octave.

That's right, in optimal meantones (with generator less than 8\19), this is the positive interval. But in meantones with the generator greater than 8\19 (which we might call "flattone"), this is negative and the doubly diminished second is positive.

> Why not give it a name, like subdiesis or something?

Because we're not sure whether it already has a name, and we ought to try reasonably hard to see if it does before giving it a new one.

Keenan

🔗Margo Schulter <mschulter@...>

Mike Battaglia wrote:

> Does anyone know of the proper name for this interval? It's the
> difference between the diesis and the chromatic semitone. (And
> "magic comma" isn't what I'm after here, this is specifically an
> interval in the meantone tuning system.)

Dear Mike,

Thank you for asking a question I believe has a quick and easy
answer: let's call it the "19-comma" or "19-diesis," maybe
depending on the shade of temperament (see below). At the end of
this article I'll give links to some articles from 1999 that
discuss this interval in detail, under the name of the "lesser
fifthtone" of 1/4-comma meantone. But if I had known the term
"19-comma" or "19-diesis" in 1999, I would have used it, and urge
that we adopt and use it now.

In a meantone context, the 19-comma is the difference between 19
fifths up and 11 octaves at 2/1. In 19-EDO, it's what gets
tempered out.

More interestingly -- and this is why I learned about, although
not under the useful name of "19-comma," back in 1999 -- it's an
interval which in meantones around 1/4-comma or the almost
identical 31-EDO can be used as a very small melodic step or
diesis, as composers such as Vicentino and Colonna did on their
31-note circulating meantone keyboards in their "enharmonic"
styles emulating the Greek enharmonic genus.

In these tunings, as you observe in your question, the chromatic
semitone gets divided in a 31-note circulating tuning into an
enharmonic diesis or 12-diesis -- a rational 128:125 or 41.059
cents in 1/4-comma meantone -- and a 19-diesis, which in
1/4-comma has a slightly smaller size of 34.990 cents.

In 31-EDO, the 12-diesis and 19-diesis are precisely equal, at
1/31 octave or 38.710 cents, and are literally fifthtones, that
is one-fifth the size of the regular major second at 5 steps.

In 1/4-comma, the 12-diesis and 19-diesis steps are slightly
unequal, as we've seen, but can still serve as approximately
equal "fifthtone" steps.

However, having both these fifthtone steps large enough to sound
like diesis steps rather than commas in the sense of not so
melodically distinct is a contrast on meantone tunings meant to
realize the enharmonic style presented in the treatises of
Vicentino (1555) and Colonna (1618). Here's a Portuguese organ
piece, I would guess maybe from Colonna's era, using these
fifthtone steps, and realized in 1/4-comma:

<http://www.bestII.com/~mschulter/Coimbra48.mp3>

From the viewpoint simply of optimizing 5/4 and 6/5, one person
might prefer 1/4-comma and another 2/7-comma, for example, But if
want to realize Vicentino's fifthtone or enharmonic music, which
depends on the division of the chromatic semitone into two equal
or near-equal steps, that takes 2/7-comma off the table, since
there we get a nice enharmonic diesis or 12-diesis at 50.276
cents, but a 19-comma at only 20.397 cents, which add up to the
chromatic semitone at a just 25/24 (70.672 cents).

Obviously a 20-cent step is too small for a Vicentino diesis or
fifthtone step, so the tempering has to be a lot less than
2/7-comma.

But getting back to our 19-comma, I'd say that that's the general
term, and one that fits especially well for 2/7-comma or the like
where the size of 20 cents does fit our idea of a typical
"comma."

However, for a tuning like 1/4-comma or 31-EDO, where both the
12-comma and 19-comma are in fact what we'd call small dieses
around 35-41 cents, why not speak of the 12-diesis and 19-diesis?
Whether to say 19-comma or 19-diesis might best be left to the
discretion of the writer, but with it understood that these are
musically equivalent terms for the same interval.

Now for the links I promised, again with the advice that my
"lesser fifthtone" be renamed the 19-comma, or in this context
better yet 19-diesis. The second part, #5621, is of special
interest here in terms of addressing this interval in 1/4-comma.

</tuning/topicId_5609.html#5609>
</tuning/topicId_5621.html#5621>
</tuning/topicId_5646.html#5646>
</tuning/topicId_5670.html#5670>

With many thanks for a great question,

Margo

🔗Mike Battaglia <battaglia01@...>

On Mon, Oct 22, 2012 at 11:43 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Put another way:
> > Minor second = diatonic semitone, if you temper this out you get 5-EDO
> > Augmented unison = chromatic semitone, if you temper this out you get 7-EDO
> > Diminished second = diesis, if you temper this out you get 12-EDO
> > Double diminished second = ?????????, if you temper this out you get 19-EDO
>
> Except the positive interval is a doubly augmented seventh, down an octave.

For some tunings of meantone. But that's a moot point, because a
doubly augmented seventh minus an octave is still a half double
diminished second going down from the tonic. You wouldn't say that a
whole step down from the root ceases to be a whole step.

> > Does anyone know of the proper name for this interval? It's the
> > difference between the diesis and the chromatic semitone. (And "magic
> > comma" isn't what I'm after here, this is specifically an interval in
> > the meantone tuning system.)
>
> Why not give it a name, like subdiesis or something?

This is a good name, though I note it's only smaller than the diesis
in tunings between 19-EDO and 31-EDO. For any tuning of meantone where
the fifth is sharper than 1/4-comma, the so-called "subdiesis" will be
larger than the diesis. For instance, in 43-EDO, the diesis is one
step, and the "subdiesis" is 2 steps. So if we name it something
relative to the size of the diesis, we're bound to fail, as their
relative sizes are tuning-sensitive.

-Mike

🔗Margo Schulter <mschulter@...>

Just one more thought on this: if I'm right, the
term "meantone double diminished second" may be
contrary to conventional interval naming, since
this is actually a negative interval in 1/4-comma,
for example! Let's see if I'm right.

Here the double dimished second starts out as
a regular _minor_ second, in 1/4-comma around
117.1 cents. So that's what we're diminishing.

And were diminishing it twice by a chromatic
semitone, each one at 76.05 cents.

Say we're at C#, and want to move up by a
double diminished second. Here's where we
are:

| m2 |
C C# D
|------------|-----------------------|
0 76.05 193.15

So we go up to D to measure the usual
minor second, 117.1 cents or so.

But now we go down two chromatic semitones,
or about 152.1 cents:

| m2 |
C C# D
|------------|-----------------------|
0 76.05 193.15
41.06
|--------------|--------------|
76.05 76.05

So we started at 76.05 cents from C, and arrive
by our "ascending double diminished second"
at 41.06 cents from C -- a _descent_ of
34.99 cents!

To avoid this kind of paradox, I suggest we
call this the 19-diesis in a context like
1/4-comma, or the 19-comma in a context
like 2/7-comma where it has a size of around
20 cents.

Best,

Margo

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Tue, Oct 23, 2012 at 6:31 AM, Margo Schulter <mschulter@...> wrote:
>
> Just one more thought on this: if I'm right, the
> term "meantone double diminished second" may be
> contrary to conventional interval naming, since
> this is actually a negative interval in 1/4-comma,
> for example! Let's see if I'm right.

Right, in 1/4-comma, to go up by a double diminished second means you
move down. This is true in general of tunings sharper than exactly
19-EDO. In tunings flatter than 19-EDO, the opposite is true. As Gene
mentioned, for tunings closer to the meantone optimal range, you
actually need to move up by a double augmented seventh, minus an
octave, to actually go up in pitch.

> To avoid this kind of paradox, I suggest we
> call this the 19-diesis in a context like
> 1/4-comma, or the 19-comma in a context
> like 2/7-comma where it has a size of around
> 20 cents.

Well, I think the 19-xxxxxxx name is a good idea for systematically
naming intervals such as these, though I suggest that changing it from
"diesis" to "comma" may be a bit confusing. For the normal diesis, for
instance, we don't usually call it "the comma" when it gets really
small, as it does in something like 1/6-comma meantone where it's ~20
cents. (Though we do tend to call it "the Pythagorean comma" when the
generator swings sharp of 12-EDO.)

-Mike

🔗Mike Battaglia <battaglia01@...>

On Tue, Oct 23, 2012 at 8:59 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > This is a good name, though I note it's only smaller than the diesis
> > in tunings between 19-EDO and 31-EDO. For any tuning of meantone where
> > the fifth is sharper than 1/4-comma, the so-called "subdiesis" will be
> > larger than the diesis.
>
> True, but I was following a system whereby the ordering of all meantone intervals is the ordering they have in 1/4-comma, since 1/4-comma is the canonical meantone tuning if there is such a thing. In 1/4 comma, the "subdiesis" is 5^(19/4)/2^11 = 34.990 cents and the diesis is 128/125 = 41.059 cents.

How about this: 15625/15552 maps to this interval. Since 15625/15552
is already called the kleisma, why doesn't this just also get the name
of (meantone) kleisma? This parallels how 128/125 is the diesis, and
the diminished second is also the (meantone) diesis.

This way, we don't have to pick a name enshrining that this interval
is smaller than the diesis. For instance, in 1/5 and 1/6-comma
meantone, it's bigger, and in 31-EDO, it's exactly the same size.

-Mike

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Tue, Oct 23, 2012 at 12:23 PM, genewardsmith
<genewardsmith@...t> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > How about this: 15625/15552 maps to this interval. Since 15625/15552
> > is already called the kleisma, why doesn't this just also get the name
> > of (meantone) kleisma?
>
> I actually thought about that, but thought wouldn't like it since you didn't want any "magic comma" stuff.

I meant that I was going after the name of the meantone interval, not
the JI interval 3125/3072 specifically. The reason I like "kleisma"
better than "magic comma" is that kleisma actually sounds like the
name of a generic interval, whereas "magic comma" is definitely named
after magic temperament.

> However, another name for 3125/3072 is "small diesis", so I decided why not "subdiesis".

We could still go with subdiesis if people prefer that to kleisma. Or
pick some prefix suggesting that it's deeper into the meantone
structure than the usual diesis (paradiesis?).

> But 15625/15552 is an important interval also, and calling the meantone version a kleisma is suggestive; it tells you it ought to vanish in 19et. I'm cool with meantone diesis and meantone kleisma, which to me at least says "vanishes in 12" and "vanishes in 19". How the hell do you say "meantone Würschmidt comma" though?

It'd be pretty classy to call it the meantone ampersand. Though while
I was typing this you suggested "meantone semicomma" as well; either
of those is cool with me.

I definitely think that that's about as deep into the structure of
meantone as anyone should ever care about; we're now talking about the
interval between the diesis and the
kleisma/subdiesis/paradiesis/ditonma/magus/whatever. Assuming for now
we go with "kleisma", there's a nice fibonacci-esque structure here:

diatonic semitone -> chromatic semitone -> diesis -> kleisma -> ampersand

which are tempered out in 5-EDO, 7-EDO, 12-EDO, 19-EDO, and 31-EDO
respectively. Furthermore, each interval is the difference of the two
intervals preceding it. Of course, the nice recurrent sequence we get
here is only because the sorts of optimal meantones we care about are
so close to golden meantone; the only other "small meantone commas" I
can think of which would of interest would be |-15 26> in the |octave
fifth> basis, which is tempered out in 26-EDO.

This is actually really useful in helping me figure out the structure
of extended meantone; it would be nice to do this exercise and name
"tempered commas" of note in all the popular regular temperaments,
especially porcupine. Working with tvals and tmonzos should make the
whole enterprise somewhat easy to automate; I'll make a tuning-math
post about it.

-Mike

🔗gdsecor <gdsecor@...>

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Mike Battaglia wrote:
>
> > Does anyone know of the proper name for this interval? It's the
> > difference between the diesis and the chromatic semitone. (And
> > "magic comma" isn't what I'm after here, this is specifically an
> > interval in the meantone tuning system.)
>
> Dear Mike,
>
> Thank you for asking a question I believe has a quick and easy
> answer: let's call it the "19-comma" or "19-diesis," maybe
> depending on the shade of temperament (see below). At the end of
> this article I'll give links to some articles from 1999 that
> discuss this interval in detail, under the name of the "lesser
> fifthtone" of 1/4-comma meantone. But if I had known the term
> "19-comma" or "19-diesis" in 1999, I would have used it, and urge
> that we adopt and use it now.
>
> In a meantone context, the 19-comma is the difference between 19
> fifths up and 11 octaves at 2/1. In 19-EDO, it's what gets
> tempered out.

In the Sagittal paper, Dave Keenan and I have established a comma nomenclature based on the prime-number content in the interval:
http://www.sagittal.org/sagittal.pdf

Observe that in Table I we have listed the most common ones, such as the 5 comma (80:81), 7 comma (63:64), 11 M-diesis (32:33), 13 L-diesis (26:27), 17 kleisma (2176:2187), 17 comma (4096:4131), 19 schisma (512:513), and 19 comma (19456:19683), as well as others consisting of combinations of primes >3, such as 5:7 kleisma (5103:5120) and 11:19 M-diesis (171:176, not shown in Table I). (The Sagittal paper is in the process of being updated, and the version presently out there has only some temporary changes. It's been slow going, because I have too many things competing for too little free time. Dave & I will make an announcement when this is completed.)

Although Dave & I have not found it necessary to identify a specific ratio for any sort of "19 diesis", the nomenclature that we have established leaves that possibility open, so I would strongly discourage using "19-comma" or "19-diesis" for the purpose you are recommending.

> ...
> Whether to say 19-comma or 19-diesis might best be left to the
> discretion of the writer, but with it understood that these are
> musically equivalent terms for the same interval.

In the course of setting up the Sagittal comma nomenclature, Dave established an objective method for establishing boundaries between the categories (schisma, kleisma, comma, diesis) of small intervals:
/tuning/topicId_56202.html#56261
Dave explains his methodology for arriving at those boundaries here:
/tuning/topicId_59383.html#59445

I was going to suggest that you consider the names "19-tone comma" or "19-tone diesis" instead, or something else. You'll need to figure out what ratio is being tempered out to determine whether "comma" or "diesis" is the proper category. (I see that 15552:15625 is currently being discussed, so "kleisma" would be appropriate.)

> ...
> With many thanks for a great question,
>
> Margo

Margo and Mike:

I noticed that you've been discussing my high-tolerance temperament (without the 5's), and I read far enough to verify that you've correctly identified the ratios being tempered out (891:892 and 351:352), and you also correctly noted that, as I have it tuned, 52:63 is exact. It would also be good to note that this makes 11:13 very nearly exact. With the addition of 5's it becomes a rank-4 temperament, for which I tune 1:2, 1:5, and 1:7 exact. Since all 15-limit consonances are either exact, tempered by ~1.66 cents, or by twice that amount, 15-limit chords are essentially proportional-beating (and also slow-beating), which gives the effect of JI with a little bit of fresh air (i.e., not absolutely stagnant). The HTT tunings that I've described (in various messages) are constant-structure mappings of sets of the 15-limit 4-D temperament into a 17-, 29-, and 41-tone octave.

Sorry -- gotta run!

--George

🔗Margo Schulter <mschulter@...>

> Although Dave & I have not found it necessary to identify a
> specific ratio for any sort of "19 diesis", the nomenclature
> that we have established leaves that possibility open, so I
> would strongly discourage using "19-comma" or "19-diesis" for
> the purpose you are recommending.

Dear George,

That's very reasonable, and, of course, the conventions should be
followed.

> In the course of setting up the Sagittal comma nomenclature, Dave established an
> objective method for establishing boundaries between the categories (schisma,
> kleisma, comma, diesis) of small intervals:
> [59]/tuning/topicId_56202.html#56261
> Dave explains his methodology for arriving at those boundaries here:
> [60]/tuning/topicId_59383.html#59445

Very helpful, and now I know for my own purposes -- see below! --
that 14.613 cents is indeed a comma! But first back to your great
solution!

> I was going to suggest that you consider the names "19-tone
> comma" or "19-tone diesis" instead, or something else.

Absolutely agreed, a friendly amendment and a brilliant
reconciliation! And the same goes for the 17-tone diesis in
Pythagorean or beyond, the one tempered out in 17-WT.

> You'll need to figure out what ratio is being tempered out to
> determine whether "comma" or "diesis" is the proper
> category. (I see that 15552:15625 is currently being discussed,
> so "kleisma" would be appropriate.)

The curious thing is that 1/4-comma, this is 34.99 cents, a bit
larger than a kleisma; but then, look at the 12-note "enharmonic
diesis" in 22-EDO at twice the size of the diatonic semitone!

> Margo and Mike:

> I noticed that you've been discussing my high-tolerance
> temperament (without the 5's), and I read far enough to verify
> that you've correctly identified the ratios being tempered out
> (891:892 and 351:352), and you also correctly noted that, as I
> have it tuned, 52:63 is exact. It would also be good to note
> that this makes 11:13 very nearly exact.

Absolutely, and I've been reflecting on how a simple typo could
cause someone to tune (44/13)^(1/3) rather than the intended
(504/13)^1/9. The intended generator is 703.579 cents, but for a
pure 13/11, or 3/2 tempered wide by (352/351)^1/3, we get 703.597
cents! And I was indeed planning to note this in a guide to
temperaments in this region, and will be sure to do so whenever I
mention your generator: we get, in effect, something virtually
equivalent to the pure 13/11 temperament in the bargain!

But now for the defining just interval of HTT, 63/52, and its
defining comma, 28672/28431 (14.613 cents). This is a comma
absolutely critical to my own use of this region since the year
2000, since it's the difference between 19683/16384 and 63/52 --
and also between 2187/2048 and 14/13, for example.

From the view of isoharmonic chords, a good example might be the
difference between the octave-less-apotome at 4096/2187, only a
3-5 schisma smaller than 15/8, and 13/7. Thus even a regular
12-MOS of the HTT generator would include good approximations of
7:11:13 (e.g. E-C-Eb).

In HTT, of course, we have specifically a generator of 3/2
tempered wide by (28672/28431)^1/9, or 1.624 cents, for a pure
63/52 -- which, to my knowledge, might be the earliest example of
tempering out this comma so as to produce a defining just
interval (like 5/4 in 1/4-comma meantone, 16/15 in 1/5-comma,
etc.).

Therefore I hereby propose that the 28672/28431, equal to the sum
(or actually product) of 896/891 and 352/351, be dubbed the
Secorian comma, or comma of Secor -- unless, of course, we know
of an earlier temperament like this.

> With the addition of 5's it becomes a rank-4 temperament, for
> which I tune 1:2, 1:5, and 1:7 exact. Since all 15-limit
> consonances are either exact, tempered by ~1.66 cents, or by
> twice that amount, 15-limit chords are essentially
> proportional-beating (and also slow-beating), which gives the
> effect of JI with a little bit of fresh air (i.e., not
> absolutely stagnant).

My only quibble is that you're underselling your accuracy
a bit here: ~1.62 cents would be the closest rounding.

But I hereby promise that whenever I explain how I took your
2.3.7.11.13 portion and have been coming up with modified
versions for the last decade, I'll be sure to mention those
4:5:6:7:9:11:13:15 ogdads!

And it's also humorous how I tend not to use term "subgroup"
except in a "When in Rome" situation: "That's not a subgroup,
that's my normal, everyday tuning!"

> The HTT tunings that I've described (in various messages) are
> constant-structure mappings of sets of the 15-limit 4-D
> temperament into a 17-, 29-, and 41-tone octave.

And great ones, as well as your Zany tuning and a fascinating
12-note set in JI from this January that I got a kind of 15+4 set
out of, and would love to share with you.

> Sorry -- gotta run!

Any appearance from you is delightful!

Best, as always,

Margo

🔗battaglia01 <battaglia01@...>

Hi George - I'm not sure why your post to the list didn't get sent out to my email; it's a good thing Margo responded or I'd have never seen it. I'll respond here...

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
>
> Observe that in Table I we have listed the most common ones, such as the 5 comma (80:81), 7 comma (63:64), 11 M-diesis (32:33), 13 L-diesis (26:27), 17 kleisma (2176:2187), 17 comma (4096:4131), 19 schisma (512:513), and 19 comma (19456:19683), as well as others consisting of combinations of primes >3, such as 5:7 kleisma (5103:5120) and 11:19 M-diesis (171:176, not shown in Table I). (The Sagittal paper is in the process of being updated, and the version presently out there has only some temporary changes. It's been slow going, because I have too many things competing for too little free time. Dave & I will make an announcement when this is completed.)

Well, I note that in this case we're referring to a meantone tempered interval, not a JI interval. Specifically we're talking about the meantone interval corresponding to all JI intervals of the form (3125/3072) * (81/80)^n.

> In the course of setting up the Sagittal comma nomenclature, Dave established an objective method for establishing boundaries between the categories (schisma, kleisma, comma, diesis) of small intervals:
> /tuning/topicId_56202.html#56261
> Dave explains his methodology for arriving at those boundaries here:
> /tuning/topicId_59383.html#59445

Yeah, so another issue is, the size of this interval is strongly sensitive to the specific tuning of meantone that we're using. In 1/3-comma it's about 7 cents, but in 1/4-comma it's about 35 cents. In tunings with the fifth being sharp of 31-EDO, this interval is larger than the usual meantone diesis, whereas in tunings flat of it, it's smaller. If you go flat enough to go past 19-EDO, then this interval reverses in size, so that it becomes negative (this is the case in 26-EDO, for instance). So it's hard to assign a canonical size to this interval.

We could arbitrarily pick some optimal tuning - TOP meantone, or POTE meantone, or 1/4-comma meantone or 31-EDO or something, but I'd rather just pick a name that's size agnostic. But I note that one JI ratio that tempers down to this interval class is the 5-limit kleisma of 15625/15552 tempers down to this, which is why I suggested just calling it the meantone kleisma.

Thanks for taking the time to weigh in,
Mike