21/20 as "septimal chromatic semitone?

I was browsing the wikipedia page for 41-tet and saw that it lists the
3rd step as the "septimal chromatic semitone," or 21/20.

Is that really correct? What's "chromatic" about it?

For example, the difference between 7/4 and 15/8 is a "septimal
diatonic semitone." If we're in C the note used to represent 7/4 would
be some kind of Bb (Bb< in HEWM) and 15/8 would be some kind of B (B-
in HEWM). So it makes sense to call that a diatonic semitone, since
the note letter actually changes.

Correspondingly, the difference between 5/4 and 21/16 is 21/20. In C,
5/4 is some kind of E (E- in HEWM) and 21/16 is probably some kind of
F (F< in HEWM)> I couldn't see how 21/16 could be anything besides an
F, really. It's 7/4 (Bb<) * 3/2.

Is this a wiki error or have people actually been calling this the
septimal chromatic semitone? For the record, Scala lists it as a
"minor semitone," which is annoying and vague.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Is that really correct? What's "chromatic" about it?

The ratio (21/20)/(25/24) = 126/125 is tempered out in septimal meantone, so if 25/24 is the chromatic semitone, 21/20 ought to be a septimal chromatic semitone. Similarly, (15/14)/(16/15) = 225/224 is tempered out, so 15/14 is reasonably called a septimal diatonic semitone.

> Correspondingly, the difference between 5/4 and 21/16 is 21/20. In C,
> 5/4 is some kind of E (E- in HEWM) and 21/16 is probably some kind of
> F (F< in HEWM)>

It's a diesis less than 4/3, so it's not an F; it's E#.

I couldn't see how 21/16 could be anything besides an
> F, really. It's 7/4 (Bb<) * 3/2.

The nomenclature is meantone based, and so the only way to extend it to talk about the 7-limit is via septimal meantone, and that makes
21/16 E# and 7/4 A#. And 21/20 a chromatic semitone.

> For example, the difference between 7/4 and 15/8 is a "septimal
> diatonic semitone." If we're in C the note used to represent 7/4 would
> be some kind of Bb (Bb< in HEWM) and 15/8 would be some kind of B (B-
> in HEWM). So it makes sense to call that a diatonic semitone, since
> the note letter actually changes.
But it doesn't change. That's why "chromatic" would make more sense here, whereas A# to B would be a diatonic step.
>
> Correspondingly, the difference between 5/4 and 21/16 is 21/20. In C,
> 5/4 is some kind of E (E- in HEWM) and 21/16 is probably some kind of
> F (F< in HEWM)> I couldn't see how 21/16 could be anything besides an
> F, really. It's 7/4 (Bb<) * 3/2.
>
> Is this a wiki error or have people actually been calling this the
> septimal chromatic semitone?
"Septimal diatonic semitone" would make more sense, as E to F is a diatonic step.
jn
>

... but in meantone, as explained by Gene above, the harmonic seventh is taken as an augmented sixth 225/128 (C A#, G E#), so "diatonic" is correct. Sorry.

jn

> > diatonic semitone." If we're in C the note used to represent 7/4 would
> > be some kind of Bb (Bb< in HEWM) and 15/8 would be some kind of B (B-
> > in HEWM). So it makes sense to call that a diatonic semitone, since
> > the note letter actually changes.
> But it doesn't change. That's why "chromatic" would make more sense here, whereas A# to B would be a diatonic step.
> >

On Fri, May 28, 2010 at 4:32 AM, genewardsmith
<genewardsmith@...> wrote:
>
> > Correspondingly, the difference between 5/4 and 21/16 is 21/20. In C,
> > 5/4 is some kind of E (E- in HEWM) and 21/16 is probably some kind of
> > F (F< in HEWM)>
>
> It's a diesis less than 4/3, so it's not an F; it's E#.

I thought that 64/63 was the "septimal comma," and 81/81 was the
syntonic comma, and the product of these two commas was 36/35, which
was the septimal diesis. Or am I getting the nomenclature wrong?

But the whole thing makes much more sense with the marvel naming
convention. So by that logic 7/4 would be a type of A# instead of Bb.
I can dig it.

Is there any consensus on how to label 14/13? If 1/1 is C, then 13/8
sounds like something between Ab and A, and then 7/4 is a type of A#.
So is 14/13 known as a type of chromatic semitone? Or is that about
when people start getting away from meantone?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, May 28, 2010 at 4:32 AM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > > Correspondingly, the difference between 5/4 and 21/16 is 21/20. In C,
> > > 5/4 is some kind of E (E- in HEWM) and 21/16 is probably some kind of
> > > F (F< in HEWM)>
> >
> > It's a diesis less than 4/3, so it's not an F; it's E#.
>
> I thought that 64/63 was the "septimal comma," and 81/81 was the
> syntonic comma, and the product of these two commas was 36/35, which
> was the septimal diesis. Or am I getting the nomenclature wrong?

No, that's right but its based on more size and on the fact that 12et tempers them all away than on meantone, I suspect, though 36/35 is in fact a septimal meantone diesis. But so is 64/63, and as far as I know that's never called a diesis.

> Is there any consensus on how to label 14/13?

There's not much sense in getting a meantone system to do it, I think.
If you run Scala and tell it to list interval names, it will claim 14/13 is a 2/3 tone, which may be as good as anything--"tridecimal 2/3 tone" should do for a name. In fact, give that 5/4 is two tones by definition, more or less, and that (5/4)/(14/13)^3 = 10985/10976, which is pretty darn small, it seems solid enough.

On Fri, May 28, 2010 at 8:48 PM, genewardsmith
<genewardsmith@...> wrote:
>
> No, that's right but its based on more size and on the fact that 12et tempers them all away than on meantone, I suspect, though 36/35 is in fact a septimal meantone diesis. But so is 64/63, and as far as I know that's never called a diesis.

Wait, are you saying that 64/63 is a diesis or that it isn't? Or is it
a comma that functions as a diesis in septimal meantone temperament?

> > Is there any consensus on how to label 14/13?
>
> There's not much sense in getting a meantone system to do it, I think.
> If you run Scala and tell it to list interval names, it will claim 14/13 is a 2/3 tone, which may be as good as anything--"tridecimal 2/3 tone" should do for a name. In fact, give that 5/4 is two tones by definition, more or less, and that (5/4)/(14/13)^3 = 10985/10976, which is pretty darn small, it seems solid enough.

That destroys everything. What's intermediate between diatonic and
chromatic? My OCD is not pleased.

-Mike

I would definitely think of the 21/20 as a septimal diatonic semitone (minor 2nd). In septimal just intonation it is the interval by which the 7th of the dominant 7th chord resolves down a half-step.

You could however, take a 5-limit 6th chord (12:15:18:20), then raise the 6th up to an augmented 6th to form a 4:5:6:7 German 6th. In that respect it would function as a septimal chromatic semitone (augmented prime).

Billy

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, May 28, 2010 at 8:48 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > No, that's right but its based on more size and on the fact that 12et tempers them all away than on meantone, I suspect, though 36/35 is in fact a septimal meantone diesis. But so is 64/63, and as far as I know that's never called a diesis.
>
> Wait, are you saying that 64/63 is a diesis or that it isn't? Or is it
> a comma that functions as a diesis in septimal meantone temperament?

I wouldn't want to call every interval which 12et tempers away but meantone calls as B#-C a diesis--that way lieth madness. You'd quickly decide the Pythagorean comma was the reciprocal of a diesis, which means the schisma must be one also. I'm opposed.

On Sat, May 29, 2010 at 3:59 AM, genewardsmith
<genewardsmith@...> wrote:
> I wouldn't want to call every interval which 12et tempers away but meantone calls as B#-C a diesis--that way lieth madness. You'd quickly decide the Pythagorean comma was the reciprocal of a diesis, which means the schisma must be one also. I'm opposed.

I think my brain just exploded. So if 64/63 isn't a diesis, then the
difference between 21/16 and 4/3 is not a diesis. But one is F and one
is E#. Is a diesis anything around ~50 cents then?

-Mike

--- In tuning@yahoogroups.com, "Billy" <billygard@...> wrote:
>
> I would definitely think of the 21/20 as a septimal diatonic semitone (minor 2nd). In septimal just intonation it is the interval by which the 7th of the dominant 7th chord resolves down a half-step.

Nope, it's how the tetrad resolves down: C-E-G-A# resolving down, not C-E-G-Bb. They are not the same in septimal meantone.

> > I thought that 64/63 was the "septimal comma," and 81/81 was the
> > syntonic comma, and the product of these two commas was 36/35, which
> > was the septimal diesis. Or am I getting the nomenclature wrong?
>
> No, that's right but its based on more size and on the fact that 12et tempers them all away than on meantone, I suspect, though 36/35 is in fact a septimal meantone diesis. But so is 64/63, and as far as I know that's never called a diesis.

Ben Johnston calls 36/35 (notated with 7 or L in his system) a "septimal chroma" (probably as the term "septimal comma" is widely used for 64/63). He doesn't use the term diesis.

jn

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

>Is a diesis anything around ~50 cents then?

It's an overworked term when naming intervals, kind of like "limma" or "Bohlen-Pierce".

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I was browsing the wikipedia page for 41-tet and saw that it lists the
> 3rd step as the "septimal chromatic semitone," or 21/20.
>
> Is that really correct? What's "chromatic" about it?

Hello, and "chromatic" and "diatonic" seem to have gotten interchanged. If you start by labeling the ratio 7/4 a septimal minor 7th, then starting from C, for example, that interval would by definition terminate in a Bb. The septimal comma 64/63 is the difference between a Pythagorean minor 7th (16/9) and the septimal minor 7th. The ratio of 21/20 would be a minor second/diatonic semitone (C-Db) while 15/14 would correspond to an augmented unison/chromatic semitone (C-C#). For some reason, the ratio 15/14 I think has been incorrectly called a diatonic vice chromatic semitone. (Perhaps it is because the 15/14 value is close the 1/4-comma meantone diatonic semitone.) If 15/14 were diatonic it would correspond to C-Db in a just tuning. This would make sense only if we start with the 7/4 ratio as an augmented 6th (C-A#) rather than a minor 7th. OTOH, you could construct a just C diatonic scale, for example, with the intervals E-F and B-C having a ratio of 15/14 but then you can end up with strange (non-customary ratios) for major 3rds, etc. Sincerely,

--- In tuning@yahoogroups.com, "ham_45242" <arl_123@...> wrote:

> Hello, and "chromatic" and "diatonic" seem to have gotten interchanged. If you start by labeling the ratio 7/4 a septimal minor 7th...

That's the key point. You don't start by doing that, since there is a system of nomenclature which is already in widespread use, and the only way to fit 7/4 into that system is as an augmented sixth, not a minor seventh. Since this system of interval naming is so widely used and understood, calling 7/4 a minor seventh can only lead to confusion.

>For some reason, the ratio 15/14 I think has been incorrectly called a diatonic vice chromatic semitone.

The reason is obvious: septimal meantone tempers out 225/224, equating 15/14 with 16/15, and septimal meantone is the only way to extend meantone to the 7-limit which makes much sense in terms of use in naming intervals. Your idea leads to "dominant temperament", and the only tuning which makes much sense for dominant is 12-equal. You've been weened on 12, and to you it makes sense to think in terms of dominant. But it doesn't really make much sense outside of a 12 context, and 12 is not the basis for the traditional Western names for intervals. Another possibility for extending to the 7-limit is flattone, by the way, which has it that 7/4 is a diminished minor seventh. Tune your keyboard to 26edo and try it out, but don't expect this naming system to catch on. It won't.

>Perhaps it is because the 15/14 value is close the 1/4-comma meantone diatonic semitone.) If 15/14 were diatonic it would correspond to C-Db in a just tuning. This would make sense only if we start with the 7/4 ratio as an augmented 6th (C-A#) rather than a minor 7th.

Give the man a cigar.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, May 29, 2010 at 3:59 AM, genewardsmith
> <genewardsmith@...> wrote:
> > I wouldn't want to call every interval which 12et tempers away but meantone calls as B#-C a diesis--that way lieth madness. You'd quickly decide the Pythagorean comma was the reciprocal of a diesis, which means the schisma must be one also. I'm opposed.
>
> I think my brain just exploded. So if 64/63 isn't a diesis, then the
> difference between 21/16 and 4/3 is not a diesis. But one is F and one
> is E#. Is a diesis anything around ~50 cents then?
>
> -Mike

63:64 is frequently called a septimal *comma* (27.264c), whereas the historical meantone *diesis* is a 5-limit interval, 125:128 (41.059c), which leads to the question: what should be the boundary between the categories "comma" and "diesis".

In the course of creating the Sagittal notation system, Dave Keenan & I noticed that, by classifying small rational intervals according to the prime-number content of their ratios, intervals having the same prime-number content (disregaring primes 2 and 3), e.g. 27 needed only to be distinguished according to their size-categories. Thus, 63:64 and 27:28, which contain only a single prime factor >3 (7), may be called a 7-comma (abbreviated 7C) and large 7-diesis (abbrev. 7L), respectively.

For an explanation of the specific categories and their boundaries, see:
/tuning/topicId_56202.html#56261
For an explanation of how these boundaries were arrived at, see:
/tuning/topicId_59383.html#59445

It's necessary to recognize that the boundaries apply only to rational intervals. Since temperaments, by their very nature, represent two or more rational intervals by a single (tempered) interval, these boundaries will not strictly apply. In the case of meantone temperament, the 7-comma (7C, 63:64) and the 125 small diesis (125S, 125:128) are represented by a single interval (an augmented 7th), which may function either as a (septimal) comma or (125) diesis.

--George

> --- In tuning@yahoogroups.com, "ham_45242" <arl_123@> wrote:
>
> > Hello, and "chromatic" and "diatonic" seem to have gotten interchanged. If you start by labeling the ratio 7/4 a septimal minor 7th...
>
> That's the key point. You don't start by doing that, since there is a system of nomenclature which is already in widespread use, and the only way to fit 7/4 into that system is as an augmented sixth, not a minor seventh. Since this system of interval naming is so widely used and understood, calling 7/4 a minor seventh can only lead to confusion.

Hello, and the problem is that in published material on JI septimal tunings the ratio 7/6 is taken as a septimal minor third, which is the difference between 7/4 and 3/2 as would exist in a 4:5:6:7 tetrad. The ratio 9/7 (3/2 divided by 7/6) is taken as a septimal major third. Therefore a C tritonic tetrad of this type would be C-E-G-Bb making the C-Bb interval a minor 7th not an augmented 6th. The only other name for the 1:7/4 interval I've seen in the literature is "harmonic 7th". I should also clarify that my previous posting was assuming just tunings (except where I mentioned meantone.)

I don't see anything wrong with defining the 7/4 interval as C-A#, for example, as long as one is consistent in the naming of the other related intervals (following the usual convention that diatonic semitones have to involve different notes (e.g. E-F, B-C, C-Db) while chromatic semitones are between a note and its sharped or flatted value. Sincerely,

--- In tuning@yahoogroups.com, "ham_45242" <arl_123@...> wrote:

> Hello, and the problem is that in published material on JI septimal tunings the ratio 7/6 is taken as a septimal minor third, which is the difference between 7/4 and 3/2 as would exist in a 4:5:6:7 tetrad. The ratio 9/7 (3/2 divided by 7/6) is taken as a septimal major third.

Those names are indeed in use, but so is "subminor third" for 7/6 and "supermajor third" for 9/7, which I think are better names in being more likely to avoid just this confusion.

Therefore a C tritonic tetrad of this type would be C-E-G-Bb making the C-Bb interval a minor 7th not an augmented 6th.

I really don't think meantone-style notation should be used unless you mean it and carry it through consistently, which would mean you are in dominant temperament and don't care to distinguish between a 8/7 and a 9/8. That doesn't work as a way of sorting out septimal names.

>The only other name for the 1:7/4 interval I've seen in the literature is "harmonic 7th".

A much better name than "minor seventh", and one that can be used in any temperament.

>I should also clarify that my previous posting was assuming just tunings (except where I mentioned meantone.)

Ah. A naming system which starts off assuming JI really works better on the whole. 7/6 could be "subminor third", 9/7 "supermajor third", 7/4 "harmonic seventh" and Scala could correct its list of names, which not only calls 9/7 a septimal major third, a bad idea, it calls it a BP third, a true naming atrocity because it drags in BP, not an important tuning system in the first place, and one that has no fifth.