Yahoo Tuning Groups Ultimate Backup tuning-math Re: Fwd: Re: A common notation for JI and ETs

--- In tuning-math@yahoogroups.com, David C Keenan <d.keenan@u...>
wrote:
> George Secor:
>
> >Filling out the rational complementation for a complete apotome
this
> >would be:
> >
> >212a: |( )|( ~|( /| |) (| (|( //| /|\ (/| (|) ~|| ~||
( )||~ ||) ||\ (||( ||~) /||) /||\ (DK)
> >
> >I have only one question: Since the 17th harmonic is so far off in
> >212-ET as to be almost midway between tones (and inconsistent
> >besides), whereas the 23rd is almost exact, would it be more
> >appropriate to substitute the 23 comma for the 17' comma symbol?
> >That would give:
> >
> >212b: |( )|( |~ /| |) (| (|( //| /|\ (/| (|) ~|| ~||
( )||~ ||) ||\ ~||) ||~) /||) /||\ (GS)
> >
> >But if you still prefer 212a for the standard set, then at least
> >Margo could use 212b as a modification, since 23 is present in her
> >tuning.
>
> 23 is present in the tuning, but not in those pitches that might be
notated
> with the 3deg212 symbol. Margo and I both chose ~|( because of its
> interpretation as a 7:13 comma, not a 17 comma. So whatever we
might decide
> for 3deg212, ~|( seems like the right symbol for Margo's tuning.

Okay, now I get it. (Sorry, I didn't read through everything as
carefully as I should have.) So you're getting two commas
represented here for the price of one.

> In determining what is best for 3deg212 I agree that 17 commas
should be
> avoided because of the inconsistency and inaccuracy, but should the
primary
> interpretation of ~|( be the 17' comma or the 7:13 comma? The only
> popularity stats we have, say that ignoring powers of 2 and 3, 17/1
is
> twice as popular as 13/7, so 17' should be the primary
interpretation.
> However these same stats say that 13/7 is slightly more popular
than 23/1,
> so perhaps we should use ~|( for that reason. 212-ET is at least
> 1,3,9,17-consistent.
>
> Is there any other advantage conferred by using |~ instead of ~|( ?

Not that I can see. It seems that the dual usage for ~|( would argue
in its favor, i.e., you should compare the combined usefulness or
popularity of 13/7 and 17 against that of 23.

So I agree with 212a as the standard symbol set.

--George

George Secor:

>Dave Keenan:
> > I'd prefer to go with )|( as the 7:11 comma since it only involves
>a 0.55
> > cent schisma. I feel that a 3 flag symbol for something under 10
>cents
> > could not be justified when a 2 flagger is within 0.98 cents.
>
>Agreed, but I would call it the 7':11 comma for reasons given below.
>
> > It seems 891:896 )|( should be called the 7:11 comma while the
>comma
> > represented by (| is called the 7:11'-comma.
>
>Before we used the colon designation for these two-prime commas, we
>were expressing them as the sum or difference of two single-prime
>commas, e.g., the 5:7 comma was the 7-5 comma. How would you do that
>for 891:896 other than as the 11'-7' comma? (However, since you
>don't like what I have for the 7' comma, see below.) Since this is
>the comma that is arrived at by invoking a new (7') comma, I think
>that this should be called the 7':11 comma.

Why not 7':11' if it is 11' - 7' ? But see my alternative suggestion below.

> > Are there any ETs in which we should now prefer )|( over some other
>symbol
> > given that it now has such a low prime-limit or low product
>complexity?
> >

I'll just note that neither of us have answered the above yet, in case the
way I edited things might have made it look like the following was
answering it, which of course it is not.

> > >They are all 7-related. In a 13-limit heptad (8:9:10:11:12:13:14)
>it
> > >is 7 that introduces scale impropriety; e.g., the fifth 5:7 is
>smaller
> > >than the fourth 7:10. Replace 14 with 15 in the heptad and I
>believe
> > >the scale is proper. So it would not be surprising that someone
>might
> > >want to respell the intervals involving 7 -- 4:7 as a sixth, 5:7
>as a
> > >fourth, 6:7 as a second, 7:9 as a fourth, 11:14 as a third, and
>13:14
> > >as an altered unison.
> > >
> > >So we would want to notate the following ratios of 7 using these
> > >commas:
> > >
> > > deg217 deg494
> > > ------ ------
> > >A# 32768:59049 ~1019.550c 185 420
> > >vs. 7/4 ~968.826c 175 399
> > >57344:59049 ~50.724c 10 21
> > >(apotome complement of 27:28 - this could be called the 7' comma)
> > >11:19 comma (|~ ~49.895c 9 21
> > >But a new symbol /|)` would represent it exactly
> > >(if the flags are added up separately 5+7+5' comma)
> >
> > I really don't think it is necessary or desirable to notate this 7'-
>comma.
> > It is larger than the standard 7-comma and it involves a longer
>chain of
> > fifths than _any_ other comma we've ever used.
>
>I think it's a matter of waiting to see if we'll have to, because I
>don't think that wanting to notate 7/4 relative to C as A-something
>would be unusual or weird.

But this one is notating it as A#-something, not A-something. Sure in
meantone you can notate 4:7 as C:A# but you'd only do it because the
something happens to vanish, it's a long way up the chain of fifths and Bb
\!/ is likely to be more convenient.

> > I think we should only accept the need for a _larger_ alternative
>comma for
> > some prime (or ratio of primes) if it involves a _shorter_ chain of
>fifths.
> >
> > >Expressed another way:
> >
> > I don't see the following quote as expressing the above quote
>another way.
> > It is a completely different 7-comma. With this comma a 4:7 above C
>would
> > be a kind of A, not A#.
>
>Okay, then call its apotome complement, 27:28, the 7' comma and use
>the 13'-5' symbol (|\' to represent it (replacing ' with whatever we
>eventually agree on for the 5' comma). I notice that this is the
>next symbol I proposed:
>
> > A 16:27
> > vs. 4:7
> >
> > >F 3:4 ~498.045c 90 205
> > >vs. 9/7 ~435.084c 79 179
> > >27:28 ~62.961c 11 26
> > >symbol )|| 12 26
> > >But a new symbol (|\' would represent it exactly
> >
> > It is very large,
>
>But it's still smaller than the 13' comma, ~65.3c, so this doesn't
>take us outside our upper boundary for single-shaft symbols.

True.

> > and the absolute value of its power of 3 is still larger
> > than that of the standard 7 comma, although only by 1. I'm not
>convinced
> > there's any need for it.
>
>As I said, let's wait and see. The nice thing about this is that it
>doesn't require any new flags other than the 5' (for which it offers
>further justification for having that new flag or whatever) and that
>it's exact. Come to think of it, I seem to recall that Margo wrote
>me a couple of weeks ago that she wanted a JI symbol for 27:28 -- you
>must admit that this is not a weird or unusual interval.

Certainly not an unusual interval, and we can already notate it. 27:28 from
C is of course Db!). But I understand you mean a symbol that represents it
as a modified unison. I can see that this might be useful, although I don't
think its apotome complement will be of much use.

> If there is
>any problem with this, I think it is that we need to be able to
>represent the 5' comma in such a way that the symbols in which it is
>used don't look weird.

That would be nice, but I would still want to avoid its use as much as
possible.

> > >F# 512:729 ~611.730c 111 252
> > >vs. 7/5 ~582.512c 105 240
> > >3584:3645 ~29.218c 6 12
> > >This is the 5:7' comma, or 7+5' comma, or 7'-5 comma
> > >A new symbol |)` would represent it exactly
> >
> > This contains 3^6 while the standard 5:7-comma has 3^-6 so I think
> > there could be some demand for this one. I think the proposed
>symbol is
> > good, being only 2 flags, however I'd like it even better if we
>could come
> > up with some way that the 5'-comma (ordinary schisma) could be
>notated as a
> > modification of the shaft rather than as a flag, or if the two
>flags were
> > not on the same side.
>
>We need to find a good way to represent *both* the 5' and -5'
>alterations that involves something other than a flag -- something
>laterally aligned with the shaft, if not a modification to the shaft
>itself. (So back to the drawing board!)

Agreed.

> > But in any case, it seems we should avoid using it if possible
>because of
> > its containing that very unfamiliar flag. It's kind of strange if
>we should
> > need to use this obscurte new flag as low as the 7-limit. You
>should leave
> > it out of the XH18 paper.
>
>I have a feeling that the 5' comma is going to be useful for notating
>all sorts of things regardless of the prime limit (we have already
>proposed incorporating it into the diaschisma, Pythagorean comma, and
>5-diesis symbols), particularly if it will indicate intervals
>exactly, so I wouldn't call this an obscure flag -- just a very small
>one. And the idea of using something other than a lateral flag to
>symbolize it strikes me as highly appropriate -- just so long as it
>looks good (and therein lies the problem).
>
> > >E 64:81 ~407.820c 74 168
> > >vs. 14/11 ~417.508c 75 172
> > >891:896 ~9.688c 1 4
> > >5:7+19 comma )|( ~9.136c 2 3
> >
> > Agreed.
> >
> > >C# 2048:2187 ~113.685c 21 47
> > >vs. 14/13 ~128.298c 23 53
> > >28431:28672 ~14.613c 2 6
> > >17' comma ~|( ~14.730c 3 6
> >
> > Agreed. I though we already had that one. I believe we called this
>the 7:13
> > comma while (|( is the 7:13' comma.
>
>I have been calling (|( the 7:13 comma, since it is the 13'-7 comma;
>however 28431:28672 isn't the 13-7 comma -- it's the 7'-13 comma (if
>27:28 is now the 7' comma), so I would then also call it the 7':13
>comma.
>
>If 27:28 is the 7' comma, then I would also have to rename the
>following in what I gave above:
>
>3584:3645 as the 5:7' comma or 7+5' comma (but now not the 7'-5 comma)
>891:896 as the 7':11 comma or 7'-11 comma

When naming these commas, how about we forget about the details of whether
an x:y comma is the sum or difference of x or x' and y or y' but simply
parse "x:y'-comma" as "(x:y)'-comma" meaning simply the second (and less
important) x:y comma.

I noticed recently, an ambiguity in the spoken form of these comma names.
To say "x prime comma" can be taken as merely referring to the fact that x
is a prime number. Perhaps in the spoken form it would be more useful to
say "small x comma" or "big x comma", except that this doesn't correspond
directly to primed or unprimed because we have the 5'-comma smaller than
the 5-comma while all the others have primed larger than unprimed.

We could refer to the unprimed one without using "big" or "small" and use
whichever of these applies for the primed one. If really necessary the
unprimed could be called "normal" or "standard".

>So have I sold you on a 7' comma, 27:28?

Yes. It's acceptable because it only contains 3^3. I feel there is some
sort of comma uselessness metric that increases with both the size of the
comma and the number of fifths. A first guess would be to take the product
of these. I wouldn't want to add any comma to the system that had this
"uselessness" higher than any we've already got. The highest so far is 362
for the 11' comma (|), and a close second is the 25 comma //| at 344.

The proposed 7' comma (27:28) is ok at 189, but 57344:59049 is out of the
question at 507.

I think we should also dump the 31" comma (65536:67797) at a uselessness of
411.

Actually, I think the number of fifths should feature more strongly in
uselessness, or we could have a sharp cutoff at 9 fifths.

But I'm still reluctant re 27:28 because of the added complication of the
5'-comma "flag". No matter how good we can make it look, the fact remains
that we haven't had to use it at all for anything else in the 15-limit plus
harmonics to 31. So I think it should not be considered part of the "basic"
system despite its low prime limit and low product complexity.
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

Re: Fwd: Re: A common notation for JI and ETs

🔗David C Keenan <d.keenan@uq.net.au>

🔗gdsecor <gdsecor@yahoo.com> <gdsecor@yahoo.com>

🔗David C Keenan <d.keenan@uq.net.au>

🔗gdsecor <gdsecor@yahoo.com> <gdsecor@yahoo.com>