[tuning-math] Re: A common notation for JI and ETs
- From: David C Keenan <d.keenan@xxxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxxxx
- Date: Sun, 14 Jul 2002 10:16:01 -0700
At 01:03 18/06/02 -0000, you wrote:
>--- In tuning-math@xxxx, "gdsecor" <gdsecor@xxxx> wrote:
>--- In tuning-math@xxxx, "dkeenanuqnetau" <d.keenan@xxxx> wrote:
>> --- In tuning-math@xxxx, "gdsecor" <gdsecor@xxxx> wrote:
>> > I would therefore recommend going back to the rational
>> > complementation system and doing the ET's that way as well.
>>
>> Agreed. Provided we _always_ use rational complements, whether this
>> results in matching half-apotomes or not.
>
>In other words, you would prefer having this:
>
>152 (76 ss.):
)| |~ /| |\ ~|) /|) /|\ (|)
(|\ ||~ /|| ||\ ~||) (||~ /||\
>
>instead of this:
>
>152 (76 ss.):
)| |~ /| |\ /|~ /|) /|\ (|)
(|\ ||~ /|| ||\ /||~ /||) /||\
>
>even if it isn't as easy to remember.
OK. I think you've got me there. :-) Remember I said I thought we shouldn't let
complements cause us to choose an inferior set of single-shaft symbols, because
some people won't use the purely saggital complements. I think we both agree
that /|~ is a better choice for 5deg152 than ~|) since it introduces fewer new
flags and puts the ET value closer to the rational value.
I don't think we have defined a rational complement for /|~ because it isn't
needed for rational tunings. But if we look at complements consistent with
494-ET (as all the rational complements are) the only complement for /|~ is
~||(. So we end up with
152 (76 ss.):
)| |~ /| |\ /|~ /|) /|\ (|)
(|\ ~||( /|| ||\ ~||) /||) /||\
But this is bad because the flag sequence is different in the two half-apotomes
_and_ ~||( = 10deg152 is inconsistent _and_ too many flag types. So you're
right. I don't want to use strict rational complements for this, particularly
with its importance in representing 1/3 commas. I'd rather have
>152 (76 ss.):
)| |~ /| |\ /|~ /|) /|\ (|)
(|\ ||~ /|| ||\ /||~ /||) /||\
I note that 76-ET can also be notated using its native fifth, as you give (and
I agree) below.
>I suggest that you try some more ET's before insisting on rational
>complements across the board. In addition to less memorable symbol
>sequences, strict rational complementation will also result in some
>bad symbol arithmetic in instances where the complement symbols are
>not consistent in some ET's. I will accept some symbol arithmetic
>inconsistency (e.g., with ||) in 72-ET), if it isn't too
>disorienting, but I think that users will need all the help they can
>get to keep the symbols straight in the larger ET's, and too many
>flags and bad symbol arithmetic aren't going to help.
Point taken.
>> > I would be agreeable to doing all of the ET's (with the rational
>> > complementation scheme) using the symbols that we agreed on in
>> > message #4443.
>>
>> OK.
>
>I erroneously stated that everything that we last agreed on (using
>what I would call "inverse complements") would stay the same.
>However, there is one exception. This:
>
>32: )| /|\ (|) (||\ /||\ (DK - inverse complements
>
>would become this:
>
>32: )| /|\ (|) (||~ /||\ (rational complements)
>
>To this I am agreeable.
That's ok with me too.
Now to start on the others with 6 or less steps per apotome.
I won't necessarily include the double-shaft symbols from here on. You should
assume they correspond to the rational complements.
We are really having problems with 1deg48 aren't we?
You wrote:
>I think that 48 and 55 have sufficiently different properties that
>there would be no reason to insist on doing them alike. Since I
>would do 96 this way:
>
>96: /| |) /|) /|\ (|\ ||) ||\ /||\
>
>I wouldn't see any problem with doing 48 as a subset of 96,
>particularly since 7 and 11 are among the best factors represented in
>48:
>
>48: |) /|\ ||) /||\
We agree 48 should be every second step of 96, but we haven't agreed on 96 yet.
I agree 48 doesn't _need_ to be the same as either 41 or 55, but it would be
good to minimise the number of different notations for all the scales with 4
steps to the apotome.
Both ~|) and ~|\ are consistently 1 degree of 48, 55 and 62-ET, but of these
only ~|) is also 2 degrees of 96-ET.
That's one reason why I favour ~|).
But lets forget 55 and 62 for now. You propose to use |) which is certainly
correct as the 7-comma for both 48 and 96-ET. Why would I want to add the ~|
17-flag to it when this is zero steps?
One problem is that we're already using |) as one degree of 36-ET and 2 degrees
of 72-ET. People will naturally attach the meaning of 1/3 semitone to it in
this application, and may find it confusing if 48 and 96-ET use it for 1/4
semitone.
This opens a whole other can of worms regarding notation relative to 12-ET.
Lots of people would like to notate their tunings (even those which are not
n*12-ETs) as deviations from 12-ET, rather than as deviations from a chain of
the tuning's own native fifths (or it may have none).
Since people are going to try to do it anyway, shouldn't we look at
standardising a consistent way of doing it? Some time ago I investigated this
in depth and I now offer a first pass at a spreadsheet that does it
semi-automatically. And, you guessed it, it requires 1deg48 and 2deg96 to be
~|).
http://groups.yahoo.com/group/tuning-math/files/Dave/Notating12ETDeviations.xls.zip
If you examine the formulae in this spreadsheet you will see that the principle
is that each symbol is given, in a lookup table, a range of cents deviations
that it covers. In general the ranges overlap, but there is a strict order of
precedence to resolve the cases where more than one symbol could notate the
same degree. Determining the ranges was quite tedious, but the main requirement
is to ensure that the symbols actually agree with their comma values, given
12-ET fifths. e.g. the changover between one symbol and the next, at the same
precedence level, occurs at the point equidistant from their two comma values
relative to a chain of 12-ET fifths.
But how did I choose which symbols to use in the first place? It's so long ago
I've almost forgotten, but the basic idea was for example, to look at all the
n*12-ETs that contained a 25c step and find which symbol corresponded to 25
cents in all of them, and so on.
Here's what it gives for all the n*12-ETs whose best fifth is the 12-ET fifth.
The dots indicate degrees that cannot be notated.
12:
24: /|\
36: |)
48: ~|) /|\
60: /| |\
72: /| |) /|\
84: /| |) /|)
96: /| ~|) |\ /|\
108: /| /|( |) /|)
120: /| (| |) |\ /|\
132: ~|( /| |) |\ /|)
144: ~|( /| ~|) |) /|) /|\
156: ~|( /| ~|) |) |\ /|)
168: ~|( /| /|( |) |\ /|) /|\
180: ~|( /| (| ~|) |) |\ /|)
192: ~|( /| (| ~|) |) |\ /|) /|\
204: ~|( /| (| ~|) |) |\ (|) /|\
216: ~|( /| (| /|( ~|) |) |\ /|) /|\
228: ~|( |( /| /|( ~|) |) |\ /|) (|) /|\
240: ~|( |( /| (| ~|) |) ~|\ |\ /|) /|\
252: ~|( |( /| (| ~|) |) ~|\ |\ /|) (|) /|\
264: ~|( |( /| (| /|( ~|) |) |\ . /|) /|\
276: ~|( |( /| (| /|( ~|) |) ~|\ |\ /|) (|) /|\
288: ~|( |( /| . (| ~|) |) ~|\ |\ /|) . (|)
300: ~|( |( /| . (| ~|) |) ~|\ |\ /|) . (|) /|\
Notice that this scheme only uses 6 types of flag since it doesn't go beyond
17-limit. Of course one has to get used to the fact that ~| is negative (-5.0
cents).
Notice that 276-ET is the largest that can be fully notated, and that
12,24,36,72 are as previously agreed. We haven't agreed on 60-ET yet, but the
proposal above is different from what either of us suggested recently.
Notice that 144-ET has bad flag arithmetic, since /| and |) [7 flag] are 2 and
4 steps respectively and thereby agree with 72-ET, but /|) is 5 steps and must
be interpreted as the 13 flag. If we are not willing to do this, then we must
accept that 144-ET cannot be fully notated in a manner consistent with 72-ET,
simply because we don't have a separate symbol for the 13-comma, and the
13-schisma doesn't vanish.
>Now 55 is a real problem, because nothing is really very good for
>1deg. The only single flags that will work are |( (17'-17) or (| (as
>the 29 comma), and the only primes that are 1,3,5,n-consistent are
>17, 23, and 29.
>
>If I wanted to minimize the number of flags, I could do it by
>introducing only one new flag:
>
>55: ~|\ /|\ ~|| /||\
>
>so that 1deg55 is represented by the larger version of the 23' comma
>symbol. Or doing it another way would introduce only two new flags:
>
>55: ~|~ /|\ ~||~ /||\
>
>The latter has for 1deg the 17+23 symbol, and its actual size (~25.3
>cents) is fairly close to 1deg55 (~21.8 cents). Besides, the symbols
>are very easy to remember. So this would be my choice.
I would not use a 23 comma to notate this when it can be done in 17-limit.
Luckily ~|\ works for 1 step as the 17+(11-5) comma (which also agrees with 2
steps of 110-ET). So I go for your first (min flags) suggestion:
55: ~|\ /|\
>What was your reason for choosing ~|)?
Probably only because I could make it agree with 48-ET.
>> 62: |) /|\ (|\ /||\ [13-commas]
>
>Considering that 7 is so well represented in this division, I would
>hesitate to use |) in the notation if it isn't being used as the 7
>comma. In fact, I don't think I would want to use |) for a symbol
>unless it *did* represent the 7 comma (lest the notation be
>misleading), although I would allow its use it in combination with
>other flags.
Good point.
> So I would prefer this:
>
>62: /|) /|\ (|\ /||\ [13-commas]
Agreed.
>> 69,76: |) ?? (|\ /||\ [13-comma]
>
>Again, I wouldn't use |) by itself defined as a 13-comma symbol, but
>would choose /|) instead:
>
>69,76: /|) )|\ (|\ /||\ [13-commas]
>
>For 2deg of either 69 or 76, )|\ is about the right size.
Agreed.
I note that 62, 69 and 76 are all 1,3,9-inconsistent and might also be notated
as subsets of 2x or 3x ETs.
We should take a look at the n*19-ET family now that it is complete.
19: /||\
38: /|\ /||\
57: /|) (|\ /||\ [13-commas]
76: /|) )|\ (|\ /||\ [13-commas]
>> 60: /| |) ||) ||\ /||\
>
>I notice that 13 is much better represented than 7, so I would prefer
>this (in which the JI symbols also more closely approximate the ET
>intervals):
>
>60: /| /|) (|\ ||\ /||\
As described above, this would not work in with the other n*12-ETs. My current
proposal uses neither 7 nor 13 comma symbols.
60: /| |\ /|| ||\ /||\
>> 67,74: ~|) /|) (|\ ~||( /||\
>
>I'm certainly in agreement with the 2deg and 3deg symbols, and if you
>must do both ET's alike, then what you have for 1deg would be the
>only choice (apart from (| as the 29 comma). We both previously
>chose )|) for 1deg74 (see message #4412), presumably because it's the
>smallest symbol that will work, and I chose |( for 1deg67 (in #4346),
>which would give this:
>
>67: |( /|) (|\ /||) /||\
>74: )|) /|) (|\ (||( /||\
>
>So what do you prefer?
I prefer yours, but I'm uncertain about the complement used for 4 steps of 74.
>> 81,88: )|) /|) (|\ (||( /||\ [13-commas]
>
>This is exactly what I have for 74, above. Should we do 67 as I did
>it above and do 74, 81, and 88 alike?
Yes.
>On the other hand, why wouldn't 88 be done as a subset of 176?
I have a reason to do both 81 and 88 as subsets, apart from the fact that they
are 1,3,9-inconsistent. When using their native fifths they need a single shaft
symbol for 4 steps and none is available.
>It is with some surprise that I find that |( is 1deg in both 67 and
>81, so 81 could also be done the same way as I have for 67, above.
Better to do it the same as 74 and 88 (or as a subset).
>> 6 steps per apotome
>> 37,44,51: )| /| /|) ||\ (||\ /||\ [13-commas]
>> or
>> 37,44,51: |) )|) /|) (||( ||) /||\ [13-commas]
>
>For 51 I had something a bit simpler (using lower primes):
So are you agreeing to one of these for 37 and 44? Presumably not the second
one because of |) not being the 7-comma. And with rational complements?
>51: |) /| /|) ||\ ||) /||\
OK.
>> 58: /| |\ /|\ /|| ||\ /||\
>> or
>> 58: /| |) /|\ ||) ||\ /||\ [13-comma]
>
>I think I would avoid your version2 -- this is another instance where
>it's too easy to be misled into thinking that |) is the 7 comma. If
>we wanted to avoid the confusability of all straight flags, we could
>try:
>
>58: /| /|) /|\ (|\ ||\ /||\
>
>Here |) would be kept as the 7 comma and (| would be the 11'-7 comma
>of 2deg58. However, I think that it would be too easy to forget
>that /|) and (|\ aren't representing ratios of 13. So I think that
>the safest choice is version 1 -- all straight flags.
Agreed:
58: /| |\ /|\ /|| ||\ /||\
>> 86,93,100: )|) |) )|\ (|\ (||( /||\ [13-commas]
>> or
>> 86,100: )|( |) )|\ (|\ (||) /||\ [13-commas]
>> 93: |( |) )|\ (|\ /||) /||\ [13-commas]
>
>I would do 93-ET and 100-ET as subsets of 186-ET and 200-ET,
>respectively.
I can agree to that for 100-ET since there is no single-shaft symbol for 5
steps, but it is of course 2*50, and 93 is 3*31, so the fifth sizes are quite
acceptable.
>For 86, I wouldn't use |) by itself as anything other than the 7
>comma, as explained above,
I totally agree we should avoid this in all cases.
> but would use convex flags for symbols
>that are actual ratios of 13. So this is how I would do it:
>
>86: ~|~ /|) (|~ (|\ ~||~ /||\ [13-commas and 23-comma]
>
>The two best primes are 13 and 23, so there is some basis for
>defining |~ as the 23 flag. In any event, I believe that (|~ can be
>a strong candidate for half an apotome if neither /|\ nor /|) nor (|\
>can be used.
I have no argument about the even steps (they agree with 43 and 50-ET). But
again I don't see the need to use a 23-comma. We have already used )|\ for a
half-apotome in the case of 69 and 76-ETs. It works here too. 86-ET is
1,3,7,13,19-consistent. So why not:
86,93,100: )|) /|) )|\ (|\ ?? /||\ [13-commas]
We can now consider the 31-ET family.
31: /|\ /||\
62: /|) /|\ (|\ /||\ [13-commas]
93: )|) /|) )|\ (|\ ?? /||\ [13-commas]
and compare it to the 19-ET family
19: /||\
38: /|\ /||\
57: /|) (|\ /||\ [13-commas]
76: /|) )|\ (|\ /||\ [13-commas]
Whew!
With that I must sadly inform you that I will not be able to contribute to this
discussion again for quite some time. I need to get seriously involved in an
electronic design project for some months now. The trouble is I'm a tuning
theory addict. I can't have just a little.
George, I strongly encourage you to present what we've agreed upon so far, to
the wider community for comment.
Regards,
-- Dave Keenan
Brisbane, Australia
http://uq.net.au/~zzdkeena
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