Notating 28-EDO (was: Comments about Isacoffs book and notation)

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:
> On the notation front, I spent some time thinking about it and
> then started looking at one of the temperments that has caught
> my attention lately, 28. I believe 28 is doubly (or quadruply)
> negative. What is interesting from a notational standpoint is
> that the chain of fifths closes at 7-equal, resulting in
> # = b = 0. (i.e., C = C# = Cx etc...)
>
> So I said, "hey, I'll just PRETEND that # = -b = 2" and then
> fill the remaining blanks with arrows".
>
> C C^ C# Dv D D^ D# Ev E E^ E# Fv F etc...
> Db Eb Fb
>
> This seems like such a practical notation for 28, preserves
> 12 identities (C# = Db, Cx = D, except at E-F and B-C) and
> probably has nothing to do with any commas.

It's fine except maybe someone's gonna wonder why B:F# and Bb:F aren't the
same as the other "perfect fifths". But I guess we've got to deal with the
fact that B:F _is_ a "perfect fifth" anyway.

There are two ways that make sense to me, for notating 28-tET. One where
you take the "perfect fifth" to be 4/7 oct (686 c) as you have done, and
the other where you notate it as every 3rd step of 84-tET (whose P5 happens
to be the 12-tET perfect fifth since 7*12=84). In the first you have only a
rather poor 1:3:5 (16 c errors) to base the notation on. In the second, the
notation shows up the rather better 1:5:11:19 and 7:9:13 relationships
(6.5c errors) and it can also be read as indicating deviations from 12-tET.

The problem with the first one in Gene's and my scheme, is the need to base
the accidentals on 1,3,p-inconsistent commas (because all the consistent
ones work back-to-front). But having accepted that, we can at least spell
the 1:11s and 7:9s correctly by using the 11-diesis [] for one step and the
septimal comma <> for two steps.

D D] D> E[ E E] E> F[ F F] F> G[ G G] G> A[ A A] A> B[ B B] B> C[ C C] C> D[ D

An approx 1:5:3 (4:5:6) chord is spelled rather oddly as e.g. C:E]:G.

The second notation looks like this:

D D} Ebv Eb> E< E^ F{ F F} F#v F#> G< G^ G#{ G# Ab} Av A> Bb< Bb^ B{ B B}
Cv C> C#< C#^ D{ D

Now an approx 1:5:3 chord is spelled e.g. F:Av:Cv which tells the truth
about the fifth really being quite narrow.

In this notation, A to G, # and b agree with 12-tET, and the other
accidentals can be read as:
v^ 14 cents
<> 29 cents
{} 43 cents

You can also use [] enharmonics for correct spelling of approx ratios of 11.

[] 57 cents

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

> From: David C Keenan <d.keenan@uq.net.au>
> Subject: Notating 28-EDO (was: Comments about Isacoffs book and notation)
>
> > What is interesting from a notational standpoint is
> > that the chain of fifths closes at 7-equal, resulting in
> > # = b = 0. (i.e., C = C# = Cx etc...)
> >
> > So I said, "hey, I'll just PRETEND that # = -b = 2" and then
> > fill the remaining blanks with arrows".
> >
> > C C^ C# Dv D D^ D# Ev E E^ E# Fv F etc...
> > Db Eb Fb
> >
> > This seems like such a practical notation for 28, preserves
> > 12 identities (C# = Db, Cx = D, except at E-F and B-C) and
> > probably has nothing to do with any commas.
>
> It's fine except maybe someone's gonna wonder why B:F# and Bb:F aren't the
> same as the other "perfect fifths". But I guess we've got to deal with the
> fact that B:F _is_ a "perfect fifth" anyway.

Got me with that one and I should spank myself. If God makes # = b = 0,
who am I to futz with it. Yes, if you are going to read it, then B-F# and
Bb-F should be a fifth, and the fact that Bb-F#, and B-F are also fifths
is just gravy on the cake.

>
> There are two ways that make sense to me, for notating 28-tET. One where
> you take the "perfect fifth" to be 4/7 oct (686 c) as you have done, and
> the other where you notate it as every 3rd step of 84-tET (whose P5 happens
> to be the 12-tET perfect fifth since 7*12=84). In the first you have only a
> rather poor 1:3:5 (16 c errors) to base the notation on. In the second, the
> notation shows up the rather better 1:5:11:19 and 7:9:13 relationships
> (6.5c errors) and it can also be read as indicating deviations from 12-tET.
>
> The problem with the first one in Gene's and my scheme, is the need to base
> the accidentals on 1,3,p-inconsistent commas (because all the consistent
> ones work back-to-front). But having accepted that, we can at least spell
> the 1:11s and 7:9s correctly by using the 11-diesis [] for one step and the
> septimal comma <> for two steps.
>
> D D] D> E[ E E] E> F[ F F] F> G[ G G] G> A[ A A] A> B[ B B] B> C[ C C] C> D[ D
>
> An approx 1:5:3 (4:5:6) chord is spelled rather oddly as e.g. C:E]:G.

Well, its odd but correct in expressing something that 28 must be about. The
"major third" (4*P5) in 28 is flat of the "best 5/4" as we are in an extremely
negative universe.

I like this one. My mistake was forgetting that #=b=0 is a very important
identity.

Now, if I want to figure out what the marks mean, based on a nifty spreadsheet
I picked up which gives me a hint that 28 is {1,3,5,15,19} consistent,

I would say that > = 20:19 (conflated with 25:24)
] = 76:75

The really good chain-crossing identity is that 3 steps = 16:15, is preserved
in these choices.

>
> The second notation looks like this:
>
> D D} Ebv Eb> E< E^ F{ F F} F#v F#> G< G^ G#{ G# Ab} Av A> Bb< Bb^ B{ B B}
> Cv C> C#< C#^ D{ D
>
> Now an approx 1:5:3 chord is spelled e.g. F:Av:Cv which tells the truth
> about the fifth really being quite narrow.

The whole set looks awful, but the chord does match what a 12et-er would
perhaps be able to comprehend "A is pinched down, heck I'll tune it to
a just third, C is pinched down too, I'll do it the same as the A".

>
> In this notation, A to G, # and b agree with 12-tET, and the other
> accidentals can be read as:
> v^ 14 cents
> <> 29 cents
> {} 43 cents
>
> You can also use [] enharmonics for correct spelling of approx ratios of 11.
>
> [] 57 cents

Have to think about this. Thank as a lot,

Bob Valentine

>
> -- Dave Keenan
> Brisbane, Australia
> http://dkeenan.com

--- In tuning@y..., David C Keenan <d.keenan@u...> wrote:

/tuning/topicId_34935.html#34935

>
> In this notation, A to G, # and b agree with 12-tET, and the other
> accidentals can be read as:
> v^ 14 cents
> <> 29 cents
> {} 43 cents
>
> You can also use [] enharmonics for correct spelling of approx
ratios of 11.
>
> [] 57 cents
>

***As usual, I've been half asleep when reading some of these posts,
but what comma does that 43 cents {} refer to (again) and where does
it occur in scales...??

Sorry for the retro recap...

jp

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., David C Keenan <d.keenan@u...> wrote:
>
> /tuning/topicId_34935.html#34935
>
> >
> > In this notation, A to G, # and b agree with 12-tET, and the other
> > accidentals can be read as:
> > v^ 14 cents
> > <> 29 cents
> > {} 43 cents
> >
> > You can also use [] enharmonics for correct spelling of approx
> ratios of 11.
> >
> > [] 57 cents
> >
>
> ***As usual, I've been half asleep when reading some of these
posts,
> but what comma does that 43 cents {} refer to (again) and where
does
> it occur in scales...??

dave was talking about 84-tone equal temperament (harald waage's
system).

43 cents is simply 3 steps of 84-equal . . .

84 = 7*12, so it's 7 'bicycle chains' . . .

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

/tuning/topicId_34935.html#34974

>
> dave was talking about 84-tone equal temperament (harald waage's
> system).
>
> 43 cents is simply 3 steps of 84-equal . . .
>
> 84 = 7*12, so it's 7 'bicycle chains' . . .

***Oh... Got it. So, it's not a specific "comma" per se...

jp

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> ***As usual, I've been half asleep when reading some of these posts,
> but what comma does that 43 cents {} refer to (again) and where does
> it occur in scales...??

Joseph, Paul answered most of this. Thanks Paul.

The comma is a 13-limit one that expresses the difference between a
(octave reduced) chain of 4 perfect fourths (3:4) and an 8:13 neutral
sixth. In Paul's notation 1024;1053

There is some question over whether the proposed common JI/ET notation
should use this 13-limit comma or the larger one that expresses the
difference between 3 perfect fifths (2:3) and the 8:13 neutral sixth.
26;27.

Just to muddy the water further :-) there's another even smaller one
that is the difference between 8 fifths and the 8:13. 6561;6656 But 8
fifths seems like too many to me.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

/tuning/topicId_34935.html#35017

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > ***As usual, I've been half asleep when reading some of these
posts,
> > but what comma does that 43 cents {} refer to (again) and where
does
> > it occur in scales...??
>
> Joseph, Paul answered most of this. Thanks Paul.
>
> The comma is a 13-limit one that expresses the difference between a
> (octave reduced) chain of 4 perfect fourths (3:4) and an 8:13
neutral
> sixth. In Paul's notation 1024;1053
>
> There is some question over whether the proposed common JI/ET
notation
> should use this 13-limit comma or the larger one that expresses the
> difference between 3 perfect fifths (2:3) and the 8:13 neutral
sixth.
> 26;27.
>
> Just to muddy the water further :-) there's another even smaller
one
> that is the difference between 8 fifths and the 8:13. 6561;6656 But
8
> fifths seems like too many to me.

***Thanks, Dave, for the recap!

jp