re: Porcupine notation

>I take Porcupine temperament to be an approximately 163c generator
>with primes 3,5,7,11 mapping to -3,-5,6,-4 generators, (11-limit sans
>9's). I'd notate the 7 note MOS as
>
>A Bv C^ D Ev F^ G
>
>v and ^ represent both the syntonic comma and the undecimal diesis and
>correspond to 1 step of either 15 or 22-tET, and 2-steps of 37-tET.
>Feel free to substitute your favourite comma symbols, everyone else
>does :-)

Is this equivalent to my suggestion?

> E#
> / \ / \ / / / .
> C#--G#--D#--A#^-E#^-
> / \ / \ / \ / \ / \ ,
> A---E---B---F#^-C#^-G#^
> / \ / \ / \ / \ / \ /
>F---C---G---D---A^--E^
>
>So in tunings with positive fifths, the # raises two steps
>and the ^ raises 1, and with normal fifths the # works as
>normal and you ignore the ^. The progression is then:
>
>15 (E, G#min, C#, E#=E^, A^, C#^, F#^, B, E)

I've been looking at Porky as a fifth generator... maybe
Gene can tell us where the fifth and the 163-cent generators
diverge (as far as series of ets).

Don't forget about 29, Dave!

-Carl

>The progression is then:
>
>15 (E, G#min, C#, E#=E^, A^, C#^, F#^, B, E)
>
>Right, except it actually starts on Eb. (C-E is a minor third in
>porcupine notation, so porcupine E corresponds to traditional Eb.)

I was just ignoring absolute pitch in that example. My notation, is
not the same as your original proposal (C-E is a major third), I'm
sure you realize?

>>12 (E, G#min, C#, E#=F, Bb, D, G, C, F), or
>> (E, G#min, C#, E#, A#, Cx, Fx, B#, E#), right?
>>
>>Does this make sense?
>
>Yes, that's one way of looking at it. I've used similar notations except
>with the D a fifth below A rather than the one above G. But if you're going
>to start notating commas, I think it's almost easier to use the Pythagorean
>fifths and write Eb, G-, C-, E--=Eb+, Ab+, C, F, Bb, Eb.
>
>. --- G#
>. / \
>. -- E---B
>. / \ / \
>. - C---G---D---A---E
>. \ / \ / \ / \ / \
>. start-> Eb--Bb--F---C---G
>. \ / \ / \ / \
>. + Db--Ab--Eb--Bb

So what commas do +/- and #/b control here?

>Porcupine notation is to the porcupine tunings what Graham Breed's decimal
>notation is for the MIRACLE temperament. The notes A-G are equally spaced,
>based on the generator of the porcupine scale (2 steps of 15-ET, 3 of
>22-ET, or 5 of 37-ET). Two steps up is a minor third, three steps a perfect
>fourth, and five steps a minor sixth. A step above G is notated as Ab, and
>the cycle keeps going, adding more and more flats. Similarly, a step below
>A is G#, and more sharps are added as you go down the scale.
>
>. G---D---A E#--B# Fx--Cx
>. \ / \ \ / \
>. Eb--Bb F---C G#--D#--A# Ex--Bx
>. / \ \ / \ \ / \
>. Gb--Db--Ab E---B F#--C# Gx--Dx--Ax
>. \ / \ \ / \ \ / \
>. Ebb-Bbb Fb--Cb G---D---A E#--B# Fx--Cx
>. \ / \ \ / \ \ / \
>. Gbb-Dbb-Abb Eb--Bb F---C G#--D#--A#
>. \ / \ \ / \ \ / \
>. Fbb-Cbb Gb--Db--Ab E---B F#--C#

Hmm. Intrestin'.

-C.

--- In tuning@y..., Carl Lumma <carl@l...> wrote:
> >I take Porcupine temperament to be an approximately 163c generator
> >with primes 3,5,7,11 mapping to -3,-5,6,-4 generators, (11-limit
sans
> >9's). I'd notate the 7 note MOS as
> >
> >A Bv C^ D Ev F^ G
> >
> >v and ^ represent both the syntonic comma and the undecimal diesis
and
> >correspond to 1 step of either 15 or 22-tET, and 2-steps of 37-tET.
> >Feel free to substitute your favourite comma symbols, everyone else
> >does :-)
>
> Is this equivalent to my suggestion?
>
> > E#
> > / \ / \ / / / .
> > C#--G#--D#--A#^-E#^-
> > / \ / \ / \ / \ / \ ,
> > A---E---B---F#^-C#^-G#^
> > / \ / \ / \ / \ / \ /
> >F---C---G---D---A^--E^

Not if the above is your suggestion. D:A and B:F# are P5s in mine (not
D:A^ and B:F#^). I really don't like these Ben Johnston style
notations.

> Don't forget about 29, Dave!

I consider it a very poor porcupine temperament. You'd be using
second-best ratios of 7.

Dave Keenan wrote...
>>>I take Porcupine temperament to be an approximately 163c generator
>>>with primes 3,5,7,11 mapping to -3,-5,6,-4 generators, (11-limit
>>>sans 9's). I'd notate the 7 note MOS as
>>>
>>>A Bv C^ D Ev F^ G
>>>
>>>v and ^ represent both the syntonic comma and the undecimal diesis
>>>and correspond to 1 step of either 15 or 22-tET, and 2-steps of
>>>37-tET.

Aha! What does the lattice look like?

>> Don't forget about 29, Dave!
>
>I consider it a very poor porcupine temperament. You'd be using
>second-best ratios of 7.

The 2nd-best in 29-equal. But Gene and all this linear temperament
stuff now shows that thinking about equal temperaments as fundamental
musical entities is a mistake! They are just convenient ways to
package linear temperaments.

-Carl

--- In tuning@y..., Carl Lumma <carl@l...> wrote:
> Dave Keenan wrote...
> >>>I take Porcupine temperament to be an approximately 163c
generator
> >>>with primes 3,5,7,11 mapping to -3,-5,6,-4 generators, (11-limit
> >>>sans 9's). I'd notate the 7 note MOS as
> >>>
> >>>A Bv C^ D Ev F^ G
> >>>
> >>>v and ^ represent both the syntonic comma and the undecimal
diesis
> >>>and correspond to 1 step of either 15 or 22-tET, and 2-steps of
> >>>37-tET.
>
> Aha! What does the lattice look like?

Which lattice? Here's the 5-limit lattice for the 7 note MOS.

Ev--Bv
\ / \
G---D---A
\ / \
F^--C^

There's only one subminor seventh (4:7), that's A:G. There are three
super fourths (8:11), G:C^, F^:Bv and Ev:A.

> >> Don't forget about 29, Dave!
> >
> >I consider it a very poor porcupine temperament. You'd be using
> >second-best ratios of 7.
>
> The 2nd-best in 29-equal. But Gene and all this linear temperament
> stuff now shows that thinking about equal temperaments as
fundamental
> musical entities is a mistake! They are just convenient ways to
> package linear temperaments.

OK. So 4/29 oct is far from an optimal Porcupine generator if you want
to approximate some JI.

To see the following properly formatted on Yahoo's dopey web interface you'll need to choose Message Index, then Expand Messages.

Here are two periods of Porcupine-7.

Ev--Bv
\ / \
G---D---A
\ / \
F^--C^
\
Ev--Bv
\ / \
G---D---A
\ / \
F^--C^

Here are two periods of Porcupine-15.

Dv--Av--Ev--Bv--F#v
\ / \ / \ / \
C---G---D---A---E
\ / \ / \ / \
Bb^--F^--C^--G^--D^
\ / \ / \ / \ / \
Dv--Av--Ev--Bv--F#v
\ / \ / \ / \
C---G---D---A---E
\ / \ / \ / \
Bb^--F^--C^--G^--D^

Here's an 11-limit-sans-9s sliderule for Porcupine-15

D^ E F#v G^ A Bv C^ D Ev F^ G Av Bb^ C Dv
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
5--11-3--------1-----------------7

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

>To see the following properly formatted on Yahoo's dopey web interface
>you'll need to choose Message Index, then Expand Messages.

I'm back on e-mail, now, and it's the best thing I've ever done. :)

>Here are two periods of Porcupine-7.
>
>Ev--Bv
> \ / \
> G---D---A
> \ / \
> F^--C^
> \
> Ev--Bv
> \ / \
> G---D---A
> \ / \
> F^--C^
>
>Here are two periods of Porcupine-15.
>
>Dv--Av--Ev--Bv--F#v
> \ / \ / \ / \
> C---G---D---A---E
> \ / \ / \ / \
> Bb^--F^--C^--G^--D^
> \ / \ / \ / \ / \
> Dv--Av--Ev--Bv--F#v
> \ / \ / \ / \
> C---G---D---A---E
> \ / \ / \ / \
> Bb^--F^--C^--G^--D^
>
>
>Here's an 11-limit-sans-9s sliderule for Porcupine-15
>
>D^ E F#v G^ A Bv C^ D Ev F^ G Av Bb^ C Dv
>|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
>5--11-3--------1-----------------7

Great, Dave!

-C.