Gorgo temperament (and its 16-note scale)

Instead of commenting on scales that others have way more experience with, maybe I should stick to commenting on scales that haven't gotten as much attention. So here's some brief information on gorgo temperament (which is one that I've been playing with recently).

On paper it doesn't look like anything special; the first I saw it was as no. 90 in a list of 114 7-limit temperaments that Gene Ward Smith sent to the tuning-math list in January 2004. The first mention of the name "gorgo" appears to have been in July 2004. Here's a brief intro section of the gorgo piece I've been working on (which cuts off abruptly after 54 seconds) that illustrates some of the musical potential in this tuning system:

http://home.comcast.net/~teamouse/gorgo.mp3

Of course, this just begins to scratch the surface of what you can do with gorgo temperament, but you can get a rough idea of what it sounds like.

TOP gorgo, which is what I'm using here, has a period of 1205.82 cents and a generator of 228.2 cents; it has distributionally even scales of 1, 2, 3, 4, 5, 6, 11, 16, 21, and 37 notes. Since it's a tempered scale, there's more than one way to use Sagittal notation to notate it; here's one possibility for the basic 16-note scale (I hope I got the notation right...):

column 1: num. of generators up or down from D
column 2: sagittal notation
column 3: shorthand sagittal notation
column 4: interval from D as notated

+0 D D 1/1
-5 E!!!( Ebc 21/20
+6 E E 9/8
+1 E|) Ef 8/7
-4 F/| F/ 6/5
+7 F||\ F#\ 5/4
+2 G!) Gt 21/16
-3 G G 4/3
-8 A!!!( Abc 7/5
+3 A A 3/2
-2 A|) Af 32/21
-7 B!!/ Bb/ 8/5
+4 B\! B\ 5/3
-1 C!) Ct 7/4
-6 C C 16/9
+5 C|||( C#r 40/21
+0 D D 2/1

If you had a generalized keyboard, you might lay it out something like this (view with a fixed-width font):

. E F#\
. D Ef Gt A B\ C#r E F#\
. Ebc F/ G Af Ct D Ef Gt A B\ C#r
. Abc Bb/ C Ebc F/ G Af Ct D
. Abc Bb/ C

Arranging the notes in this way helps to understand the structure of the tuning system and the musical resources (harmonic and melodic) of this 16-note scale. For instance, you can see that it has a lot of fifths (go right three spaces from any note, except C#r, E, and F#\) and a reasonable number of major and minor thirds. You can also see the melodic arrangement of the five large steps (98.56 cents), which go diagonally up to the right skipping a row (Ebc-E, F/-F#\, Abc-A, Bb/-B\, and C-C#r), and the 11 small steps (64.82 cents). You could look at the mathematical definition of the tuning and calculate what intervals are tempered out from that, but it's so much easier just to look at a chart like this and visualize the chord patterns (once you become familiar with the basic intervals). Here's an octave of an extended 37-note gorgo keyboard with the actual Sagittal symbols for reference:

http://home.comcast.net/~teamouse/gorgo.png

Sounds very good to me.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Instead of commenting on scales that others have way more
experience
> with, maybe I should stick to commenting on scales that haven't
gotten
> as much attention. So here's some brief information on gorgo
temperament
> (which is one that I've been playing with recently).
>
> On paper it doesn't look like anything special; the first I saw it
was
> as no. 90 in a list of 114 7-limit temperaments that Gene Ward
Smith
> sent to the tuning-math list in January 2004. The first mention of
the
> name "gorgo" appears to have been in July 2004. Here's a brief
intro
> section of the gorgo piece I've been working on (which cuts off
abruptly
> after 54 seconds) that illustrates some of the musical potential
in this
> tuning system:
>
> http://home.comcast.net/~teamouse/gorgo.mp3
>
> Of course, this just begins to scratch the surface of what you can
do
> with gorgo temperament, but you can get a rough idea of what it
sounds like.
>
> TOP gorgo, which is what I'm using here, has a period of 1205.82
cents
> and a generator of 228.2 cents; it has distributionally even
scales of
> 1, 2, 3, 4, 5, 6, 11, 16, 21, and 37 notes. Since it's a tempered
scale,
> there's more than one way to use Sagittal notation to notate it;
here's
> one possibility for the basic 16-note scale (I hope I got the
notation
> right...):
>
> column 1: num. of generators up or down from D
> column 2: sagittal notation
> column 3: shorthand sagittal notation
> column 4: interval from D as notated
>
> +0 D D 1/1
> -5 E!!!( Ebc 21/20
> +6 E E 9/8
> +1 E|) Ef 8/7
> -4 F/| F/ 6/5
> +7 F||\ F#\ 5/4
> +2 G!) Gt 21/16
> -3 G G 4/3
> -8 A!!!( Abc 7/5
> +3 A A 3/2
> -2 A|) Af 32/21
> -7 B!!/ Bb/ 8/5
> +4 B\! B\ 5/3
> -1 C!) Ct 7/4
> -6 C C 16/9
> +5 C|||( C#r 40/21
> +0 D D 2/1
>
> If you had a generalized keyboard, you might lay it out something
like
> this (view with a fixed-width font):
>
> . E F#\
> . D Ef Gt A B\ C#r E F#\
> . Ebc F/ G Af Ct D Ef Gt A B\ C#r
> . Abc Bb/ C Ebc F/ G Af Ct D
> . Abc Bb/ C
>
> Arranging the notes in this way helps to understand the structure
of the
> tuning system and the musical resources (harmonic and melodic) of
this
> 16-note scale. For instance, you can see that it has a lot of
fifths (go
> right three spaces from any note, except C#r, E, and F#\) and a
> reasonable number of major and minor thirds. You can also see the
> melodic arrangement of the five large steps (98.56 cents), which
go
> diagonally up to the right skipping a row (Ebc-E, F/-F#\, Abc-A,
Bb/-B\,
> and C-C#r), and the 11 small steps (64.82 cents). You could look
at the
> mathematical definition of the tuning and calculate what intervals
are
> tempered out from that, but it's so much easier just to look at a
chart
> like this and visualize the chord patterns (once you become
familiar
> with the basic intervals). Here's an octave of an extended 37-note
gorgo
> keyboard with the actual Sagittal symbols for reference:
>
> http://home.comcast.net/~teamouse/gorgo.png
>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> http://home.comcast.net/~teamouse/gorgo.mp3

Hi Hermann. Wot Fun!

> TOP gorgo, which is what I'm using here, has a period of 1205.82 cents
> and a generator of 228.2 cents; it has distributionally even scales of
> 1, 2, 3, 4, 5, 6, 11, 16, 21, and 37 notes. Since it's a tempered
scale,
> there's more than one way to use Sagittal notation to notate it; here's
> one possibility for the basic 16-note scale (I hope I got the notation
> right...):
>
> column 1: num. of generators up or down from D
> column 2: sagittal notation
> column 3: shorthand sagittal notation
> column 4: interval from D as notated
>
> +0 D D 1/1
> -5 E!!!( Ebc 21/20
> +6 E E 9/8
> +1 E|) Ef 8/7
> -4 F/| F/ 6/5
> +7 F||\ F#\ 5/4
> +2 G!) Gt 21/16
> -3 G G 4/3
> -8 A!!!( Abc 7/5
> +3 A A 3/2
> -2 A|) Af 32/21
> -7 B!!/ Bb/ 8/5
> +4 B\! B\ 5/3
> -1 C!) Ct 7/4
> -6 C C 16/9
> +5 C|||( C#r 40/21
> +0 D D 2/1

While these are technically correct, don't you think this is more of a
notation for that particular JI scale, which as you say is one of many
that is approximated?

Don't you think that a notation for a regular temperament should
_show_ that regularity?

e.g. as follows:

I use the 7-comma symbols for notating gorgo below, rather than the
5-comma because the 5-comma is negative in this temperament.

+0 D D
-5 D|) Df
+6 E E
+1 E|) Ef
-4 F!) Ft
+7 F|||) F#f
+2 G!) Gt
-3 G G
-8 G|) Gf
+3 A A
-2 A|) Af
-7 B!!!) Bbt
+4 B|) Bf
-1 C!) Ct
-6 C C
+5 D!) Dt
+0 D D

And then we just memorise the fact that in this temperament
|\ (5C up) is the same as !) (7C down).

Whereas in JI (and in most temperaments where the 5-comma does not
vanish or become negative) the major third up from C is notated E\! or
E\ , in gorgo it is notated E|) or Ef .

-- Dave

Two corrections:

That should have been

"Hi Herman. Wot fun!"

and

"And then we just memorise the fact that in this temperament
/| (5C up) is the same as !) (7C down)."

:-)

-- Dave

On Thu, 21 Dec 2006 06:42:56 -0000, "Dave Keenan" <d.keenan@bigpond.net.au>
wrote:

>--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>> http://home.comcast.net/~teamouse/gorgo.mp3
>
>Hi Hermann. Wot Fun!
>
>While these are technically correct, don't you think this is more of a
>notation for that particular JI scale, which as you say is one of many
>that is approximated?
>
>Don't you think that a notation for a regular temperament should
>_show_ that regularity?

I haven't actually tried that collection of notes as a JI scale, so I don't
know if it makes much sense in JI, but certainly there wouldn't be any way
to tell just looking at a score that it's something other than a JI scale.
And in some cases, notating regular temperaments as JI can require notating
ratios that don't have direct Sagittal notation (I'm not at home now, so I
don't have the details, but I think 5-limit Luna was one of those that
can't easily be notated as JI).

In general, the issue of notating regular temperaments is a very
interesting problem, and there are advantages and drawbacks to different
approaches. I wanted something that I could use for the generalized
keyboard charts that wouldn't require a lot of additional explanation
(beyond the Sagittal article), so I'm notating the notes as JI for that
purpose, but one of these days I'm going to write up something about some
of the alternatives.