Yahoo Tuning Groups Ultimate Backup tuning 12 tuning systems -- notation ideas? music?

Hi everyone . . . I'm bored so here's some actual tuning material to
hopefully spur some discussion (or better yet, music) on this
list . . .

Carl noticed on tuning-math that there are 12 tuning systems Gene and
I both included in our lists of best (simple & accurate) 7-
limit 'linear' temperaments. I've constructed horagrams ('clock
diagrams' with concentric rings representing 'distributionally even'
scales) for these under the assumption that octave-repetition is used
(but that these octaves are tempered). I'd love to hear diverse
opinions on how to notate these. Unfortunately, Dave Keenan and
George Secor have left, but hopefully Mark Gould, Carl Lumma, and
some newbies and oldbies would like to offer ideas and/or
questions . . . Even more, I'd love to hear music in any of these
systems. Joseph Pehrson has composed a bit in Blackjack, etc. but a
few of these tuning systems are unexplored territory . . .

I mention the 4:5:6:7 chord but these tuning systems are designed to
represent the entire 7-prime-limit, so all kinds of other chords can
and should be used as well.

Three of the tuning systems are fairly familiar. The simplest is
called 'dominant sevenths':
/tuning/files/perlich/domsev.gif
as it forms a familiar chain of fifths and approximates 4:5:6:7 with
a dominant seventh chord formed from this chain. As Kalle pointed
out, this is suitable for a static form of tonality based on the
mixolydian (and possibly dorian, with a m6 chord as tonic) mode. It
seems most likely that one would use conventional 7-letter notation
for this, and the rough 4:5:6:7 chords would be notated as D-F#-A-C,
C-E-G-Bb, etc. The first ring in the horagram which contains any of
these chords is the 7-note ring, which has one (since the diatonic
scale has one dominant seventh chord); the 12-note ring has six.

Then we have meantone:
/tuning/files/perlich/meantone.gif
with the 4:5:6:7 approximated by a German augmented sixth chord -- an
idea that may have originated with Huygens. One would probably stick
with conventional notation, and 4:5:6:7s occur as D-F#-A-B#, C-E-G-
A#, Ab-C-Eb-F#, etc. The first ring in the horagram which contains
any of these chords is the 12-note ring, which has two (since a 12-
note meantone chromatic has two augmented sixth chords).

The most accurate of the familiar three is called 'schismatic',
or 'schismic' by Graham:
/tuning/files/perlich/schi7.gif
It's formed like Pythagorean tuning, but optimally the fifth and
octave are both stretched by less than 1 cent. Again, conventional 7-
letter notation seems ideal, but here the 4:5:6:7 is obtained with D-
Gb-A-Dbb, C-Fb-G-Cbb, D#-G-A#-Db, etc. Though there's a 12-note ring
present, the first ring in the horagram which contains any of these
chords is the 17-note ring, which has three. I believe Margo Schulter
has discussed this arragement, though she tends to use 24 notes for
most of her tuning systems.

The other 9 systems have a generator that is not a perfect fifth
(fourth) and/or a period that is not an octave, so they're more
unconventional.

The system I essentially wrote the following paper on:
http://lumma.org/tuning/erlich/erlich-decatonic.pdf
is now called Pajara:
/tuning/files/perlich/pajar.gif
since Paul used to torture his friends Jeff and Ara with endless
fantasies on keyboard tuned with this arrangement. The paper suggests
using the digits 1-9 and 0 as a notation for the 10-note ring (which
contains four of the approximate 4:5:6:7 chords, while the 12-note
ring contains six), and little triangles for "chromatic" alterations.
Earlier, I had used the letters P through Y for notation, and this
might be preferable because numbers mean so many things already that
confusion tends to arise when they are also used for note names.

The paper also hints at another system that I now call Injera:
/tuning/files/perlich/injera.gif
The 14-note ring mentioned is in the paper has six approximate
4:5:6:7s, while the 12-note ring has four.

Reflecting some 20th century usages of 12-equal, but tuned more
specifically, are the augmented system:
/tuning/files/perlich/aug7.gif
which has three rough 4:5:6:7 chords in the 9-note ring, and six in
the 12-note ring;
and the diminished system:
/tuning/files/perlich/dimsev.gif
which has four rough 4:5:6:7 chords in the 8-note ring, and eight in
the 12-note ring.

The other 5 systems have no 12-note ring, so are the most exotic.

Firstly, a system that should be familiar to many on this list (there
were thousands of messages on it in 2001, before anyone noticed
George Secor discovered it in 1975) is called Miracle:
/tuning/files/perlich/
It's the most accurate of these 12 systems. Graham has suggested
notating the 10-note ring with the digits 0-9, and using the symbols
^ and v to get the 21-note "chromatic" scale, called Blackjack, which
contains eight near-just 4:5:6:7 chords.

The least accurate is called Blackwood, since Easley B. used and
discussed the 10-note ring (though he tuned it in 15-equal):
/tuning/files/perlich/miracle.gif
The 10-note ring contains five rough 4:5:6:7 chords, and the 15-note
ring, ten.

Graham Breed came up with the name Magic for this:
/tuning/files/perlich/magic.gif
The 19-note ring has seven approximate 4:5:6:7 chords; as always, a
ring with N fewer (more) notes will have N fewer (more) 4:5:6:7
chords (until the number would drop below zero), so for example the
13-note ring has only one approximate 4:5:6:7 chord.

Gene Ward Smith came up with the name Orwell for this:
/tuning/files/perlich/orwell.gif
The 13-note ring has two approximate 4:5:6:7 chords; the 22-note ring
has eleven.

Last but certainly not least, there's the little-known 'Semisixths'
(since its generator is about half of a major sixth) system:
/tuning/files/perlich/semisixths.gif
Here the 19-note ring contains six approximate 4:5:6:7 chords.

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

Even more, I'd love to hear music in any of these
> systems. Joseph Pehrson has composed a bit in Blackjack, etc. but a
> few of these tuning systems are unexplored territory . . .

I've fiddled a little with both nonkleismic and hemiwuerschmidt,
neither of which make your list, alas. Maybe it's the rotten names I
gave them. Would a hemiwuerschmidt piece suit your mood? Or I could
try the waters of semisixths, which I haven't played with.

By the way, has anyone tried Canasta yet?

> I mention the 4:5:6:7 chord but these tuning systems are designed
to
> represent the entire 7-prime-limit, so all kinds of other chords
can
> and should be used as well.
>
> Three of the tuning systems are fairly familiar. The simplest is
> called 'dominant sevenths':
> /tuning/files/perlich/domsev.gif
> as it forms a familiar chain of fifths and approximates 4:5:6:7
with
> a dominant seventh chord formed from this chain.

You can always listen to common-practice music in 12-equal and call
it dominant sevenths.

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Herman Miller <hmiller@IO.COM>

wallyesterpaulrus wrote:
> Carl noticed on tuning-math that there are 12 tuning systems Gene and > I both included in our lists of best (simple & accurate) 7-
> limit 'linear' temperaments. I've constructed horagrams ('clock > diagrams' with concentric rings representing 'distributionally even' > scales) for these under the assumption that octave-repetition is used > (but that these octaves are tempered). I'd love to hear diverse > opinions on how to notate these.

For notating the "superpelog" scale (14 steps of a 261 cent generator) (http://www.io.com/~hmiller/music/superpelog.html), I came up with the idea of using consecutive letters of the alphabet from A to M, corresponding to the number of generators to reach the note from A. But it might make more sense to label the notes of the 9-note scale from A to I (A F B G C H D I E in order of pitch), and use an accidental for the small step between J and A. With that scheme, the 14-note scale would be A F Bb B G Cb C H Db D I Eb E Ab. This scheme for naming notes is easily adaptable to other linear temperaments, even highly warped ones, but might be best reserved for those (like superpelog) that don't easily map to traditional notation. It's useful for keeping track of the notes in Cakewalk piano rolls and event lists, at any rate.

> The paper also hints at another system that I now call Injera:
> /tuning/files/perlich/injera.gif > The 14-note ring mentioned is in the paper has six approximate > 4:5:6:7s, while the 12-note ring has four.

Naming every note of Injera[12] seems like the logical thing to do, but that's a lot of notes. The thing is that there's a large number of steps all the same size, without any smaller steps. Since it's a 12-note scale, standard notation might be the most convenient, reserving Cb and E# for the extra two notes of Injera[14].

> The least accurate is called Blackwood, since Easley B. used and > discussed the 10-note ring (though he tuned it in 15-equal):
> /tuning/files/perlich/miracle.gif > The 10-note ring contains five rough 4:5:6:7 chords, and the 15-note > ring, ten.

For scales with 5 or more periods in the octave, probably the most reasonable thing to do is notate the periods and use accidentals for the generator.

> Graham Breed came up with the name Magic for this:
> /tuning/files/perlich/magic.gif > The 19-note ring has seven approximate 4:5:6:7 chords; as always, a > ring with N fewer (more) notes will have N fewer (more) 4:5:6:7 > chords (until the number would drop below zero), so for example the > 13-note ring has only one approximate 4:5:6:7 chord.

Another scale with lots of identically-sized steps and no smaller ones until you get to Magic[22]. Like Injera, probably the best thing to do is use traditional notation for Magic[19], and use double sharps or double flats for the few extra notes you need to represent Magic[22].

> Gene Ward Smith came up with the name Orwell for this:
> /tuning/files/perlich/orwell.gif > The 13-note ring has two approximate 4:5:6:7 chords; the 22-note ring > has eleven.

The 9-note ring would make a good letter notation, with a similar scheme to my Superpelog notation: in pitch order, A F B G C H D I E. Then the next iterator of the generator after I gives you A#, and so on. For the 13-note ring, you have A A# F B B# G C C# H D D# I E. The 22-note ring gets into double sharps, or you could take the generator downwards from A to add flats: A A# Fb F Bb B B# Gb G Cb C C# Hb H Db D D# Ib I Eb E Ab. For the outer rings, another smaller set of accidentals would be useful to have.

> Last but certainly not least, there's the little-known 'Semisixths' > (since its generator is about half of a major sixth) system:
> /tuning/files/perlich/semisixths.gif
> Here the 19-note ring contains six approximate 4:5:6:7 chords.

The 8-note ring would be good for notation: A D G B E H C F. Then you start adding flats: A D G Bb B E H Cb C F Ab, then for the 19-note scale (going down a couple of generators to add sharps): A Db D Gb G G# Bb B Eb E Hb H H# Cb C Fb F F# Ab. Another set of accidentals would be needed for the smaller intervals of Semisixths[27] and Semisixths[46].

🔗Herman Miller <hmiller@IO.COM>

🔗Graham Breed <graham@microtonal.co.uk>

🔗monz <monz@attglobal.net>

hi Herman,

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> Gene Ward Smith wrote:
>
> > By the way, has anyone tried Canasta yet?
>
> Canasta is Miracle[31], right? Or is it Miracle[41]?
> I've used Miracle[31], but I haven't done much with it.
> It sounds great, though.

yes, canasta is MIRACLE[31].

(i still use capitals for MIRACLE because it is an acronym
as well as a regular word.)

we've dubbed MIRACLE[41] "studloco".

of all the MIRACLE tunings, i'm most interested myself in
canasta, but have been so busy with webpages and software
that i haven't yet composed any music with it.
soon, hopefully ... our software will make it easy.

see:

http://tonalsoft.com/enc/miracle.htm
http://tonalsoft.com/monzo/blackjack/blackjack.htm
http://tonalsoft.com/enc/canasta.htm
http://tonalsoft.com/enc/studloco.htm

-monz

🔗hstraub64 <hstraub64@telesonique.net>

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> Hi everyone . . . I'm bored so here's some actual tuning material
> to hopefully spur some discussion (or better yet, music) on this
> list . . .
>
> Carl noticed on tuning-math that there are 12 tuning systems Gene
> and I both included in our lists of best (simple & accurate) 7-
> limit 'linear' temperaments. I've constructed horagrams ('clock
> diagrams' with concentric rings representing 'distributionally
> even' scales) for these under the assumption that octave-repetition
> is used (but that these octaves are tempered).

I am haveing a little trouble understanding these diagrams. is there
an explanation available somewhere?

> I'd love to hear diverse
> opinions on how to notate these. Unfortunately, Dave Keenan and
> George Secor have left, but hopefully Mark Gould, Carl Lumma, and
> some newbies and oldbies would like to offer ideas and/or
> questions . . . Even more, I'd love to hear music in any of these
> systems. Joseph Pehrson has composed a bit in Blackjack, etc. but a
> few of these tuning systems are unexplored territory . . .
>
> I mention the 4:5:6:7 chord but these tuning systems are designed
> to represent the entire 7-prime-limit, so all kinds of other chords
> can and should be used as well.
>

<snip>

> The system I essentially wrote the following paper on:
> http://lumma.org/tuning/erlich/erlich-decatonic.pdf
> is now called Pajara:
> /tuning/files/perlich/pajar.gif
> since Paul used to torture his friends Jeff and Ara with endless
> fantasies on keyboard tuned with this arrangement.

Good to know! Got to rememgber that name.

> The paper suggests
> using the digits 1-9 and 0 as a notation for the 10-note ring
(which
> contains four of the approximate 4:5:6:7 chords, while the 12-note
> ring contains six), and little triangles for "chromatic"
> alterations.
> Earlier, I had used the letters P through Y for notation, and this
> might be preferable because numbers mean so many things already
> that confusion tends to arise when they are also used for note
> names.
>

I am currently experimenting with 22-equal on my synthesizer. The
notation I use is sort of dictated by the synth's limitations: it only
supporting per-octave detuning, I have to use two mappings on 12 notes
per octave, assigned to separate split areas of the keyboard. Hard to
play :-(... But at least it enables me to use conventional music
notation - I just need two systems indicating what to play on the
upper and the lower split area.

BTW, has anyone tried to play blues with one of these systems? Appears
a natural idea to me since the 4:5:6:7 chord has a quite "bluesy"
character. I am trying to do that now.

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> Even more, I'd love to hear music in any of these
> > systems. Joseph Pehrson has composed a bit in Blackjack, etc. but
a
> > few of these tuning systems are unexplored territory . . .
>
> I've fiddled a little with both nonkleismic and hemiwuerschmidt,
> neither of which make your list, alas. Maybe it's the rotten names
I
> gave them.

No, it had nothing to do with the names. Did you see Carl's post on
tuning-math I was referring to?

> > Three of the tuning systems are fairly familiar. The simplest is
> > called 'dominant sevenths':
> > /tuning/files/perlich/domsev.gif
> > as it forms a familiar chain of fifths and approximates 4:5:6:7
> with
> > a dominant seventh chord formed from this chain.
>
> You can always listen to common-practice music in 12-equal and call
> it dominant sevenths.

You could, but it would make more sense to say common-practice music,
having a triadic standard of consonance, uses dominant sevenths as
dissonant sonorities that set up a strong voice-leading expectation
which in turn defines the major and minor modes as the tonal ones.

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> wallyesterpaulrus wrote:
> > Carl noticed on tuning-math that there are 12 tuning systems Gene
and
> > I both included in our lists of best (simple & accurate) 7-
> > limit 'linear' temperaments. I've constructed horagrams ('clock
> > diagrams' with concentric rings representing 'distributionally
even'
> > scales) for these under the assumption that octave-repetition is
used
> > (but that these octaves are tempered). I'd love to hear diverse
> > opinions on how to notate these.
>
> For notating the "superpelog" scale (14 steps of a 261 cent
generator)
> (http://www.io.com/~hmiller/music/superpelog.html), I came up with
the
> idea of using consecutive letters of the alphabet from A to M,
> corresponding to the number of generators to reach the note from A.
But
> it might make more sense to label the notes of the 9-note scale
from A
> to I (A F B G C H D I E in order of pitch),

An intriguing "ordering" of the letters . . .

> It's useful for keeping track of the
> notes in Cakewalk piano rolls and event lists, at any rate.

That's good to keep in mind. I'd like to hear from Manuel Op de Coul
and Joe Monzo about some of the notations their software supports and
to what extent one can make use of them when composing sequences.

> > The paper also hints at another system that I now call Injera:
> > /tuning/files/perlich/injera.gif
> > The 14-note ring mentioned is in the paper has six approximate
> > 4:5:6:7s, while the 12-note ring has four.
>
> Naming every note of Injera[12] seems like the logical thing to do,
but
> that's a lot of notes. The thing is that there's a large number of
steps
> all the same size, without any smaller steps. Since it's a 12-note
> scale, standard notation might be the most convenient, reserving Cb
and
> E# for the extra two notes of Injera[14].

Hmm . . . not sure how much sense that would make, given that this is
essentially two interlaced diatonic scales. My own notation uses the
standard letter-names for one diatonic scale and upside-
down/backwards letters for the notes a half-octave away from the
corresponding normal letters.

> > Last but certainly not least, there's the little-
known 'Semisixths'
> > (since its generator is about half of a major sixth) system:
> > /tuning/files/perlich/semisixths.gif
> > Here the 19-note ring contains six approximate 4:5:6:7 chords.
>
> The 8-note ring would be good for notation: A D G B E H C F.

Whoa. Where does this sequence of letters come from?

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:

> I am haveing a little trouble understanding these diagrams. is there
> an explanation available somewhere?

'Horagrams' like these were published by Erv Wilson some time ago,
but I don't think he wrote up an explanation. They're pretty much
self-explanatory. Look at the meantone example, and you'll see your
familiar pentatonic, diatonic, and chromatic scales as
adjacent 'rings'. Let me know if you have any specific questions
about the meantone case; by addressing those, I'll hopefully be able
to resolve most of your confusion with all of them. While Erv always
applied the generator to the same end of the chain, here I applied it
alternately to the two ends of the chain(s).

> I am currently experimenting with 22-equal on my synthesizer. The
> notation I use is sort of dictated by the synth's limitations: it
only
> supporting per-octave detuning, I have to use two mappings on 12
notes
> per octave, assigned to separate split areas of the keyboard. Hard
to
> play :-(...

There are three or four 12-out-of-22 scales I could recommend for you
for the time being . . . someday, we'll all have generalized
keyboards :)

> BTW, has anyone tried to play blues with one of these systems?

Yeah, various mutant blues-like sounds . . .

>Appears
> a natural idea to me since the 4:5:6:7 chord has a quite "bluesy"
> character. I am trying to do that now.

The characteristic "blue notes" are neutral thirds, which some feel
are approximating 11/9, and neutral sevenths (11/6?). I've played
bluesy stuff with the following scale in 22-equal: 0 6 9 10 13 19
(22) -- the neutral seventh and that "quartertone" really make things
sonically flavorful.

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

/tuning/topicId_52534.html#52549

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > Even more, I'd love to hear music in any of these
> > > systems. Joseph Pehrson has composed a bit in Blackjack, etc.
but
> a
> > > few of these tuning systems are unexplored territory . . .
> >
> > I've fiddled a little with both nonkleismic and hemiwuerschmidt,
> > neither of which make your list, alas. Maybe it's the rotten
names
> I
> > gave them.
>
> No, it had nothing to do with the names. Did you see Carl's post
on
> tuning-math I was referring to?
>
> > > Three of the tuning systems are fairly familiar. The simplest
is
> > > called 'dominant sevenths':
> > > /tuning/files/perlich/domsev.gif
> > > as it forms a familiar chain of fifths and approximates
4:5:6:7
> > with
> > > a dominant seventh chord formed from this chain.
> >
> > You can always listen to common-practice music in 12-equal and
call
> > it dominant sevenths.
>
> You could, but it would make more sense to say common-practice
music,
> having a triadic standard of consonance, uses dominant sevenths as
> dissonant sonorities that set up a strong voice-leading
expectation
> which in turn defines the major and minor modes as the tonal ones.

***Yes, this is the "traditional" interpretation...

🔗czhang23@aol.com

🔗Herman Miller <hmiller@IO.COM>

wallyesterpaulrus wrote:
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> >>wallyesterpaulrus wrote:
>>>Last but certainly not least, there's the little-
> > known 'Semisixths' > >>>(since its generator is about half of a major sixth) system:
>>>/tuning/files/perlich/semisixths.gif
>>>Here the 19-note ring contains six approximate 4:5:6:7 chords.
>>
>>The 8-note ring would be good for notation: A D G B E H C F.
> > > Whoa. Where does this sequence of letters come from?

The semisixths generator is two small steps and one large step of the 8-note MOS (in TOP tuning: L=180.98 cents, s=131.09 cents, L+2s=443.16 cents). Here's the full 8-note scale:

! semisixths-8.scl
!
8-note MOS of Semisixths [7, 9, 13, -2, 1, 5] temperament, TOP tuning
8
!
131.09135
262.18270
443.16029
574.25164
705.34299
886.32059
1017.41193
1198.38953

The sequence of letters comes from successive iterations of the L+2s generator, wrapped around to fit within the octave period:

A B C D E F G H
s s L s s L s L s s L s s L s L s s L s s L s L
A D G B E H C F A D G B E H C F A D G B E H C F A

If you applied the same idea to meantone notation, it'd look like this:

A B C D E F G
s L L s L L L s L L s L L L s L L s L L L
A F D B G E C A F D B G E C A F D B G E C A
B E A D G C F (traditional notation)

Note that the semitones are between A-F and B-G (up 5 generators), while the whole steps are down 2 generators (F-D, D-B, etc.) You can tell the minor thirds from the major thirds the same way: minor = up 3 generators, major = down 4.

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> wallyesterpaulrus wrote:
> > --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> >
> >>wallyesterpaulrus wrote:
> >>>Last but certainly not least, there's the little-
> >
> > known 'Semisixths'
> >
> >>>(since its generator is about half of a major sixth) system:
> >>>/tuning/files/perlich/semisixths.gif
> >>>Here the 19-note ring contains six approximate 4:5:6:7 chords.
> >>
> >>The 8-note ring would be good for notation: A D G B E H C F.
> >
> >
> > Whoa. Where does this sequence of letters come from?
>
> The semisixths generator is two small steps and one large step of
the
> 8-note MOS (in TOP tuning: L=180.98 cents, s=131.09 cents,
L+2s=443.16
> cents). Here's the full 8-note scale:
>
> ! semisixths-8.scl
> !
> 8-note MOS of Semisixths [7, 9, 13, -2, 1, 5] temperament, TOP
tuning
> 8
> !
> 131.09135
> 262.18270
> 443.16029
> 574.25164
> 705.34299
> 886.32059
> 1017.41193
> 1198.38953
>
> The sequence of letters comes from successive iterations of the
L+2s
> generator, wrapped around to fit within the octave period:
>
> A B C D E F G H
> s s L s s L s L s s L s s L s L s s L s s L s L
> A D G B E H C F A D G B E H C F A D G B E H C F A
>
> If you applied the same idea to meantone notation, it'd look like
this:
>
> A B C D E F G
> s L L s L L L s L L s L L L s L L s L L L
> A F D B G E C A F D B G E C A F D B G E C A
> B E A D G C F (traditional notation)

What is the advantage of this over traditional notation -- this?:

> Note that the semitones are between A-F and B-G (up 5 generators),
while
> the whole steps are down 2 generators (F-D, D-B, etc.) You can tell
the
> minor thirds from the major thirds the same way: minor = up 3
> generators, major = down 4.

🔗Herman Miller <hmiller@IO.COM>

🔗Manuel Op de Coul <manuel.op.de.coul@eon-benelux.com>

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> wallyesterpaulrus wrote:
> > --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> >>If you applied the same idea to meantone notation, it'd look like
> >
> > this:
> >
> >>A B C D E F G
> >> s L L s L L L s L L s L L L s L L s L L L
> >>A F D B G E C A F D B G E C A F D B G E C A
> >>B E A D G C F (traditional notation)
> >
> >
> > What is the advantage of this over traditional notation -- this?:
>
> The advantage is that the same scheme is generally applicable to
other
> linear temperaments. In the case of meantone, traditional notation
has
> the advantage of centuries of use and familiarity, so this system
would
> just be confusing if applied to meantone, schismic, pelogic, or
other
> fourth-based temperaments. But there isn't a long-established
tradition
> of notation for semisixths or other linear temperaments, and this
scheme
> makes it easier to find your way around an unfamiliar tuning.

It seems to make it harder to find your way around melodically, only
perhaps easier harmonically.

> It's a
> logical extension of my porcupine notation, which uses the letters
A-G
> for the basic 7-note MOS.

Porcupine happens to be generated by a "step", so this notation
manages to make melodic sense in porcupine.

🔗hstraub64 <hstraub64@telesonique.net>

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
>
> 'Horagrams' like these were published by Erv Wilson some time ago,
> but I don't think he wrote up an explanation. They're pretty much
> self-explanatory. Look at the meantone example, and you'll see
your
> familiar pentatonic, diatonic, and chromatic scales as
> adjacent 'rings'. Let me know if you have any specific questions
> about the meantone case; by addressing those, I'll hopefully be
able
> to resolve most of your confusion with all of them.

Now it's getting clearer. The numbers are all cents, right?
But then I have another question about Pajara - the horagram
apparently is not 22-equal, in contrary to the paper of the
decatonic scales. So what am I missing here?

>
> > I am currently experimenting with 22-equal on my synthesizer. The
> > notation I use is sort of dictated by the synth's limitations:
it
> only
> > supporting per-octave detuning, I have to use two mappings on 12
> notes
> > per octave, assigned to separate split areas of the keyboard.
Hard
> to
> > play :-(...
>
> There are three or four 12-out-of-22 scales I could recommend for
you
> for the time being . . .

Yes, please do recommend! My current mapping (c and f# the same
above and below, and the other keys differing by 1 unit) is quite
uncomfortable; I am still looking for a better one.
--
Hans Straub

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> >
> > 'Horagrams' like these were published by Erv Wilson some time
ago,
> > but I don't think he wrote up an explanation. They're pretty much
> > self-explanatory. Look at the meantone example, and you'll see
> your
> > familiar pentatonic, diatonic, and chromatic scales as
> > adjacent 'rings'. Let me know if you have any specific questions
> > about the meantone case; by addressing those, I'll hopefully be
> able
> > to resolve most of your confusion with all of them.
>
> Now it's getting clearer. The numbers are all cents, right?

Yes -- sorry I didn't say so.

> But then I have another question about Pajara - the horagram
> apparently is not 22-equal, in contrary to the paper of the
> decatonic scales.

There's no contradiction -- 22-equal is used through most, but not
all, of the paper. Other tunings of the scale are also discussed in
the paper, though using tempered octaves was not explored there. All
these horagrams represent non-ET tunings.

You'll see a 22-tone ring in the horagram, which would be equally-
spaced had the 22-equal version of the generator been used.

> > > I am currently experimenting with 22-equal on my synthesizer.
The
> > > notation I use is sort of dictated by the synth's limitations:
> it
> > only
> > > supporting per-octave detuning, I have to use two mappings on
12
> > notes
> > > per octave, assigned to separate split areas of the keyboard.
> Hard
> > to
> > > play :-(...
> >
> > There are three or four 12-out-of-22 scales I could recommend for
> you
> > for the time being . . .
>
> Yes, please do recommend!

Here's one: If you put 2 steps of 22 between each pair of keys except
1 step between E and F and between B and C, you'll have
a 'hexachordal dodecatonic' scale, within which there are several
decatonic scales. This was the mapping Ara used for 'Decatonic
Swing'. Other mappings I've used are available in Scala; are you
using Scala?

> My current mapping (c and f# the same
> above and below, and the other keys differing by 1 unit) is quite
> uncomfortable;

Don't understand . . .