Magic[16]

Thinking about how to get 16 notes to an octave to decently circulate, it occurred to me to look at 3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2. A nice circle of consonances, which as it turns out is Magic[16]. I don't recall magic being discussed much in the context of its 16 note MOS, but possessing the above circle, and the fact that the step sizes are not too terribly mismatched, suggests more consideration ought to be given to it.

http://xenharmonic.wikispaces.com/magic16

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Thinking about how to get 16 notes to an octave to decently circulate, it occurred to me to > look at 3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2.
> nice circle of consonances, which as it turns out is Magic[16]. I don't recall magic being
> discussed much in the context of its 16 note MOS, but possessing the above circle, and the
> fact that the step sizes are not too terribly mismatched, suggests more consideration ought
> to be given to it.

Not too shabby...quite far removed from 16-TET, though. What about Lemba[16]? Or Gorgo[16]? Lemba in 16-TET actually sounds pretty good, if you omit the 3/2's. I play in Lemba[6] a lot, using alternating 1/1-5/4-7/4 and 1/1-13/10-11/6 triads (the latter has got to be some kind of essentially-tempered chord). They even expand concordantly to tetrads, 1/1-5/4-7/4-35/16, 1/1-5/4-7/4-16/7 (another probably essentially-tempered chord, to make an 11/6 between 5/4 and 16/7), and 1/1-13/10-11/6-16/7 (also essentially-tempered to make a 7/4 between 13/10 and 16/7 somehow?). Whoa, I didn't realize I was using so many essentially-tempered chords in 16-TET before, but that's got to be what they are.

-Igs

If we look at 2.3.5.7.13 lemba, http://x31eq.com/cgi-bin/rt.cgi?ets=10%2C16&limit=2.3.5.7.13, we have (I think) a very good candidate for well-tempering. Since 65/64 equates 8/5 with 13/8, we have a whole range between 814 and 840 cents that will be more-or-less concordant, and we can adjust the generator chain to favor 8/5 at one end and 13/8 at the other, ranging from a 240-cent generator (of 10-ED2) to a 225-cent generator (of 16-ED2), centered of course on the ~232-cent generator that's TE-optimal. I'm not quite sure how to apply this in scale-making, but supposing we wanted to come up with some kind of 16-tone music theory, this would be a good candidate that could include 16-ED2 as a "simple but inaccurate" tuning (analogous to meantone in 12-TET), 26-ED2 as a "more accurate but more complex" tuning (analogous to meantone in 19-TET), 10-ED2 as an "over-simplified" tuning (analogous to meantone in 7-TET), and of course a host of well-temperaments and adaptive JI schemes to really dial in the concordance.

This is probably the best we'll be able to do for a system that has some plausible relation to 16-ED2. I suppose there's Gorgo[16] as well, but it's more complex and not so accurate if we insist on including the 13th harmonic (which I think we should, as it's pretty essential to use in 16-TET harmony).

Well, there's also this 2.5.7.13.19 16&20 temperament, http://x31eq.com/cgi-bin/rt.cgi?ets=16%2C+20&limit=2.5.7.13.19, which would be great for fans of the "Hendrix chord" (8:10:14:19). Sort of an alternative way of looking at the diminished scale in 16 that might also be a worthwhile temperament in its own right.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> > Thinking about how to get 16 notes to an octave to decently circulate, it occurred to me to > look at 3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2.
> > nice circle of consonances, which as it turns out is Magic[16]. I don't recall magic being
> > discussed much in the context of its 16 note MOS, but possessing the above circle, and the
> > fact that the step sizes are not too terribly mismatched, suggests more consideration ought
> > to be given to it.
>
> Not too shabby...quite far removed from 16-TET, though. What about Lemba[16]? Or Gorgo[16]? Lemba in 16-TET actually sounds pretty good, if you omit the 3/2's. I play in Lemba[6] a lot, using alternating 1/1-5/4-7/4 and 1/1-13/10-11/6 triads (the latter has got to be some kind of essentially-tempered chord). They even expand concordantly to tetrads, 1/1-5/4-7/4-35/16, 1/1-5/4-7/4-16/7 (another probably essentially-tempered chord, to make an 11/6 between 5/4 and 16/7), and 1/1-13/10-11/6-16/7 (also essentially-tempered to make a 7/4 between 13/10 and 16/7 somehow?). Whoa, I didn't realize I was using so many essentially-tempered chords in 16-TET before, but that's got to be what they are.
>
> -Igs
>

16 notes of Magic give you nine limit harmony with modulation. It isn't the obvious place to stop because there are too many notes for a diatonic and they tend to be too unevenly spaced to compete with 19 as an enharmonic. But 16 is surely enough to make music with.
I don't know how to get them to circulate.

Graham

------Original message------
From: genewardsmith <genewardsmith@...>
To: <tuning@yahoogroups.com>
Date: Friday, January 13, 2012 4:13:21 PM GMT-0000
Subject: [tuning] Magic[16]

Thinking about how to get 16 notes to an octave to decently circulate, it occurred to me to look at 3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2. A nice circle of consonances, which as it turns out is Magic[16]. I don't recall magic being discussed much in the context of its 16 note MOS, but possessing the above circle, and the fact that the step sizes are not too terribly mismatched, suggests more consideration ought to be given to it.

http://xenharmonic.wikispaces.com/magic16

------------------------------------

You can configure your subscription by sending an empty email to one
of these addresses (from the address at which you receive the list):
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - leave the group.
tuning-nomail@yahoogroups.com - turn off mail from the group.
tuning-digest@yahoogroups.com - set group to send daily digests.
tuning-normal@yahoogroups.com - set group to send individual emails.
tuning-help@yahoogroups.com - receive general help information.
Yahoo! Groups Links

No doubt it was explained somewhere else - despite that - what does it mean
" to get them to circulate."?

On Fri, Jan 13, 2012 at 3:17 PM, gbreed@gmail.com <gbreed@...> wrote:

> **
>
>
> 16 notes of Magic give you nine limit harmony with modulation. It isn't
> the obvious place to stop because there are too many notes for a diatonic
> and they tend to be too unevenly spaced to compete with 19 as an
> enharmonic. But 16 is surely enough to make music with.
> I don't know how to get them to circulate.
>
> Graham
>
> ------Original message------
> From: genewardsmith <genewardsmith@sbcglobal.net>
> To: <tuning@yahoogroups.com>
> Date: Friday, January 13, 2012 4:13:21 PM GMT-0000
> Subject: [tuning] Magic[16]
>
> Thinking about how to get 16 notes to an octave to decently circulate, it
> occurred to me to look at
> 3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2. A
> nice circle of consonances, which as it turns out is Magic[16]. I don't
> recall magic being discussed much in the context of its 16 note MOS, but
> possessing the above circle, and the fact that the step sizes are not too
> terribly mismatched, suggests more consideration ought to be given to it.
>
> http://xenharmonic.wikispaces.com/magic16
>
> ------------------------------------
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>

I mean that it works as a circulating temperament. You can transpose from one degree to another and not do too much violence to the music. 16 from Magic seen too unequal for this. (As do 21 from Miracle, mentioned way upthread.)

Graham

------Original message------
From: Chris Vaisvil <chrisvaisvil@...>
To: <tuning@yahoogroups.com>
Date: Friday, January 13, 2012 4:32:13 PM GMT-0500
Subject: Re: [tuning] Magic[16]

No doubt it was explained somewhere else - despite that - what does it mean
" to get them to circulate."?

On Fri, Jan 13, 2012 at 3:17 PM, gbreed@... <gbreed@...> wrote:

> **
>
>
> 16 notes of Magic give you nine limit harmony with modulation. It isn't
> the obvious place to stop because there are too many notes for a diatonic
> and they tend to be too unevenly spaced to compete with 19 as an
> enharmonic. But 16 is surely enough to make music with.
> I don't know how to get them to circulate.
>
> Graham
>
> ------Original message------
> From: genewardsmith <genewardsmith@...>
> To: <tuning@yahoogroups.com>
> Date: Friday, January 13, 2012 4:13:21 PM GMT-0000
> Subject: [tuning] Magic[16]
>
> Thinking about how to get 16 notes to an octave to decently circulate, it
> occurred to me to look at
> 3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2. A
> nice circle of consonances, which as it turns out is Magic[16]. I don't
> recall magic being discussed much in the context of its 16 note MOS, but
> possessing the above circle, and the fact that the step sizes are not too
> terribly mismatched, suggests more consideration ought to be given to it.
>
> http://xenharmonic.wikispaces.com/magic16
>
> ------------------------------------
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>

ok, thanks Graham - that makes sense!

On Fri, Jan 13, 2012 at 4:46 PM, gbreed@... <gbreed@...> wrote:

> **
>
>
> I mean that it works as a circulating temperament. You can transpose from
> one degree to another and not do too much violence to the music. 16 from
> Magic seen too unequal for this. (As do 21 from Miracle, mentioned way
> upthread.)
>
> Graham
>
> -
>

In case you're confused, btw, a "circulating temperament" is the same thing
as a "well temperament" - it means that some keys are "more in tune" than
others.

-Mike

On Fri, Jan 13, 2012 at 5:59 PM, Chris Vaisvil <chrisvaisvil@...>wrote:

> **
>
>
> ok, thanks Graham - that makes sense!
>
>
> On Fri, Jan 13, 2012 at 4:46 PM, gbreed@... <gbreed@...>wrote:
>
>> **
>>
>>
>> I mean that it works as a circulating temperament. You can transpose from
>> one degree to another and not do too much violence to the music. 16 from
>> Magic seen too unequal for this. (As do 21 from Miracle, mentioned way
>> upthread.)
>>
>> Graham
>>
>> -
>>
>
>
>

On 1/13/2012 1:46 PM, cityoftheasleep wrote:
> If we look at 2.3.5.7.13 lemba,
> http://x31eq.com/cgi-bin/rt.cgi?ets=10%2C16&limit=2.3.5.7.13, we have
> (I think) a very good candidate for well-tempering. Since 65/64
> equates 8/5 with 13/8, we have a whole range between 814 and 840
> cents that will be more-or-less concordant, and we can adjust the
> generator chain to favor 8/5 at one end and 13/8 at the other,
> ranging from a 240-cent generator (of 10-ED2) to a 225-cent generator
> (of 16-ED2), centered of course on the ~232-cent generator that's
> TE-optimal. I'm not quite sure how to apply this in scale-making,
> but supposing we wanted to come up with some kind of 16-tone music
> theory, this would be a good candidate that could include 16-ED2 as a
> "simple but inaccurate" tuning (analogous to meantone in 12-TET),
> 26-ED2 as a "more accurate but more complex" tuning (analogous to
> meantone in 19-TET), 10-ED2 as an "over-simplified" tuning (analogous
> to meantone in 7-TET), and of course a host of well-temperaments and
> adaptive JI schem! es to really dial in the concordance.
>
> This is probably the best we'll be able to do for a system that has
> some plausible relation to 16-ED2. I suppose there's Gorgo[16] as
> well, but it's more complex and not so accurate if we insist on
> including the 13th harmonic (which I think we should, as it's pretty
> essential to use in 16-TET harmony).
>
> Well, there's also this 2.5.7.13.19 16&20 temperament,
> http://x31eq.com/cgi-bin/rt.cgi?ets=16%2C+20&limit=2.5.7.13.19, which
> would be great for fans of the "Hendrix chord" (8:10:14:19). Sort of
> an alternative way of looking at the diminished scale in 16 that
> might also be a worthwhile temperament in its own right.
>
> -Igs

There's also diminished, not to mention the most obvious 16-note temperament: armodue!

Here is a chromatic piece in magic 16 (the tuning referenced by Gene)

I did a fair amount of editing and the playback is 2x for a better flow.

http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3

On Fri, Jan 13, 2012 at 11:13 AM, genewardsmith <genewardsmith@...
> wrote:

> **
>
>
> Thinking about how to get 16 notes to an octave to decently circulate, it
> occurred to me to look at
> 3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2. A
> nice circle of consonances, which as it turns out is Magic[16]. I don't
> recall magic being discussed much in the context of its 16 note MOS, but
> possessing the above circle, and the fact that the step sizes are not too
> terribly mismatched, suggests more consideration ought to be given to it.
>
> http://xenharmonic.wikispaces.com/magic16
>
>
>

On Fri, Jan 13, 2012 at 8:44 PM, Herman Miller <hmiller@...> wrote:
>
> There's also diminished, not to mention the most obvious 16-note
> temperament: armodue!

I was going to suggest this, but I didn't feel like getting into
another argument about how everyone's just pretending to like armodue
to be cool, or whatever it was.

-Mike

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

> There's also diminished, not to mention the most obvious 16-note
> temperament: armodue!

Diminished[16] doesn't make a lot of sense, since it's so close to optimal in 12-TET. The 2.5.7.13.19 16&20 temperament makes more sense and leads to the same scale in 16-TET. Armodue is probably too controversial; a lot of people like it, a lot of people think the fifths are too far off. Even I think the fifths are too far off, although I occasionally get in a gamelanesque mood and enjoy the beating.

-Igs

On Fri, Jan 13, 2012 at 11:13 AM, genewardsmith
<genewardsmith@...t> wrote:
>
> Thinking about how to get 16 notes to an octave to decently circulate, it occurred to me to look at 3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2. A nice circle of consonances, which as it turns out is Magic[16]. I don't recall magic being discussed much in the context of its 16 note MOS, but possessing the above circle, and the fact that the step sizes are not too terribly mismatched, suggests more consideration ought to be given to it.
>
> http://xenharmonic.wikispaces.com/magic16

You know, I never realized before that this isn't actual magic.

Is there any way we could get a lesfip out of this one like we did
with porcupine? What happens if you lesfip starting with this guy
right here?

Although given what happened with porcupine, it might be better to
start with a generator even flatter than usual, because lesfipping
will undo some of the damage and make it all circulate better. So if
magic[16] with ~380 cents doesn't work well as a generator to seed the
lesfip, ~378 or ~376 might do the trick.

-Mike

On Thu, Jan 26, 2012 at 3:51 AM, Mike Battaglia <battaglia01@...> wrote:
> On Fri, Jan 13, 2012 at 11:13 AM, genewardsmith
> <genewardsmith@sbcglobal.net> wrote:
>>
>> Thinking about how to get 16 notes to an octave to decently circulate, it occurred to me to look at 3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2-3/2-10/7-3/2. A nice circle of consonances, which as it turns out is Magic[16]. I don't recall magic being discussed much in the context of its 16 note MOS, but possessing the above circle, and the fact that the step sizes are not too terribly mismatched, suggests more consideration ought to be given to it.
>>
>> http://xenharmonic.wikispaces.com/magic16
>
> You know, I never realized before that this isn't actual magic.

I mean, that it is actual magic. I thought it was a well temperament.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Although given what happened with porcupine, it might be better to
> start with a generator even flatter than usual, because lesfipping
> will undo some of the damage and make it all circulate better. So if
> magic[16] with ~380 cents doesn't work well as a generator to seed the
> lesfip, ~378 or ~376 might do the trick.

Starting with 376 I got all sorts of strange things. I'm not sure what you want--my point was that because of the alternation of 3/2 with 10/7, Magic[16] already alternates in a manner of speaking.

On Thu, Jan 26, 2012 at 5:51 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Although given what happened with porcupine, it might be better to
> > start with a generator even flatter than usual, because lesfipping
> > will undo some of the damage and make it all circulate better. So if
> > magic[16] with ~380 cents doesn't work well as a generator to seed the
> > lesfip, ~378 or ~376 might do the trick.
>
> Starting with 376 I got all sorts of strange things.

How strange are we talking here?

> I'm not sure what you want--my point was that because of the alternation of 3/2 with 10/7, Magic[16] already alternates in a manner of speaking.

I was just curious what'd come out. The best 3/2's would be about as
good as these, and then the second-best ones would be close to 16-EDO,
and then the worst ones would be flat. I'm curious if there's a way
for even the flat ones to be around 666 cents or so, where 9-EDO is.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > Starting with 376 I got all sorts of strange things.
>
> How strange are we talking here?

For example:

! strange16.scl
!
Lesfip scale, 11-limit diamond, 10 cents tolerance
16
!
116.94577
177.73850
266.17058
322.76186
381.86836
498.63344
557.73994
614.33123
702.76331
763.55603
880.50180
936.15732
996.30671
1084.19509
1144.34448
1200.00000

> > I'm not sure what you want--my point was that because of the alternation of 3/2 with 10/7, Magic[16] already alternates in a manner of speaking.
>
> I was just curious what'd come out. The best 3/2's would be about as
> good as these, and then the second-best ones would be close to 16-EDO,
> and then the worst ones would be flat. I'm curious if there's a way
> for even the flat ones to be around 666 cents or so, where 9-EDO is.

If I start nearer to Magic[16], it seems to me the 3/2 and 10/7s in a circle, among other things, are going to pull it into sticking pretty close to Magic[16].

On Thu, Jan 26, 2012 at 7:32 AM, genewardsmith
<genewardsmith@...> wrote:
>
> For example:
>
> ! strange16.scl
> !
> Lesfip scale, 11-limit diamond, 10 cents tolerance
> 16
> !
> 116.94577
> 177.73850
> 266.17058
> 322.76186
> 381.86836
> 498.63344
> 557.73994
> 614.33123
> 702.76331
> 763.55603
> 880.50180
> 936.15732
> 996.30671
> 1084.19509
> 1144.34448
> 1200.00000

That is kind of strange. But also promising.

You know, another nice way to deal with 16-EDO is that the 4/3's are
right between actual 4/3's and 11/8. As a result, two of them is an
almost perfect 12/11. It might be interesting to consider circulations
of that sort as well, with 8 4/3's and 8 11/8's. I think that'd
basically just be demolished[16] though.

-Mike

got it - thank you!

On Thu, Jan 26, 2012 at 7:32 AM, genewardsmith
<genewardsmith@...>wrote:

> **
>
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > Starting with 376 I got all sorts of strange things.
> >
> > How strange are we talking here?
>
> For example:
>
> ! strange16.scl
> !
> Lesfip scale, 11-limit diamond, 10 cents tolerance
> 16
> !
> 116.94577
> 177.73850
> 266.17058
> 322.76186
> 381.86836
> 498.63344
> 557.73994
> 614.33123
> 702.76331
> 763.55603
> 880.50180
> 936.15732
> 996.30671
> 1084.19509
> 1144.34448
> 1200.00000
>
>
> > > I'm not sure what you want--my point was that because of the
> alternation of 3/2 with 10/7, Magic[16] already alternates in a manner of
> speaking.
> >
> > I was just curious what'd come out. The best 3/2's would be about as
> > good as these, and then the second-best ones would be close to 16-EDO,
> > and then the worst ones would be flat. I'm curious if there's a way
> > for even the flat ones to be around 666 cents or so, where 9-EDO is.
>
> If I start nearer to Magic[16], it seems to me the 3/2 and 10/7s in a
> circle, among other things, are going to pull it into sticking pretty close
> to Magic[16].
>
>
>