The farthest from meantone/pythagorean

Hi everybody! I've been dormant in the microsphere for awhile, and
this post probably isn't going to herald a comeback, but I wanted to
share a simple but extremely xenharmonic piece of music I recently
scraped together. It's at my soundclick site:
http://www.soundclick.com/bands/1/cityoftheasleepmusic.htm
It's called "Breakdown Hives" and is basically in 11-tET. Well, the
intro is actually based on the idea of two 11-tET guitars tuned 1
step of 22-tET apart, but the rest is straight-up 11. I have
honestly come to a point where I'm absolutely tired of the usually
approximated 3- and 5-limit sonorities and I want to play something
that's really jarring and "off" to the western ear. I'm also getting
tired of navigating all of these tightly-packed frets! So I think
for my next instrument, I'm setting the following limitations:

1. fewer than 19 tones per 2/1 (or approximate 2/1)
2. Approximate 3/2's or 4/3's must be AT LEAST 15 cents off
(preferably more).
3. NO 5/4's or 6/5's closer to Just than those of 12-equal

I'm only considering equal temperaments, since J.I. on guitar is more
of a bother than I want to deal with.

I'm curious as to what anyone on this forum might suggest. Right
now, I'm thinking either 13, 18, 10, or 14 equal divisions of the
octave, or maybe even something non-octave.

Does anyone here have any suggestions on the matter? I'm going for
maximum weirdness here.

Thanks!

-Igs

> 1. fewer than 19 tones per 2/1 (or approximate 2/1)
> 2. Approximate 3/2's or 4/3's must be AT LEAST 15 cents off
> (preferably more).
> 3. NO 5/4's or 6/5's closer to Just than those of 12-equal
>
> I'm only considering equal temperaments, since J.I. on guitar is more
> of a bother than I want to deal with.
>
> I'm curious as to what anyone on this forum might suggest. Right
> now, I'm thinking either 13, 18, 10, or 14 equal divisions of the
> octave, or maybe even something non-octave.
>
> Does anyone here have any suggestions on the matter? I'm going for
> maximum weirdness here.

13 or 16 or 18 might be evil enough for you. Look also at equal
divisions of the 4/1 - for example 21st root of 4, aka
10-and-a-half-tet. There's also that scale that Hanuman Zhang calls "a
very WOLF-colour-saturated version/mutant perversion of 11tET/EDO,"
or, occasionally, "throbbin chthonic bleu": 7th root of 20/13. With a
name like TCB, it's gotta be good.

Hi Igs,

I remember Gary Morrison going for shock value with 88-CET.
It was the 9:7 thirds, in particular, that did the trick he
said.

-C.

>Hi everybody! I've been dormant in the microsphere for awhile, and
>this post probably isn't going to herald a comeback, but I wanted to
>share a simple but extremely xenharmonic piece of music I recently
>scraped together. It's at my soundclick site:
>http://www.soundclick.com/bands/1/cityoftheasleepmusic.htm
>It's called "Breakdown Hives" and is basically in 11-tET. Well, the
>intro is actually based on the idea of two 11-tET guitars tuned 1
>step of 22-tET apart, but the rest is straight-up 11. I have
>honestly come to a point where I'm absolutely tired of the usually
>approximated 3- and 5-limit sonorities and I want to play something
>that's really jarring and "off" to the western ear. I'm also getting
>tired of navigating all of these tightly-packed frets! So I think
>for my next instrument, I'm setting the following limitations:
>
>1. fewer than 19 tones per 2/1 (or approximate 2/1)
>2. Approximate 3/2's or 4/3's must be AT LEAST 15 cents off
>(preferably more).
>3. NO 5/4's or 6/5's closer to Just than those of 12-equal
>
>I'm only considering equal temperaments, since J.I. on guitar is more
>of a bother than I want to deal with.
>
>I'm curious as to what anyone on this forum might suggest. Right
>now, I'm thinking either 13, 18, 10, or 14 equal divisions of the
>octave, or maybe even something non-octave.
>
>Does anyone here have any suggestions on the matter? I'm going for
>maximum weirdness here.
>
>Thanks!
>
>-Igs
>
>
>
>
>
>
>Yahoo! Groups Links
>
>
>
>

hi Igs,

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones" <igliashon@s.
..> wrote:

> <snip> ... I have honestly come to a point where I'm
> absolutely tired of the usually approximated 3- and 5-limit
> sonorities and I want to play something that's really
> jarring and "off" to the western ear. I'm also getting
> tired of navigating all of these tightly-packed frets!
> So I think for my next instrument, I'm setting the
> following limitations:
>
> 1. fewer than 19 tones per 2/1 (or approximate 2/1)
> 2. Approximate 3/2's or 4/3's must be AT LEAST 15 cents off
> (preferably more).
> 3. NO 5/4's or 6/5's closer to Just than those of 12-equal
>
> I'm only considering equal temperaments, since J.I. on
> guitar is more of a bother than I want to deal with.
>
> I'm curious as to what anyone on this forum might suggest.
> Right now, I'm thinking either 13, 18, 10, or 14 equal
> divisions of the octave, or maybe even something non-octave.
>
> Does anyone here have any suggestions on the matter? I'm
> going for maximum weirdness here.

i just recently posted some observations about prime-space
approximations of 11-edo and 13-edo, on the tuning list:

/tuning/topicId_58502.html#58502

/tuning/topicId_58502.html#58503

i have a webpage about "EDO prime error" which shows you
the relative accuracy of EDOs in representing prime-factors:

http://tonalsoft.com/enc/edo-prime-error.htm

by "relative", i mean that the error is shown as a
percentage of one EDO degree, rather than an "absolute"
error in, say, cents.

you can see that 11-edo is pretty good at representing
prime-factor 7, and quite good for 11 and 17. so in 11-edo
you're probably able to perceive the approximations to
ratios containing these prime-factors.

13-edo does a really good job with prime-factor 11,
and then is mediocre (but not really bad) for a bunch
of other primes: 13, 17, 23, 29.

14-edo is really good with 29, 41, and 43, but doesn't
do any of the lower primes well.

and 18-edo doesn't really give serviceable approximations
of any of the primes in my graphs, that is, any of them up
to 43. so maybe that's the one you want.

-monz
http://tonalsoft.com
microtonal music software

Igliashon,

You wrote:
> I'm curious as to what anyone on this forum might suggest. Right
> now, I'm thinking either 13, 18, 10, or 14 equal divisions of the
> octave, or maybe even something non-octave.
>
> Does anyone here have any suggestions on the matter? I'm going for
> maximum weirdness here.

I haven't done the maths (yet), but I suspect you'd get
"weird" if you tried 14 equal divisions of the 7/4.
One reason this appeals to me is that your period is
a "short octave"; makes it easier to sing to over
several "octaves".

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.308 / Virus Database: 266.11.6 - Release Date: 6/5/05

Something I like to stress, probably to the point where peoplewould
be throwing tomatoes at me if they could, is that humans make the
music, not the tunings.Or put another way, I have a 13-tet tenor uke
and a 2O-tet guitar and so on and so forth, and early on I found that
these types of non teritan tunings could easily make "pretty" music
if _I_ wanted them to. Same goes for 3 and 5 limit tunings....if
they're what I've got to work with and I want "ugly" or "brutal",
then the tuning sure as hell ain't gonnastop me much! That
said,there's an obvious truth to the fact that different tunings are
different.....but that's tuning,not music and this is a difference!
oK,SOrry. You know it seems we really have some similar tendencies,
because I agree with you that many guitarist--not all, but many--used
to the standard fretting layout are almost always going to find
higher ETs or complex just frettings somewhat cramped and crowded.
Interface is important, and this is one reason why I'd mention
fretless here. The interface is pretty liberating and the
possibilities for punishing discord and alien pandemonium are many!
Anyway, keep up the good work and I'm sure you'll find something you
like and I'll try to check your 11-teter next time I drop by the
library....btw, I'm have a 14-tet classical guitar in the works right
now .

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
> Hi everybody! I've been dormant in the microsphere for awhile, and
> this post probably isn't going to herald a comeback, but I wanted
to
> share a simple but extremely xenharmonic piece of music I recently
> scraped together. It's at my soundclick site:
> http://www.soundclick.com/bands/1/cityoftheasleepmusic.htm
> It's called "Breakdown Hives" and is basically in 11-tET. Well,
the
> intro is actually based on the idea of two 11-tET guitars tuned 1
> step of 22-tET apart, but the rest is straight-up 11. I have
> honestly come to a point where I'm absolutely tired of the usually
> approximated 3- and 5-limit sonorities and I want to play something
> that's really jarring and "off" to the western ear. I'm also
getting
> tired of navigating all of these tightly-packed frets! So I think
> for my next instrument, I'm setting the following limitations:
>
> 1. fewer than 19 tones per 2/1 (or approximate 2/1)
> 2. Approximate 3/2's or 4/3's must be AT LEAST 15 cents off
> (preferably more).
> 3. NO 5/4's or 6/5's closer to Just than those of 12-equal
>
> I'm only considering equal temperaments, since J.I. on guitar is
more
> of a bother than I want to deal with.
>
> I'm curious as to what anyone on this forum might suggest. Right
> now, I'm thinking either 13, 18, 10, or 14 equal divisions of the
> octave, or maybe even something non-octave.
>
> Does anyone here have any suggestions on the matter? I'm going for
> maximum weirdness here.
>
> Thanks!
>
> -Igs

Ahoy, Dan!

Just want to clear up a little bit that by "jarring/off" I didn't
necessarily mean "nasty" or "dissonant". It's very much the case
that all of my current micro-gitarras are capable of this, but
basically I just want an instrument upon which it is *impossible* to
play the usual sonorities.

Do any of your songs feature the 13-EDO uke? 13 is inching out ahead
as the top candidate, though 18 would be easier to refret to since I
could leave every other fret-slot from 12 (I'm attempting this one
myself...to heck with paying $500 for a new neck!). I need to hear
more music featuring them. I've thought about going fretless, but
frankly I enjoy the constraints of frets. Playing fretless opens up
too many possibilities all at once for me, and I really enjoy playing
with numbers and scale arrangements in an ET framework. I have a
fretless bass, but I really don't find it as inspiring as any of my
micro-guitars.

Let me know how your 14-tET turns out!

-Igs

--- In MakeMicroMusic@yahoogroups.com, "daniel_anthony_stearns"
<daniel_anthony_stearns@y...> wrote:
> Something I like to stress, probably to the point where peoplewould
> be throwing tomatoes at me if they could, is that humans make the
> music, not the tunings.Or put another way, I have a 13-tet tenor
uke
> and a 2O-tet guitar and so on and so forth, and early on I found
that
> these types of non teritan tunings could easily make "pretty" music
> if _I_ wanted them to. Same goes for 3 and 5 limit tunings....if
> they're what I've got to work with and I want "ugly" or "brutal",
> then the tuning sure as hell ain't gonnastop me much! That
> said,there's an obvious truth to the fact that different tunings
are
> different.....but that's tuning,not music and this is a difference!
> oK,SOrry. You know it seems we really have some similar tendencies,
> because I agree with you that many guitarist--not all, but many--
used
> to the standard fretting layout are almost always going to find
> higher ETs or complex just frettings somewhat cramped and crowded.
> Interface is important, and this is one reason why I'd mention
> fretless here. The interface is pretty liberating and the
> possibilities for punishing discord and alien pandemonium are many!
> Anyway, keep up the good work and I'm sure you'll find something
you
> like and I'll try to check your 11-teter next time I drop by the
> library....btw, I'm have a 14-tet classical guitar in the works
right
> now .
>
>
>
> --- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
> <igliashon@s...> wrote:
> > Hi everybody! I've been dormant in the microsphere for awhile,
and
> > this post probably isn't going to herald a comeback, but I wanted
> to
> > share a simple but extremely xenharmonic piece of music I
recently
> > scraped together. It's at my soundclick site:
> > http://www.soundclick.com/bands/1/cityoftheasleepmusic.htm
> > It's called "Breakdown Hives" and is basically in 11-tET. Well,
> the
> > intro is actually based on the idea of two 11-tET guitars tuned 1
> > step of 22-tET apart, but the rest is straight-up 11. I have
> > honestly come to a point where I'm absolutely tired of the
usually
> > approximated 3- and 5-limit sonorities and I want to play
something
> > that's really jarring and "off" to the western ear. I'm also
> getting
> > tired of navigating all of these tightly-packed frets! So I
think
> > for my next instrument, I'm setting the following limitations:
> >
> > 1. fewer than 19 tones per 2/1 (or approximate 2/1)
> > 2. Approximate 3/2's or 4/3's must be AT LEAST 15 cents off
> > (preferably more).
> > 3. NO 5/4's or 6/5's closer to Just than those of 12-equal
> >
> > I'm only considering equal temperaments, since J.I. on guitar is
> more
> > of a bother than I want to deal with.
> >
> > I'm curious as to what anyone on this forum might suggest. Right
> > now, I'm thinking either 13, 18, 10, or 14 equal divisions of the
> > octave, or maybe even something non-octave.
> >
> > Does anyone here have any suggestions on the matter? I'm going
for
> > maximum weirdness here.
> >
> > Thanks!
> >
> > -Igs

>7th root of 20/13. With a name like TCB, it's gotta be good.

Whoa, that's a strange one! I think I might like that one better than
88-cET! Do you know of any music that has been written in it?

Hmm...do I want to be the first kid on my block to have a guitar with
no octaves or fourths/fifths? That could be worth it for the shock-
value alone, but then again it could be a little too much. Decisions,
decisions!

-Igs

I like 16 because it has a scale the opposite of the major with its llsllls being s s l s s s l

Igliashon Jones wrote:

>Ahoy, Dan!
>
>Just want to clear up a little bit that by "jarring/off" I didn't >necessarily mean "nasty" or "dissonant". It's very much the case >that all of my current micro-gitarras are capable of this, but >basically I just want an instrument upon which it is *impossible* to >play the usual sonorities.
>
>Do any of your songs feature the 13-EDO uke? 13 is inching out ahead >as the top candidate, though 18 would be easier to refret to since I >could leave every other fret-slot from 12 (I'm attempting this one >myself...to heck with paying $500 for a new neck!). I need to hear >more music featuring them. I've thought about going fretless, but >frankly I enjoy the constraints of frets. Playing fretless opens up >too many possibilities all at once for me, and I really enjoy playing >with numbers and scale arrangements in an ET framework. I have a >fretless bass, but I really don't find it as inspiring as any of my >micro-guitars.
>
>Let me know how your 14-tET turns out!
>
>-Igs
>
>--- In MakeMicroMusic@yahoogroups.com, "daniel_anthony_stearns" ><daniel_anthony_stearns@y...> wrote:
> >
>>Something I like to stress, probably to the point where peoplewould >>be throwing tomatoes at me if they could, is that humans make the >>music, not the tunings.Or put another way, I have a 13-tet tenor >> >>
>uke > >
>>and a 2O-tet guitar and so on and so forth, and early on I found >> >>
>that > >
>>these types of non teritan tunings could easily make "pretty" music >>if _I_ wanted them to. Same goes for 3 and 5 limit tunings....if >>they're what I've got to work with and I want "ugly" or "brutal", >>then the tuning sure as hell ain't gonnastop me much! That >>said,there's an obvious truth to the fact that different tunings >> >>
>are > >
>>different.....but that's tuning,not music and this is a difference! >>oK,SOrry. You know it seems we really have some similar tendencies, >>because I agree with you that many guitarist--not all, but many--
>> >>
>used > >
>>to the standard fretting layout are almost always going to find >>higher ETs or complex just frettings somewhat cramped and crowded. >>Interface is important, and this is one reason why I'd mention >>fretless here. The interface is pretty liberating and the >>possibilities for punishing discord and alien pandemonium are many! >>Anyway, keep up the good work and I'm sure you'll find something >> >>
>you > >
>>like and I'll try to check your 11-teter next time I drop by the >>library....btw, I'm have a 14-tet classical guitar in the works >> >>
>right > >
>>now .
>>
>>
>>
>>--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones" >><igliashon@s...> wrote:
>> >>
>>>Hi everybody! I've been dormant in the microsphere for awhile, >>> >>>
>and > >
>>>this post probably isn't going to herald a comeback, but I wanted >>> >>>
>>to >> >>
>>>share a simple but extremely xenharmonic piece of music I >>> >>>
>recently > >
>>>scraped together. It's at my soundclick site:
>>>http://www.soundclick.com/bands/1/cityoftheasleepmusic.htm
>>>It's called "Breakdown Hives" and is basically in 11-tET. Well, >>> >>>
>>the >> >>
>>>intro is actually based on the idea of two 11-tET guitars tuned 1 >>>step of 22-tET apart, but the rest is straight-up 11. I have >>>honestly come to a point where I'm absolutely tired of the >>> >>>
>usually > >
>>>approximated 3- and 5-limit sonorities and I want to play >>> >>>
>something > >
>>>that's really jarring and "off" to the western ear. I'm also >>> >>>
>>getting >> >>
>>>tired of navigating all of these tightly-packed frets! So I >>> >>>
>think > >
>>>for my next instrument, I'm setting the following limitations: >>>
>>>1. fewer than 19 tones per 2/1 (or approximate 2/1)
>>>2. Approximate 3/2's or 4/3's must be AT LEAST 15 cents off >>>(preferably more).
>>>3. NO 5/4's or 6/5's closer to Just than those of 12-equal
>>>
>>>I'm only considering equal temperaments, since J.I. on guitar is >>> >>>
>>more >> >>
>>>of a bother than I want to deal with.
>>>
>>>I'm curious as to what anyone on this forum might suggest. Right >>>now, I'm thinking either 13, 18, 10, or 14 equal divisions of the >>>octave, or maybe even something non-octave.
>>>
>>>Does anyone here have any suggestions on the matter? I'm going >>> >>>
>for > >
>>>maximum weirdness here.
>>>
>>>Thanks!
>>>
>>>-Igs
>>> >>>
>
>
>
>
>
> >Yahoo! Groups Links
>
>
>
> >
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

> Hi Igs,
>
> I remember Gary Morrison going for shock value with 88-CET.
> It was the 9:7 thirds, in particular, that did the trick he
> said.

Do you know if there are any recordings of his music? I've really
noticed how the fact that I'm a guitarist affects my approach to
microtonality. In rock music, you don't find a lot of complex vertical
structures, so it really frees you up to use less harmonic tunings. If
I recall, 88-CET is actually pretty good for harmony--I think
its "shock value" is probably higher for more orchestrated music than
it would be for my weird brand of rock, since 88-CET does have good
3/2's (which I'm trying to avoid, since 3/2 is THE rock interval).
Then again, the rest of 88-CET is about as far from standard-practice
stuff as you can get...I really need to hear more. A worthy suggestion
though, to be sure.

-Igs

>> Hi Igs,
>>
>> I remember Gary Morrison going for shock value with 88-CET.
>> It was the 9:7 thirds, in particular, that did the trick he
>> said.
>
>Do you know if there are any recordings of his music?

Sure; I have several.

>since 88-CET does have good 3/2's (which I'm trying to avoid,

Gary created to hit consonances, actually.

It strikes me that a reversal of the tuning-math stuff might
be interesting... Gene could probably give you some candidates
for least-just tunings.

-Carl

> i have a webpage about "EDO prime error" which shows you
> the relative accuracy of EDOs in representing prime-factors:
>
> http://tonalsoft.com/enc/edo-prime-error.htm
>

Excellent. When you say "prime-factors", you're referring
exclusively to ratios with prime numerators and denominators of 1,
right?

> and 18-edo doesn't really give serviceable approximations
> of any of the primes in my graphs, that is, any of them up
> to 43. so maybe that's the one you want.

I think it's useful (in my case) to think not just in terms of prime-
factors but also in terms of the "common" Just 12-tone scale (1:1
16:15 9:8 6:5 5:4 4:3 [some type of tritone] 3:2 8:5 5:3 16:9 15:8
2:1). Basically I'm trying to get as far from this scale as possible
without getting more than 19 notes in a/an approximate 2:1. In that
sense, 13 and 14 seem to be the prime candidates, and 18 doesnt' seem
so ideal since 1 out of every 3 intervals in 18-tET can be found in
12-tET.

Decisions, decisions!

-Igs

> I like 16 because it has a scale the opposite of the major with its
> llsllls being s s l s s s l

Whoa, interesting! I never noticed that about 16 before. That's a
very nice melodic scale, too! Maybe 16 is worth more consideration
than I've given it...

-Igs

> It strikes me that a reversal of the tuning-math stuff might
> be interesting... Gene could probably give you some candidates
> for least-just tunings.

Indeed, that's certainly an interesting approach I have yet to see. Do
you think I should cross-post to the Tuning list?

-Igs

On 5/11/05, Igliashon Jones <igliashon@...> wrote:
>
> > It strikes me that a reversal of the tuning-math stuff might
> > be interesting... Gene could probably give you some candidates
> > for least-just tunings.
>
> Indeed, that's certainly an interesting approach I have yet to see. Do
> you think I should cross-post to the Tuning list?

It looks like it's still here.

I have been playing with this. I calculated over some 5-limit ratios,
with the largest integer no greater than 8, weighted by the geometric
mean. Taking the RMS error, any x.5-edo scale looks weird, and scales
look less weird the more notes they have to the octave. 13.5 and 15.5
edo stand out a bit, but not much.

Taking the smallest, weighted error gives more interesting results.
The worst in the range 8-18 is 8.3 edo (13.2), then 12.9 (7.0), 10.7
(4.9), 13.8 (4.6), 17.3 (4.2) and 16.6 (3.7). Numbers in parentheses
are the (dimensionless) smallest weighted error multiplied by a
million.

For comparison, the 7th root of 20/13 scale is 11.3 edo and 88 cet is 13.6 edo.

Caveats apply, of course -- there's no guarantee they will sound weird
because the theory says so and there are plenty of arbitrary features
in the theory.

Graham

On Wednesday 11 May 2005 3:09 am, Igliashon Jones wrote:
> > It strikes me that a reversal of the tuning-math stuff might
> > be interesting... Gene could probably give you some candidates
> > for least-just tunings.
>
> Indeed, that's certainly an interesting approach I have yet to see. Do
> you think I should cross-post to the Tuning list?

Igs, by my recollection, 11 and 13 are very good candidates for maximum
un-tradtional sound. You also might want to farm scala for some really off
the wall irrational number tunings (like those based on phi, which at least
in theory, are maximully vague or complex)

Cheers,
Aaron.

At 07:31 AM 5/11/2005, you wrote:
>On Wednesday 11 May 2005 3:09 am, Igliashon Jones wrote:
>> > It strikes me that a reversal of the tuning-math stuff might
>> > be interesting... Gene could probably give you some candidates
>> > for least-just tunings.
>>
>> Indeed, that's certainly an interesting approach I have yet to see. Do
>> you think I should cross-post to the Tuning list?
>
>Igs, by my recollection, 11 and 13 are very good candidates for maximum
>un-tradtional sound.

They still have excellent octaves.

>You also might want to farm scala for some really off the wall
>irrational number tunings (like those based on phi, which at least
>in theory, are maximully vague or complex)

Phi makes for the 'least rational' dyad, but due to TOLERANCE
it actually makes a passable 13:8 IIRC.

-Carl

On Phi, as with most recurrent sequences , it has a consonance of it its own since the difference tone generated are different tones in the scale. So there is a limit to ow dissonant one can get. This can be heard in "the Stolen Stars" that dispute all the different subscales, it all gels together.
If dissonance is what you what, this is the wrong direction to look.

Carl Lumma wrote:

>At 07:31 AM 5/11/2005, you wrote:
> >
>>On Wednesday 11 May 2005 3:09 am, Igliashon Jones wrote:
>> >>
>>>>It strikes me that a reversal of the tuning-math stuff might
>>>>be interesting... Gene could probably give you some candidates
>>>>for least-just tunings.
>>>> >>>>
>>>Indeed, that's certainly an interesting approach I have yet to see. Do
>>>you think I should cross-post to the Tuning list?
>>> >>>
>>Igs, by my recollection, 11 and 13 are very good candidates for maximum >>un-tradtional sound.
>> >>
>
>They still have excellent octaves.
>
> >
>>You also might want to farm scala for some really off the wall
>>irrational number tunings (like those based on phi, which at least >>in theory, are maximully vague or complex)
>> >>
>
>Phi makes for the 'least rational' dyad, but due to TOLERANCE
>it actually makes a passable 13:8 IIRC.
>
>-Carl
>
>
>
>
> >Yahoo! Groups Links
>
>
>
> >
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

>>>You also might want to farm scala for some really off the wall
>>>irrational number tunings (like those based on phi, which at least
>>>in theory, are maximully vague or complex)
>>
>>Phi makes for the 'least rational' dyad, but due to TOLERANCE
>>it actually makes a passable 13:8 IIRC.
>
>On Phi, as with most recurrent sequences, it has a consonance of it
>its own since the difference tone generated are different tones in the
>scale. So there is a limit to ow dissonant one can get. This can be
>heard in "the Stolen Stars" that dispute all the different subscales,
>it all gels together.
> If dissonance is what you what, this is the wrong direction to look.

This is only the case if you use Phi in the particular way you're
referring to.

-Carl

hi Igs,

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones" <igliashon@s.
..> wrote:

>
> > i have a webpage about "EDO prime error" which shows you
> > the relative accuracy of EDOs in representing prime-factors:
> >
> > http://tonalsoft.com/enc/edo-prime-error.htm
> >
>
> Excellent. When you say "prime-factors", you're referring
> exclusively to ratios with prime numerators and denominators
> of 1, right?

yes, and also the reverse, prime denominators with numerators
of 1, because these graphs show absolute values.

> > and 18-edo doesn't really give serviceable approximations
> > of any of the primes in my graphs, that is, any of them up
> > to 43. so maybe that's the one you want.
>
> I think it's useful (in my case) to think not just in terms
> of prime-factors but also in terms of the "common" Just
> 12-tone scale (1:1 16:15 9:8 6:5 5:4 4:3 [some type of tritone]
> 3:2 8:5 5:3 16:9 15:8 2:1). Basically I'm trying to get as
> far from this scale as possible without getting more than
> 19 notes in a/an approximate 2:1. In that sense, 13 and
> 14 seem to be the prime candidates, and 18 doesnt' seem
> so ideal since 1 out of every 3 intervals in 18-tET can
> be found in 12-tET.

well, in that case you want to look at this one:

http://tonalsoft.com/enc/edo-11-odd-limit-error.htm

this is different from the other in that it shows
all the combinations of ratios in the 11-limit, and
also gives positive and negative error values, not
absolutes.

-monz
http://tonalsoft.com
microtonal music software

It is a acoustical phenomenon of the tuning, in the same way it we do a piece just on the harmonic series, one is only going to be able to get so dissonant.

Carl Lumma wrote:

>>>>You also might want to farm scala for some really off the wall
>>>>irrational number tunings (like those based on phi, which at least >>>>in theory, are maximully vague or complex)
>>>> >>>>
>>>Phi makes for the 'least rational' dyad, but due to TOLERANCE
>>>it actually makes a passable 13:8 IIRC.
>>> >>>
>>On Phi, as with most recurrent sequences, it has a consonance of it
>>its own since the difference tone generated are different tones in the >>scale. So there is a limit to ow dissonant one can get. This can be >>heard in "the Stolen Stars" that dispute all the different subscales,
>>it all gels together.
>>If dissonance is what you what, this is the wrong direction to look.
>> >>
>
>This is only the case if you use Phi in the particular way you're
>referring to.
>
>-Carl
>
>
>
>
> >Yahoo! Groups Links
>
>
>
> >
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

>>>>Phi makes for the 'least rational' dyad, but due to TOLERANCE
>>>>it actually makes a passable 13:8 IIRC.
>>>
>>>On Phi, as with most recurrent sequences, it has a consonance of it
>>>its own since the difference tone generated are different tones in the
>>>scale. So there is a limit to ow dissonant one can get. This can be
>>>heard in "the Stolen Stars" that dispute all the different subscales,
>>>it all gels together.
>>>If dissonance is what you what, this is the wrong direction to look.
>>
>>This is only the case if you use Phi in the particular way you're
>>referring to.
>
>It is a acoustical phenomenon of the tuning, in the same way it we do a
>piece just on the harmonic series, one is only going to be able to get
>so dissonant.

I'm not arguing with that. I was just saying that there is more
than one way to use Phi to get a scale or tuning.

-Carl

> If dissonance is what you what, this is the wrong direction to
look.

Not dissonance, just weirdness. Maybe this Phi thing is worth
looking into...anyone have any Scala files that I could use to check
it out?

-igs

> Carl Lumma wrote:
>
> >At 07:31 AM 5/11/2005, you wrote:
> >
> >
> >>On Wednesday 11 May 2005 3:09 am, Igliashon Jones wrote:
> >>
> >>
> >>>>It strikes me that a reversal of the tuning-math stuff might
> >>>>be interesting... Gene could probably give you some candidates
> >>>>for least-just tunings.
> >>>>
> >>>>
> >>>Indeed, that's certainly an interesting approach I have yet to
see. Do
> >>>you think I should cross-post to the Tuning list?
> >>>
> >>>
> >>Igs, by my recollection, 11 and 13 are very good candidates for
maximum
> >>un-tradtional sound.
> >>
> >>
> >
> >They still have excellent octaves.
> >
> >
> >
> >>You also might want to farm scala for some really off the wall
> >>irrational number tunings (like those based on phi, which at
least
> >>in theory, are maximully vague or complex)
> >>
> >>
> >
> >Phi makes for the 'least rational' dyad, but due to TOLERANCE
> >it actually makes a passable 13:8 IIRC.
> >
> >-Carl
> >
> >
> >
> >
> >
> >Yahoo! Groups Links
> >
> >
> >
> >
> >
> >
> >
> >
> >
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Monz, you da man. Just what the doctor ordered!

-Igs

--- In MakeMicroMusic@yahoogroups.com, "monz" <monz@t...> wrote:
> hi Igs,
>
> --- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s.
> ..> wrote:
>
> >
> > > i have a webpage about "EDO prime error" which shows you
> > > the relative accuracy of EDOs in representing prime-factors:
> > >
> > > http://tonalsoft.com/enc/edo-prime-error.htm
> > >
> >
> > Excellent. When you say "prime-factors", you're referring
> > exclusively to ratios with prime numerators and denominators
> > of 1, right?
>
>
> yes, and also the reverse, prime denominators with numerators
> of 1, because these graphs show absolute values.
>
>
> > > and 18-edo doesn't really give serviceable approximations
> > > of any of the primes in my graphs, that is, any of them up
> > > to 43. so maybe that's the one you want.
> >
> > I think it's useful (in my case) to think not just in terms
> > of prime-factors but also in terms of the "common" Just
> > 12-tone scale (1:1 16:15 9:8 6:5 5:4 4:3 [some type of tritone]
> > 3:2 8:5 5:3 16:9 15:8 2:1). Basically I'm trying to get as
> > far from this scale as possible without getting more than
> > 19 notes in a/an approximate 2:1. In that sense, 13 and
> > 14 seem to be the prime candidates, and 18 doesnt' seem
> > so ideal since 1 out of every 3 intervals in 18-tET can
> > be found in 12-tET.
>
>
> well, in that case you want to look at this one:
>
> http://tonalsoft.com/enc/edo-11-odd-limit-error.htm
>
>
> this is different from the other in that it shows
> all the combinations of ratios in the 11-limit, and
> also gives positive and negative error values, not
> absolutes.
>
>
>
> -monz
> http://tonalsoft.com
> microtonal music software

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
> Just want to clear up a little bit ...
> basically I just want an instrument upon which it is *impossible*
to
> play the usual sonorities.
>
> ... I need to hear
> more music featuring them. ...
>
> -Igs

If you want to hear a sampling of some really unusual tunings, then
you should listen to some of Andrew Heathwaite's compositions:

http://www.soundclick.com/bands/3/andrewheathwaitemusic.htm

There's one in there (near the bottom) called "Impress the Children
in 18tET.

--George

No 13uke online yet, soon. The only 13 I have on line is here:

http://meowing.memh.uc.edu/~chris/micromp3s/suite.html

(note that it's a TONAL piece in 13)

good luck

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
> Ahoy, Dan!
>
> Just want to clear up a little bit that by "jarring/off" I didn't
> necessarily mean "nasty" or "dissonant". It's very much the case
> that all of my current micro-gitarras are capable of this, but
> basically I just want an instrument upon which it is *impossible*
to
> play the usual sonorities.
>
> Do any of your songs feature the 13-EDO uke? 13 is inching out
ahead
> as the top candidate, though 18 would be easier to refret to since
I
> could leave every other fret-slot from 12 (I'm attempting this one
> myself...to heck with paying $500 for a new neck!). I need to hear
> more music featuring them. I've thought about going fretless, but
> frankly I enjoy the constraints of frets. Playing fretless opens
up
> too many possibilities all at once for me, and I really enjoy
playing
> with numbers and scale arrangements in an ET framework. I have a
> fretless bass, but I really don't find it as inspiring as any of my
> micro-guitars.
>
> Let me know how your 14-tET turns out!
>
> -Igs
>
> --- In MakeMicroMusic@yahoogroups.com, "daniel_anthony_stearns"
> <daniel_anthony_stearns@y...> wrote:
> > Something I like to stress, probably to the point where
peoplewould
> > be throwing tomatoes at me if they could, is that humans make the
> > music, not the tunings.Or put another way, I have a 13-tet tenor
> uke
> > and a 2O-tet guitar and so on and so forth, and early on I found
> that
> > these types of non teritan tunings could easily make "pretty"
music
> > if _I_ wanted them to. Same goes for 3 and 5 limit tunings....if
> > they're what I've got to work with and I want "ugly" or "brutal",
> > then the tuning sure as hell ain't gonnastop me much! That
> > said,there's an obvious truth to the fact that different tunings
> are
> > different.....but that's tuning,not music and this is a
difference!
> > oK,SOrry. You know it seems we really have some similar
tendencies,
> > because I agree with you that many guitarist--not all, but many--
> used
> > to the standard fretting layout are almost always going to find
> > higher ETs or complex just frettings somewhat cramped and
crowded.
> > Interface is important, and this is one reason why I'd mention
> > fretless here. The interface is pretty liberating and the
> > possibilities for punishing discord and alien pandemonium are
many!
> > Anyway, keep up the good work and I'm sure you'll find something
> you
> > like and I'll try to check your 11-teter next time I drop by the
> > library....btw, I'm have a 14-tet classical guitar in the works
> right
> > now .
> >
> >
> >
> > --- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
> > <igliashon@s...> wrote:
> > > Hi everybody! I've been dormant in the microsphere for awhile,
> and
> > > this post probably isn't going to herald a comeback, but I
wanted
> > to
> > > share a simple but extremely xenharmonic piece of music I
> recently
> > > scraped together. It's at my soundclick site:
> > > http://www.soundclick.com/bands/1/cityoftheasleepmusic.htm
> > > It's called "Breakdown Hives" and is basically in 11-tET.
Well,
> > the
> > > intro is actually based on the idea of two 11-tET guitars tuned
1
> > > step of 22-tET apart, but the rest is straight-up 11. I have
> > > honestly come to a point where I'm absolutely tired of the
> usually
> > > approximated 3- and 5-limit sonorities and I want to play
> something
> > > that's really jarring and "off" to the western ear. I'm also
> > getting
> > > tired of navigating all of these tightly-packed frets! So I
> think
> > > for my next instrument, I'm setting the following limitations:
> > >
> > > 1. fewer than 19 tones per 2/1 (or approximate 2/1)
> > > 2. Approximate 3/2's or 4/3's must be AT LEAST 15 cents off
> > > (preferably more).
> > > 3. NO 5/4's or 6/5's closer to Just than those of 12-equal
> > >
> > > I'm only considering equal temperaments, since J.I. on guitar
is
> > more
> > > of a bother than I want to deal with.
> > >
> > > I'm curious as to what anyone on this forum might suggest.
Right
> > > now, I'm thinking either 13, 18, 10, or 14 equal divisions of
the
> > > octave, or maybe even something non-octave.
> > >
> > > Does anyone here have any suggestions on the matter? I'm going
> for
> > > maximum weirdness here.
> > >
> > > Thanks!
> > >
> > > -Igs

> No 13uke online yet, soon. The only 13 I have on line is here:
>
> http://meowing.memh.uc.edu/~chris/micromp3s/suite.html
>

This link doesn't seem to work.

yes quite right

>
>I'm not arguing with that. I was just saying that there is more
>than one way to use Phi to get a scale or tuning.
>
>-Carl
>
>
>
>
> >Yahoo! Groups Links
>
>
>
> >
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

>You also might want to farm scala for some really off
> the wall irrational number tunings (like those based on phi, which at
least
> in theory, are maximully vague or complex)

After some exploration, I've noticed two fascinating things about Phi,
in relation to music:

1) The PIth (ðth) root of Phi is almost exactly the 7:6 subminor third.
2) A scale using the 9th root of Phi as a generator comes out to a
slightly stretched (1203-cent 2:1's) version of 13-EDO, and the 8th
root of Phi is very close to the 23rd root of 4.

Also, the 10th root of Phi creates a lovely and very unusual scale that
is somewhat similar to 14.5-EDO, with some great approximations of 5-,
7-, and 11-limit intervals in the 2nd "octave" range.

Wasn't there a thread on the tuning list a while back about mystical
numbers in tuning? Maybe I should go dig that up....

-Igs

--- In MakeMicroMusic@yahoogroups.com, Kraig Grady <kraiggrady@a...>
wrote:
> On Phi, as with most recurrent sequences , it has a consonance of it
its
> own since the difference tone generated are different tones in the
> scale. So there is a limit to ow dissonant one can get. This can be
> heard in "the Stolen Stars" that dispute all the different subscales,
it
> all gels together.
> If dissonance is what you what, this is the wrong direction to look.

In what way did you use Phi to generate these scales?

-Igs

I think the 13 tone scale that is generated by Phi has an interval of all most 100 cents.

Igliashon Jones wrote:

>>You also might want to farm scala for some really off >>the wall irrational number tunings (like those based on phi, which at >> >>
>least > >
>>in theory, are maximully vague or complex)
>> >>
>
>After some exploration, I've noticed two fascinating things about Phi, >in relation to music:
>
>1) The PIth (�th) root of Phi is almost exactly the 7:6 subminor third.
>2) A scale using the 9th root of Phi as a generator comes out to a >slightly stretched (1203-cent 2:1's) version of 13-EDO, and the 8th >root of Phi is very close to the 23rd root of 4.
>
>Also, the 10th root of Phi creates a lovely and very unusual scale that >is somewhat similar to 14.5-EDO, with some great approximations of 5-, >7-, and 11-limit intervals in the 2nd "octave" range.
>
>Wasn't there a thread on the tuning list a while back about mystical >numbers in tuning? Maybe I should go dig that up....
>
>-Igs
>
>
>
>
>
> >Yahoo! Groups Links
>
>
>
> >
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

mainly through a recurrent sequence, because i find i like the the sound of all these type of scales right before it all starts to get within a narrow range of converging. on the other hand usually the first few terms one uses to 'seed' the equation are not as usable since the basic shape is not perceivable.
Meta Mavila btw could be considered the mirror of meantone in the sense that the 'third' is found much by going up four 'fifths' but by going down 3 'fifths'
see http://www.anaphoria.com/meantone-mavila.PDF
My last (mini) CD uses this scale starting on 37. this scale comes out close to 16 ET
Igliashon Jones wrote:

>
>In what way did you use Phi to generate these scales? >
>-Igs
>
>
>
>
>
> >Yahoo! Groups Links
>
>
>
> >
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

> I think the 13 tone scale that is generated by Phi has an interval of
> all most 100 cents.

phi^(1/9)=1.05492321=92.566 cents
2^(1/13)=1.05476608=92.308 cents

They're practically the same.
How do difference tones come into play with Phi-based scales? Is it
only with certain roots of Phi?

-Igs

> mainly through a recurrent sequence,

What does "recurrent sequence" refer to here? Sequence of what?
Recurrent how?

> see http://www.anaphoria.com/meantone-mavila.PDF

I wish I could say I understood this, but it's just a bunch of numbers
and arcane symbols to me.

monz,

You wrote:
...
> i have a webpage about "EDO prime error" which shows you
> the relative accuracy of EDOs in representing prime-factors:
> http://tonalsoft.com/enc/edo-prime-error.htm

[YA]
There, you write:
"The striking thing to observe about 55edo is that, despite its status as a
"standard" tuning during the meantone era (see my webpage on Mozart's
Tuning), none of its lower-prime representations is especially good in terms
of relative error: the best approximations are about 1/5 of a 55edo degree
off. "

Perhaps this "badness" translates to a fuller sound, due to increased
prevalence of beats? Given the clarity (and apparent simplicity) of much of
WA Mozart's writing, he may have particularly sought to enrich the harmonic
texture. Then again, maybe I'm projecting ...

> by "relative", i mean that the error is shown as a
> percentage of one EDO degree, rather than an "absolute"
> error in, say, cents.

[YA]
How different, I wonder, would your conclusions be
if the errors of each EDO were given in cents?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.308 / Virus Database: 266.11.8 - Release Date: 10/5/05

Yes, the site was down last night for some reason, but it's back now:

http://meowing.memh.uc.edu/~chris/micromp3s/suite.html

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
> > No 13uke online yet, soon. The only 13 I have on line is here:
> >
> > http://meowing.memh.uc.edu/~chris/micromp3s/suite.html
> >
>
> This link doesn't seem to work.

I am taking the octave divided by Phi.
off to San francisco

Igliashon Jones wrote:

>>I think the 13 tone scale that is generated by Phi has an interval of >>all most 100 cents.
>> >>
>
>phi^(1/9)=1.05492321=92.566 cents
>2^(1/13)=1.05476608=92.308 cents
>
>They're practically the same.
>How do difference tones come into play with Phi-based scales? Is it >only with certain roots of Phi?
>
>-Igs
>
>
>
>
>
> >Yahoo! Groups Links
>
>
>
> >
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

To make this table look nice, use a fixed-width font, and turn off wrapping...

Name #pitches Description
------ -------- ------------
clampitt-phi.scl 7 David Clampitt, phi+1 mod 3phi+2, from "Pairwise
Well-Formed Scales", 1997
meangold.scl 12 Meantone scale with Blackwood's R = phi, and
diat./chrom. ST = phi, ~4/15-comma
phi1_13.scl 13 Pythagorean scale with (Phi + 1) / 2 as fifth
phi_10.scl 10 Pythagorean scale with Phi as fifth
phi_12.scl 12 Non-Octave Pythagorean scale with Phi as fourth. Jacky
Ligon TL 12-04-2001
phi_13.scl 13 Pythagorean scale with Phi as fifth
phi_13a.scl 13 Non-Octave Pythagorean scale with Phi as fifth, Jacky
Ligon TL 12-04-2001
phi_13b.scl 13 Non-Octave Pythagorean scale with 12 3/2s, Jacky Ligon,
TL 12-04-2001
phi_17.scl 17 Phi + 1 equal division by 17, Brouncker (1653)
phi_7b.scl 7 Heinz Bohlen's Pythagorean scale with Phi as fifth
(1999)
phi_7be.scl 7 36-tET approximation of phi_7b
phi_8.scl 8 Non-Octave Pythagorean scale with 4/3s, Jacky Ligon, TL
12-04-2001
phi_8a.scl 8 Non-Octave Pythagorean scale with 5/4s, Jacky Ligon, TL
12-04-2001
temes-ur.scl 5 Temes' Ur 5-tone phi scale
wilson_gh1.scl 7 Golden Horagram nr.1: 1phi+0 / 7phi+1
wilson_gh11.scl 7 Golden Horagram nr.11: 1phi+0 / 3phi+1
wilson_gh2.scl 7 Golden Horagram nr.2: 1phi+0 / 6phi+1
wilson_gh50.scl 12 Golden Horagram nr.50: 7phi+2 / 17phi+5

Sorry, trying again for the formatting....

Name #pitches Description
------ -------- ------------
clampitt-phi.scl 7 David Clampitt, phi+1 mod 3phi+2, from "Pairwise
Well-Formed Scales", 1997
meangold.scl 12 Meantone scale with Blackwood's R = phi, and
diat./chrom. ST = phi, ~4/15-comma
phi1_13.scl 13 Pythagorean scale with (Phi + 1) / 2 as fifth
phi_10.scl 10 Pythagorean scale with Phi as fifth
phi_12.scl 12 Non-Octave Pythagorean scale with Phi as fourth. Jacky
Ligon TL 12-04-2001
phi_13.scl 13 Pythagorean scale with Phi as fifth
phi_13a.scl 13 Non-Octave Pythagorean scale with Phi as fifth, Jacky
Ligon TL 12-04-2001
phi_13b.scl 13 Non-Octave Pythagorean scale with 12 3/2s, Jacky Ligon,
TL 12-04-2001
phi_17.scl 17 Phi + 1 equal division by 17, Brouncker (1653)
phi_7b.scl 7 Heinz Bohlen's Pythagorean scale with Phi as fifth
(1999)
phi_7be.scl 7 36-tET approximation of phi_7b
phi_8.scl 8 Non-Octave Pythagorean scale with 4/3s, Jacky Ligon, TL
12-04-2001
phi_8a.scl 8 Non-Octave Pythagorean scale with 5/4s, Jacky Ligon, TL
12-04-2001
temes-ur.scl 5 Temes' Ur 5-tone phi scale
wilson_gh1.scl 7 Golden Horagram nr.1: 1phi+0 / 7phi+1
wilson_gh11.scl 7 Golden Horagram nr.11: 1phi+0 / 3phi+1
wilson_gh2.scl 7 Golden Horagram nr.2: 1phi+0 / 6phi+1
wilson_gh50.scl 12 Golden Horagram nr.50: 7phi+2 / 17phi+5

>>Sorry, trying again for the formatting....
>
>Ditto.

Not that it's a big deal...

-C.

>To make this table look nice, use a fixed-width font, and turn off
>wrapping...

FYI, many e-mail servers hard wrap, since the e-mail RFC calls
for a hard return every 76 chars, I believe. Anyway, this was
hard wrapped.

-Carl

>Sorry, trying again for the formatting....

Ditto.

-C.

checked this today.Te exact, lockstep harmony isn't my favorite
thing in the world, but I really Really like everything from the
first change into the microanthemish section until the end....more
good work mista Jones.Keep it upp!

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
> Hi everybody! I've been dormant in the microsphere for awhile,
and
> this post probably isn't going to herald a comeback, but I wanted
to
> share a simple but extremely xenharmonic piece of music I recently
> scraped together. It's at my soundclick site:
> http://www.soundclick.com/bands/1/cityoftheasleepmusic.htm
> It's called "Breakdown Hives" and is basically in 11-tET. Well,
the
> intro is actually based on the idea of two 11-tET guitars tuned 1
> step of 22-tET apart, but the rest is straight-up 11. I have
> honestly come to a point where I'm absolutely tired of the usually
> approximated 3- and 5-limit sonorities and I want to play
something
> that's really jarring and "off" to the western ear. I'm also
getting
> tired of navigating all of these tightly-packed frets! So I think
> for my next instrument, I'm setting the following limitations:
>
> 1. fewer than 19 tones per 2/1 (or approximate 2/1)
> 2. Approximate 3/2's or 4/3's must be AT LEAST 15 cents off
> (preferably more).
> 3. NO 5/4's or 6/5's closer to Just than those of 12-equal
>
> I'm only considering equal temperaments, since J.I. on guitar is
more
> of a bother than I want to deal with.
>
> I'm curious as to what anyone on this forum might suggest. Right
> now, I'm thinking either 13, 18, 10, or 14 equal divisions of the
> octave, or maybe even something non-octave.
>
> Does anyone here have any suggestions on the matter? I'm going
for
> maximum weirdness here.
>
> Thanks!
>
> -Igs

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
>
> > I think the 13 tone scale that is generated by Phi has an interval of
> > all most 100 cents.
>
> phi^(1/9)=1.05492321=92.566 cents
> 2^(1/13)=1.05476608=92.308 cents
>
> They're practically the same.
> How do difference tones come into play with Phi-based scales? Is it
> only with certain roots of Phi?

Dear Igliashon,

I hope the following answers some of your questions:

When someone says "generated by" some interval, they mean take
sucessive _powers_ of that interval (not its roots) and usually
octave-reduce them. So a just fifth is said to _generate_ Pythagorean,
and a fifth narrowed by 1/6 to 1/3 of a syntonic comma generates meantone.

Of course it's easier to do this using the logarithmic measure of
cents, where the process of generating the scale then becomes one of
merely taking sucessive _multiples_ of the generator and reducing them
modulo 1200.

Just in case there is anyone reading who doesn't know what phi is,
phi = (sqrt(5)+1)/2 ~= 1.618
It can be defined as that positive real number which is one more than
its own reciprocal.

It is, in a sense, the "least rational" of all real numbers because it
needs the largest numerators and denominators to approximate it by a
ratio. Sucessively better approximations to it are obtained as the
ratio of successive pairs of numbers from the Fibonacci series
1,1,2,3,5,8,13,21,34,... where you can see what Carl mentioned, that
it approximates 8:13. However it doesn't approximate it (or any other
such ratio) sufficiently well for it to be recognisable as such.

There are at least two different meanings of "phi-generator". One is
the frequency ratio 1:phi, the other is the 1/phi octave-fraction.

In cents, the first is
ln(phi)/ln(2)*1200 ~= 833.09 cents
The second is
1200/phi ~= 741.64 cents

The first is the one that has the property Kraig mentioned, of
difference tones also being in the scale (a harmonic-series scale also
has this property of course, but is not distributionally even).

So when Kraig says he is dividing the octave by phi he seems to be
confusing the two. They both have 13 note MOS scales with intervals of
approximately 100 cents.

The first (833.09 cent generator) is the only one that has some
theoretical claim to avoiding close approximations to _all_ small
whole-number frequency ratios (except octaves), not merely those
approximated by diatonic scales. But in reality it has a 397 cent
major third (slightly better than that of 12-ET), so it isn't what
you're looking for.

The second (741.64 cent generator) not only has a good major third,
but a near-perfect perfect-fourth, so it isn't what you're looking for
either.

I think there is some confusion about what you're actually looking
for. I think you've made it clear (correct me if I'm wrong) that you
want a scale of
19 notes or less per approximate octave,
that is easy to fret a guitar for,
and you are _not_ trying to maximise dissonance,
and that you want to avoid good fourths, fifths, and major or minor
thirds (i.e. 2:3, 3:4, 4:5, 5:6),
and you're happy to include subminor and supermajor thirds (6:7, 7:9).

But what other consonant or semi-consonant diatonic intervals or
chords do you want to avoid?

Do you want to avoid octaves (1:2), major and minor sixths (3:5, 5:8),
minor sevenths (5:9), major seconds or ninths (8:9, 4:9)? What about
the super fourth or subdiminished fifth (5:7) that is somewhat implied
by the tritone? What about octave-extensions of the above ratios, such
as 1:3, 3:8, 2:5?

I suggest avoiding either
(a) all ratios between the integers 1 to 6, or
(b) all ratios between the integers 1 to 9 excluding 7.
(c) all ratios between the integers 2 to 6, or
(d) all ratios between the integers 2 to 9 excluding 7.

(a) and (b) would avoid octaves, while (c) and (d) would not.

Are there some ratios, such as those involving 7 or 11, that you might
actually like to have _good_ approximations for, while avoiding the
others?

I think Graham Breed is on the right track. He just needs to know your
specification for the above. He has assumed you want to avoid octaves
and ratios of 7, but not ratios of 9.

For guitar-fretting purposes I think his results would be better
expressed as -cETs rather than fractional -tETs although it's also
interesting to know where they fall as -tETs.

Do you also want to avoid the diatonic melodic pattern of large and
small steps LLsLLLs making up an approximate octave? That will
probably happen automatically given the other constraints.

-- Dave Keenan

Thank you, Mr. Keenan, for your clear and thorough response! Yes,
you have answered many of my questions. Let me answer some of yours,
to clarify what I'm looking for:

As far as other consonances I'm looking to avoid, I'd say 9:8's (I
like my whole tones narrow, but wide could be cool also) and 16:9's,
and also probably 7:4's since all of my current microgitarras
approximate that interval. I'd also like to avoid the usual sixths.
And yes, I want to avoid octave-equivalencies to these intervals as
well, but not necessarily 2:1's themselves. And I DEFINITELY want to
avoid the LLsLLLs pattern.

I don't have any J.I. intervals in mind that I'd like to approximate
particularly well...it's more sort the general character that I'm
after. Probably higher-limit stuff, though. I would be cool with
something close to a 9:5 minor seventh, but that's about it for the
strictly low numbers. The Phi thing really intrigued me because of
the recursivity: difference tones being notes from the scale adds a
sort of fractal dimension not found in most music. Can you think of
any other scales that display a similar property?

HTH,

-Igs

--- In MakeMicroMusic@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
> <igliashon@s...> wrote:
> >
> > > I think the 13 tone scale that is generated by Phi has an
interval of
> > > all most 100 cents.
> >
> > phi^(1/9)=1.05492321=92.566 cents
> > 2^(1/13)=1.05476608=92.308 cents
> >
> > They're practically the same.
> > How do difference tones come into play with Phi-based scales? Is
it
> > only with certain roots of Phi?
>
> Dear Igliashon,
>
> I hope the following answers some of your questions:
>
> When someone says "generated by" some interval, they mean take
> sucessive _powers_ of that interval (not its roots) and usually
> octave-reduce them. So a just fifth is said to _generate_
Pythagorean,
> and a fifth narrowed by 1/6 to 1/3 of a syntonic comma generates
meantone.
>
> Of course it's easier to do this using the logarithmic measure of
> cents, where the process of generating the scale then becomes one of
> merely taking sucessive _multiples_ of the generator and reducing
them
> modulo 1200.
>
> Just in case there is anyone reading who doesn't know what phi is,
> phi = (sqrt(5)+1)/2 ~= 1.618
> It can be defined as that positive real number which is one more
than
> its own reciprocal.
>
> It is, in a sense, the "least rational" of all real numbers because
it
> needs the largest numerators and denominators to approximate it by a
> ratio. Sucessively better approximations to it are obtained as the
> ratio of successive pairs of numbers from the Fibonacci series
> 1,1,2,3,5,8,13,21,34,... where you can see what Carl mentioned, that
> it approximates 8:13. However it doesn't approximate it (or any
other
> such ratio) sufficiently well for it to be recognisable as such.
>
> There are at least two different meanings of "phi-generator". One is
> the frequency ratio 1:phi, the other is the 1/phi octave-fraction.
>
> In cents, the first is
> ln(phi)/ln(2)*1200 ~= 833.09 cents
> The second is
> 1200/phi ~= 741.64 cents
>
> The first is the one that has the property Kraig mentioned, of
> difference tones also being in the scale (a harmonic-series scale
also
> has this property of course, but is not distributionally even).
>
> So when Kraig says he is dividing the octave by phi he seems to be
> confusing the two. They both have 13 note MOS scales with intervals
of
> approximately 100 cents.
>
> The first (833.09 cent generator) is the only one that has some
> theoretical claim to avoiding close approximations to _all_ small
> whole-number frequency ratios (except octaves), not merely those
> approximated by diatonic scales. But in reality it has a 397 cent
> major third (slightly better than that of 12-ET), so it isn't what
> you're looking for.
>
> The second (741.64 cent generator) not only has a good major third,
> but a near-perfect perfect-fourth, so it isn't what you're looking
for
> either.
>
> I think there is some confusion about what you're actually looking
> for. I think you've made it clear (correct me if I'm wrong) that you
> want a scale of
> 19 notes or less per approximate octave,
> that is easy to fret a guitar for,
> and you are _not_ trying to maximise dissonance,
> and that you want to avoid good fourths, fifths, and major or minor
> thirds (i.e. 2:3, 3:4, 4:5, 5:6),
> and you're happy to include subminor and supermajor thirds (6:7,
7:9).
>
> But what other consonant or semi-consonant diatonic intervals or
> chords do you want to avoid?
>
> Do you want to avoid octaves (1:2), major and minor sixths (3:5,
5:8),
> minor sevenths (5:9), major seconds or ninths (8:9, 4:9)? What about
> the super fourth or subdiminished fifth (5:7) that is somewhat
implied
> by the tritone? What about octave-extensions of the above ratios,
such
> as 1:3, 3:8, 2:5?
>
> I suggest avoiding either
> (a) all ratios between the integers 1 to 6, or
> (b) all ratios between the integers 1 to 9 excluding 7.
> (c) all ratios between the integers 2 to 6, or
> (d) all ratios between the integers 2 to 9 excluding 7.
>
> (a) and (b) would avoid octaves, while (c) and (d) would not.
>
> Are there some ratios, such as those involving 7 or 11, that you
might
> actually like to have _good_ approximations for, while avoiding the
> others?
>
> I think Graham Breed is on the right track. He just needs to know
your
> specification for the above. He has assumed you want to avoid
octaves
> and ratios of 7, but not ratios of 9.
>
> For guitar-fretting purposes I think his results would be better
> expressed as -cETs rather than fractional -tETs although it's also
> interesting to know where they fall as -tETs.
>
> Do you also want to avoid the diatonic melodic pattern of large and
> small steps LLsLLLs making up an approximate octave? That will
> probably happen automatically given the other constraints.
>
> -- Dave Keenan

Igs,

Why not use a phi-generator, so that you set has many of its' propeties, and
then sieve out the pitches in the set that are too close to whatever
low-limit JI intervals you want to avoid?

I'm just thinking out loud, maybe there would be some theoretical problems
with doing that. And maybe it just doesn't matter whether it 'theoretically'
works or not, if you like the sound!

Best,
Aaron.

On Thursday 12 May 2005 8:13 pm, Igliashon Jones wrote:
> Thank you, Mr. Keenan, for your clear and thorough response! Yes,
> you have answered many of my questions. Let me answer some of yours,
> to clarify what I'm looking for:
>
> As far as other consonances I'm looking to avoid, I'd say 9:8's (I
> like my whole tones narrow, but wide could be cool also) and 16:9's,
> and also probably 7:4's since all of my current microgitarras
> approximate that interval. I'd also like to avoid the usual sixths.
> And yes, I want to avoid octave-equivalencies to these intervals as
> well, but not necessarily 2:1's themselves. And I DEFINITELY want to
> avoid the LLsLLLs pattern.
>
> I don't have any J.I. intervals in mind that I'd like to approximate
> particularly well...it's more sort the general character that I'm
> after. Probably higher-limit stuff, though. I would be cool with
> something close to a 9:5 minor seventh, but that's about it for the
> strictly low numbers. The Phi thing really intrigued me because of
> the recursivity: difference tones being notes from the scale adds a
> sort of fractal dimension not found in most music. Can you think of
> any other scales that display a similar property?
>
> HTH,
>
> -Igs
>
> --- In MakeMicroMusic@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
>
> wrote:
> > --- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
> >
> > <igliashon@s...> wrote:
> > > > I think the 13 tone scale that is generated by Phi has an
>
> interval of
>
> > > > all most 100 cents.
> > >
> > > phi^(1/9)=1.05492321=92.566 cents
> > > 2^(1/13)=1.05476608=92.308 cents
> > >
> > > They're practically the same.
> > > How do difference tones come into play with Phi-based scales? Is
>
> it
>
> > > only with certain roots of Phi?
> >
> > Dear Igliashon,
> >
> > I hope the following answers some of your questions:
> >
> > When someone says "generated by" some interval, they mean take
> > sucessive _powers_ of that interval (not its roots) and usually
> > octave-reduce them. So a just fifth is said to _generate_
>
> Pythagorean,
>
> > and a fifth narrowed by 1/6 to 1/3 of a syntonic comma generates
>
> meantone.
>
> > Of course it's easier to do this using the logarithmic measure of
> > cents, where the process of generating the scale then becomes one of
> > merely taking sucessive _multiples_ of the generator and reducing
>
> them
>
> > modulo 1200.
> >
> > Just in case there is anyone reading who doesn't know what phi is,
> > phi = (sqrt(5)+1)/2 ~= 1.618
> > It can be defined as that positive real number which is one more
>
> than
>
> > its own reciprocal.
> >
> > It is, in a sense, the "least rational" of all real numbers because
>
> it
>
> > needs the largest numerators and denominators to approximate it by a
> > ratio. Sucessively better approximations to it are obtained as the
> > ratio of successive pairs of numbers from the Fibonacci series
> > 1,1,2,3,5,8,13,21,34,... where you can see what Carl mentioned, that
> > it approximates 8:13. However it doesn't approximate it (or any
>
> other
>
> > such ratio) sufficiently well for it to be recognisable as such.
> >
> > There are at least two different meanings of "phi-generator". One is
> > the frequency ratio 1:phi, the other is the 1/phi octave-fraction.
> >
> > In cents, the first is
> > ln(phi)/ln(2)*1200 ~= 833.09 cents
> > The second is
> > 1200/phi ~= 741.64 cents
> >
> > The first is the one that has the property Kraig mentioned, of
> > difference tones also being in the scale (a harmonic-series scale
>
> also
>
> > has this property of course, but is not distributionally even).
> >
> > So when Kraig says he is dividing the octave by phi he seems to be
> > confusing the two. They both have 13 note MOS scales with intervals
>
> of
>
> > approximately 100 cents.
> >
> > The first (833.09 cent generator) is the only one that has some
> > theoretical claim to avoiding close approximations to _all_ small
> > whole-number frequency ratios (except octaves), not merely those
> > approximated by diatonic scales. But in reality it has a 397 cent
> > major third (slightly better than that of 12-ET), so it isn't what
> > you're looking for.
> >
> > The second (741.64 cent generator) not only has a good major third,
> > but a near-perfect perfect-fourth, so it isn't what you're looking
>
> for
>
> > either.
> >
> > I think there is some confusion about what you're actually looking
> > for. I think you've made it clear (correct me if I'm wrong) that you
> > want a scale of
> > 19 notes or less per approximate octave,
> > that is easy to fret a guitar for,
> > and you are _not_ trying to maximise dissonance,
> > and that you want to avoid good fourths, fifths, and major or minor
> > thirds (i.e. 2:3, 3:4, 4:5, 5:6),
> > and you're happy to include subminor and supermajor thirds (6:7,
>
> 7:9).
>
> > But what other consonant or semi-consonant diatonic intervals or
> > chords do you want to avoid?
> >
> > Do you want to avoid octaves (1:2), major and minor sixths (3:5,
>
> 5:8),
>
> > minor sevenths (5:9), major seconds or ninths (8:9, 4:9)? What about
> > the super fourth or subdiminished fifth (5:7) that is somewhat
>
> implied
>
> > by the tritone? What about octave-extensions of the above ratios,
>
> such
>
> > as 1:3, 3:8, 2:5?
> >
> > I suggest avoiding either
> > (a) all ratios between the integers 1 to 6, or
> > (b) all ratios between the integers 1 to 9 excluding 7.
> > (c) all ratios between the integers 2 to 6, or
> > (d) all ratios between the integers 2 to 9 excluding 7.
> >
> > (a) and (b) would avoid octaves, while (c) and (d) would not.
> >
> > Are there some ratios, such as those involving 7 or 11, that you
>
> might
>
> > actually like to have _good_ approximations for, while avoiding the
> > others?
> >
> > I think Graham Breed is on the right track. He just needs to know
>
> your
>
> > specification for the above. He has assumed you want to avoid
>
> octaves
>
> > and ratios of 7, but not ratios of 9.
> >
> > For guitar-fretting purposes I think his results would be better
> > expressed as -cETs rather than fractional -tETs although it's also
> > interesting to know where they fall as -tETs.
> >
> > Do you also want to avoid the diatonic melodic pattern of large and
> > small steps LLsLLLs making up an approximate octave? That will
> > probably happen automatically given the other constraints.
> >
> > -- Dave Keenan
>
>
>
> Yahoo! Groups Links
>
>
>

Igliashon,

Graham Breed wrote:
> I have been playing with this. I calculated over some 5-limit ratios,
> with the largest integer no greater than 8, weighted by the geometric
> mean. Taking the RMS error, any x.5-edo scale looks weird, and scales
> look less weird the more notes they have to the octave. 13.5 and 15.5
> edo stand out a bit, but not much.

Conversely, the fewer notes the EDO scale has to the octave (or
double octave, per Graham), the weirder it may sound. Particularly
when the period doesn't divide 12, such as 2, 3, 4 and 6-EDO, we get
some nice (easily-fingered) scales that don't remind us too inexorably
of common repertoire music.

For instance, 7-EDO is the theoretical Thai court music scale, and
that sounds particularly rootless or restless (to me at least).

I've long believed that any artist worthy of the name needs to be
able to make the most of limited resources - as they are often a
fact of life - and besides, the mental discipline of doing so is also
a real challenge to creativity. In any scale whatsoever, it's a
wonderful exercise to try to make engaging music with, say, only
three or four distinct pitches.

For some nicely weird 11-EDO music, I recommend a listen to Bill
Sethares' "The Turquoise Dabo Girl", from Xentonality.

6-EDO is of course the whole-tone scale that Debussy exploited so
brilliantly. But how about 5-EDO? With its very limited gamut, and
no point of harmonic repose, that should stretch musicianship and
inventiveness pretty far ...

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.308 / Virus Database: 266.11.8 - Release Date: 10/5/05

--- In MakeMicroMusic@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:

> 6-EDO is of course the whole-tone scale that Debussy exploited so
> brilliantly. But how about 5-EDO? With its very limited gamut, and
> no point of harmonic repose, that should stretch musicianship and
> inventiveness pretty far ...

There's Andrew Heathwaite's Pinta Penta one soundclick:

http://www.soundclick.com/bands/3/andrewheathwaite.htm

If all the different versions of that are not enough for you, here
is his 5-et Pinta Penta, reconstitututed a marvelous dwarf scale.
So with 5-et, you can start out with the limited gamut you describe,
and then subsequently remove the limits.

http://66.98.148.43/~xenharmo/ogg/gene/marvpenta.ogg

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
> Thank you, Mr. Keenan, for your clear and thorough response! Yes,
> you have answered many of my questions. Let me answer some of yours,
> to clarify what I'm looking for:
...
> The Phi thing really intrigued me because of
> the recursivity: difference tones being notes from the scale adds a
> sort of fractal dimension not found in most music. Can you think of
> any other scales that display a similar property?

Possibly any generator that represents a frequency ratio of phi^n
where n is a whole number. i.e. 1:phi, 1:phi^2, 1:phi^3 .... I'm
running on math intuition here. I haven't checked it.

But look no further. Aaron Johnson's post prompted me to look again. I
totally agree with you Aaron. I was wrong when I said the 13 note
1:phi (833.09 cent) generated scale was disqualified. There is only
one single instance of a 397 cent interval in the scale since it takes
the full chain of 12 generators before it is produced. It will be very
easy to avoid.

Please note that the octave-inversion of the generator can just as
well be considered as _the_ generator, i.e 1200 - 833.09 = 366.91
cents. This is just like we can consider either the fifth or the
fourth as the generator for meantone.

You know how in meantone you have a single wolf fifth? Well in this
maximally-wolfy golden ratio scale, generated by a maximally-wolfy
third of 367 cents, the single 397 cent third is the anti-wolf!
Poetic, don't you think?

This would be brilliant for guitar fretting since the guitar could
have all its strings separated by 367 cent generating interval, and
have 13 straight frets to the octave.

I assume you would tune it up electronically and give it a listen
before comitting to wood. I expect it to sound like the devil's own scale.

-- Dave Keenan

> This would be brilliant for guitar fretting since the guitar could
> have all its strings separated by 367 cent generating interval, and
> have 13 straight frets to the octave.

367 cents is pretty narrow for guitar tuning. To maintain proper
string tension I'd have to use thicker strings for the high-end. And
wouldn't I get a lot of out-of-scale intervals on the higher strings
with straight frets? In fact, wouldn't there also be some "anti-
wolf" fifths and fourths as well, intervals that would be right about
on par with 12-equal?

> I assume you would tune it up electronically and give it a listen
> before comitting to wood. I expect it to sound like the devil's own
>scale.

Maybe it's my ears, but it really doesn't sound as bad as you'd
think. In fact it's very very close to subset of 36-equal, which
doesn't seem like the most dissonant tuning out there.

Hmm...I dunno about this one. It's a cool idea, to be sure, but
maybe this recursive combinational tone idea is a little too
impractical for this current project of mine.

However, I did just recently notice that 23-EDO contains very good
approximations to both Phi and Pi, as well as a host of other bizarre
intervals. However, it violates my first criterion (less than 19
steps per approximate 2:1), so maybe 11.5-EDO? i'll have to play
around a bit.

Many thanks for your math help, though. These ideas all warrant
future exploration, and I hope I'm not the only one interested.

-Igs

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
> 367 cents is pretty narrow for guitar tuning. To maintain proper
> string tension I'd have to use thicker strings for the high-end.

Sure. But that's easy enough to do (compared to refretting a guitar).

> And
> wouldn't I get a lot of out-of-scale intervals on the higher strings
> with straight frets? In fact, wouldn't there also be some "anti-
> wolf" fifths and fourths as well, intervals that would be right
> about on par with 12-equal?

Yes. That would be a problem. And having thought about it some more,
we can do much better than 367 cents between all strings. It should
instead go
466 367 367 466 367 or
367 466 367 367 466 or
367 367 466 367 367

Since 367c is +1 generator and 466c is -2 generators we need never
depart from the reference scale by more than +-1 generator and so we
stop just short of hitting those 12-equal fourths and fifths.

See
http://dkeenan.com/Music/MicroGuitar.pdf

> > I assume you would tune it up electronically and give it a listen
> > before comitting to wood. I expect it to sound like the devil's
> > own scale.
>
> Maybe it's my ears, but it really doesn't sound as bad as you'd
> think.

I trust your ears. But I've got bad news for you. ;-) I don't think it
gets any badder unless you either:
(a) have fewer notes per octave, or
(b) give up octaves,
or both.

The more notes you have per octave the easier it gets to find
something vaguely diatonic. If you stopped at 10 notes of this
phi-generated scale, then it would be better.

But I think you're not only trying to avoid 5-limit JI approximations,
but also approximations to intervals of 12-equal. And these scales do
have 100 cent steps, and sometimes two in a row.

> In fact it's very very close to subset of 36-equal, which
> doesn't seem like the most dissonant tuning out there.

Yes it is essentially a subset of 36-equal (and so of 72-equal), but
as Erv Wilson notes, on the last page of
http://www.anaphoria.com/tres.PDF
72-ET contains both the best and the worst.

The golden generator 50/72 oct (or equivalently 22/72 oct) just
happens to be the one whose multiples take the longest to hit a
consonance and that hits the fewest consonances per note. George
Secor's miracle generator of 7/72 is the opposite.

> Hmm...I dunno about this one. It's a cool idea, to be sure, but
> maybe this recursive combinational tone idea is a little too
> impractical for this current project of mine.
>
> However, I did just recently notice that 23-EDO contains very good
> approximations to both Phi and Pi, as well as a host of other bizarre
> intervals. However, it violates my first criterion (less than 19
> steps per approximate 2:1), so maybe 11.5-EDO? i'll have to play
> around a bit.

You could try using 7/23 oct as the generator (~365 cents) and stop at
10 notes. But this gives you better 8:9s than the 11/36 oct generator.

If you wanted primarily to get as far away from any 12-equal intervals
as possible then I'd advise scrapping the octave and using a generator
which is some noble multiple of the 12-equal semitone. The simplest
are 100*phi = 161.8-cET (7.4-ET) and 100/phi = 61.8-cET (19.4-ET) but
these probably have too few and too many notes respectively.

This is interesting
http://mathworld.wolfram.com/SilverRatio.html
but it gives us noble ratios _further_ from 1, whereas we need them
closer to 1.

Please read
"The Noble Mediant: Complex ratios and metastable musical intervals"
by Margo Schulter and myself. See
http://dkeenan.com/Music/NobleMediant.txt

The next noblest number closer to 1 is probably the noble mediant of
1/1 and 3/2 = (1+3phi)/(1+2*phi) ~= 1.382
so you could try 138.2-cET (8.7-ET) and 72.4-cET (16.6-ET)

I note that Graham Breed also suggested 16.6-ET, based on a completely
different method, in
/makemicromusic/topicId_9723.html#9741

-- Dave Keenan

Dave,

Has anyone named this scale yet?

I like "Max Wolf Gold" ... :-)

Indeed, sounds like the very devil,
just playing it on Scala.

Regards,
Yahya

-----Original Message-----
Dave Keenan wrote:

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
> Thank you, Mr. Keenan, for your clear and thorough response! Yes,
> you have answered many of my questions. Let me answer some of yours,
> to clarify what I'm looking for:
...
> The Phi thing really intrigued me because of
> the recursivity: difference tones being notes from the scale adds a
> sort of fractal dimension not found in most music. Can you think of
> any other scales that display a similar property?

Possibly any generator that represents a frequency ratio of phi^n
where n is a whole number. i.e. 1:phi, 1:phi^2, 1:phi^3 .... I'm
running on math intuition here. I haven't checked it.

But look no further. Aaron Johnson's post prompted me to look again. I
totally agree with you Aaron. I was wrong when I said the 13 note
1:phi (833.09 cent) generated scale was disqualified. There is only
one single instance of a 397 cent interval in the scale since it takes
the full chain of 12 generators before it is produced. It will be very
easy to avoid.

Please note that the octave-inversion of the generator can just as
well be considered as _the_ generator, i.e 1200 - 833.09 = 366.91
cents. This is just like we can consider either the fifth or the
fourth as the generator for meantone.

You know how in meantone you have a single wolf fifth? Well in this
maximally-wolfy golden ratio scale, generated by a maximally-wolfy
third of 367 cents, the single 397 cent third is the anti-wolf!
Poetic, don't you think?

This would be brilliant for guitar fretting since the guitar could
have all its strings separated by 367 cent generating interval, and
have 13 straight frets to the octave.

I assume you would tune it up electronically and give it a listen
before comitting to wood. I expect it to sound like the devil's own scale.

-- Dave Keenan

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Igliashon and all,

Wrote the following Saturday morning, when I had plenty of time.
Had meant to revisit the unfinished business - the relative tuning of
the open strings - before sending this, but time flies when you work
for a living. So rather than wait till the thread grows cold, I'm sending
this preliminary report to you now. YA

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

Here's my report on some simple experiments on a guitar. (If you're
past this stage, there's probably not much new in it for you.) I'm
pleased to say they were non-destructive esperiments, and can easily
be repeated without any real expense.

I wrote, earlier:
> ... the fewer notes the EDO scale has to the octave (or
> double octave, per Graham), the weirder it may sound. Particularly
> when the period doesn't divide 12, such as 2, 3, 4 and 6-EDO, we get
> some nice (easily-fingered) scales that don't remind us too inexorably
> of common repertoire music.
>
> For instance, 7-EDO is the theoretical Thai court music scale, and
> that sounds particularly rootless or restless (to me at least).
>
> I've long believed that any artist worthy of the name needs to be
> able to make the most of limited resources - as they are often a
> fact of life - and besides, the mental discipline of doing so is also
> a real challenge to creativity. In any scale whatsoever, it's a
> wonderful exercise to try to make engaging music with, say, only
> three or four distinct pitches.
>
> For some nicely weird 11-EDO music, I recommend a listen to Bill
> Sethares' "The Turquoise Dabo Girl", from Xentonality.
>
> 6-EDO is of course the whole-tone scale that Debussy exploited so
> brilliantly. But how about 5-EDO? With its very limited gamut, and
> no point of harmonic repose, that should stretch musicianship and
> inventiveness pretty far ...

Lying in bed this morning, I had a silly idea. What if the fingerboard
of my guitar were made in rubber, and I stretched it, uniformly, by
say 10%? - but left the bridge in the same place. The frets would
become 10% further apart, and the notes 10% higher (in relative
frequencies 100 cents --> 110 cents). Or equivalently, and much more
usefully, move the bridge one-eleventh of the length closer to the nut?

The numbers seemed quite good - to avoid any GOOD consonances
(which are bad in this context), you'd want the minor and major third
and sixth, as well as the fourth, fifth and octave, to be significantly
off. That is, the notes from frets 3, 4, 5, 7, 9, 10 and 12 shouldn't be
too consonant. With a 10% stretch, these turn into 330, 400, 550,
770, 990, 1100 and 1320 respectively. Only the last three are near to
whole 12-EDO semitones, and maybe fiddling with the stretch factor
could fix that.

But how easily could I adapt the guitar, perhaps by adding another
bridge nearer the sound hole?

So I got out of bed, got a glass of water and went to take my tablets
before grabbing the guitar. Unfortunately, I slipped in a puddle on
the tile floor in the kitchen (dog getting old, weak bladder) which
sort of delayed proceedings while we cleaned up broken glass ... :-(

When I got the guitar, I saw I could easily slip a pen underneath the
strings to act as a movable bridge, and tension it by jamming a thin
bit of plastic (actually, the handle of a pink feather duster!) under
one end. The pen was a cheap gel one, with an almost straight plastic
barrel, from K-Mart. A ball-point would do just as well.

Success! In no time, I was picking out "tunes" in my approximate
11-EDO.

With the bridge almost at the edge of the soundhole (this is a flat-
top Spanish guitar), I had very close to 10-EDO, and I'm sorry to
say, this was very tuneful indeed. Only a couple of the intervals
were bad enough for Igliashon's purpose, of having no readily
available consonances. I jotted down one of the tunes I picked, in
what I can only think of as 10-EDO E minor; only the thirds are not
really melodic. I may convert that to a Scala .seq file to post to
tuning-files to show what I mean about the thirds.

With a 20% stretch, the consonances turn into 360, 480, 600, 840,
1080, 1200 and 1440 respectively. But this (10-EDO) scale also
includes 720, which is a passable fifth, and does give a perfect
octave.

The 11-EDO setting (10% stretch) was more dissonant than the
10-EDO, by far. The fourths and fifths are really BAD, and that's
good :-) Fact, the only thing that sounds halfway good is playing
diminished chord positions on the top four strings; they sound
rather like ordinary diminished chords, but stretched (I can't call
them "augmented", can I?). All the major, minor and dom seventh
chord (fingerings, that is) that I tried do sound quite bad. The
minor chords were much better (therefore worse) than the others.
(But see Caveat below.)

To avoid the octave completely, I tried 10.5-EDO. This was even
more successful than 11-EDO. The minor chords were still
acceptable, particularly in speed, but everything else was "off".
(Also see Caveat below.)

With 9-EDO, the resonance is weakened greatly with the bridge
moved so far over the sound-hole. However, I found it made for
interesting arpeggio figures when shifting a diminished chord
position or a barre chromatically up and down a few frets.
(Also see Caveat below.)

One thing I did notice in all tunings was that fast melodic passages
sound pretty good; again, especially in "minor keys". I suspect
that only slow music, that emphasises the most discordant intervals,
will sound "weird" enough for Igliashon's purposes.

Caveat:
I should mention that in all of the above, I did not alter the basic
tuning of the guitar, which would have become stretched fourths
and a stretched third. For example, in 9-EDO, the strings are
tuned, not 5, 5, 5, 4 and 5 steps apart, but about 4, 4, 4, 3.5 and
4 steps apart. Unless the strings are an exact number of steps
of your gamut apart, the chords are not drawn from the same gamut,
since each string has its own tuning. So the results above don't
accurately represent 10-EDO, 10.5-EDO or 11-EDO, do they?
Except on one string at a time. Although, by chance, the strings
happened to be almost in tune in 9-EDO. (Actually, it's not chance,
but maths - 4 steps of 9 is 444 cents, while 5 steps of 12 is 417.)

Therefore the conclusions I drew above about chords are
probably wrong. I need to retune the strings to notes of the
n-EDO gamut I'm using to be able to hear what chords really
sound like in it.

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

The point of the standard guitar tuning is to maximise playability;
without shifting hand position, you can use up all your fingers (and
no more!) to get different notes from one string, then move to
the next string to continue the sequence of notes.

So I tried tuning the strings to the standard number of steps apart,
in 9-EDO, by leaving the top E string fixed and dropping the others.
All the strings were quite playable. The chords I played then were
at least all drawn from the same gamut.

(Side note: after this retuning, by removing the movable bridge, I
found that I had a usable non-harmonic tuning, with the notes tuned
about 6.25, 6.25, 6.25, 5 and 6.25 steps apart. This produces no
recognisable harmonies, no matter how I finger chords, and the
consonances you get in melodies played in close positions of the hand
are limited to those of one tetrachord (fourth) at a time. We might
call this chance discovery an accidentally non-harmonic tuning - if
that didn't cause any confusion.)

And yes, 9-EDO IS weird - especially melodically. I hardly know how
to make a tune in it. Of course, almost every chord is wrong: the
gamut consists of 0, 133, 267, 400, 533, 677, 800, 933, 1067, 1200
cents above Doh. So you get an equally-tempered augmented chord
as your most consonant triad. An interesting scale pattern is -
l s l s l s
Its inverse is -
s l s l s l
In 9-EDO, s=1 and l=2.

Translating these into 12-EDO, with s=1 and l=3 we get the scales:
C D# E G Ab B C and C Db E F G# A C. (The second scale is
actually just a transposition of a mode of the first.) Both have a
number of harmonic chords, as follows -

The first has major triads - C E G and Ab C D# and E G# B,
minor triads - C D# G and E G B and Ab B D#,
and augmented triads - C E Ab and G B D#.

The second has major triads - Db F G# and F A C and A Db E,
minor triads - Db E G# and F G# C and A C E,
and augmented triads - C E G# and Db F A.
(As expected, these are transpositions of those of the first.)

The relevance of this to 9-EDO is that its third step, 400 cents,
is exactly the fourth step of 12-EDO, a (rough) minor third. So
this interval is always available in 9-EDO, which makes it rather
(sub-)harmonic. This doesn't meet Igliashon's desire to eliminate
all the usual consonances, as it leaves minor thirds and major
sixths.

Back in 9-EDO, with s=1 and l=2, those two scales become:
C D#- E G- Ab B- C and C Db+ E F+ G# A+ C,
where by + and - I mean a pitch shift of 33 cents up or down.
(With this convention we could notate the entire 9-EDO gamut as -
C Db+ D#- E F+ G- G#/Ab A+ B- C
0 133 267 400 533 677 800 933 1067 1200.)
The harmonic chords of these scales in 12-EDO become -

First scale - major triads - C E G- and Ab C D#- and E G# B-,
minor triads - C D#- G- and E G- B- and Ab B- D#-,
and augmented triads - C E Ab and G- B- D#-.

Second scale - major triads - Db+ F+ G# and F+ A+ C and A+ Db+ E ,
minor triads - Db+ E G# and F+ G# C and A+ C E,
and augmented triads - C E G# and Db+ F+ A+.

The major and minor triads all have -
o a fairly good major third of 400 cents
(well, we're used to hearing it as good!),
o a bad minor third of 267 cents, and
o a bad fifth of 677 cents.

The augmented triads are exactly as good as in 12-EDO.

It seems that 9-EDO is harmonically bad, compared with
12-EDO. For Igliashon's purposes, it's better than 12-EDO,
but still probably not weird enough.

Now, I know that some people may say "There are easier
ways of showing this", and they may be right. But for me,
musically, there's no better way of finding something out
than by trying it.

---

What about 5-EDO? It's just every second note from 10-EDO;
however, having the guitar tuned in 10-EDO doesn't necessarily
make it easy to finger 5-EDO. The notes are 240, 480, 720, 960
and 1200. The second note is a flattish fourth and the third is a
sharpish fifth; altogether too consonant, I think. This is, of course,
the "Javese Salendro, assumed" scale reported by Ellis as number
95 in his table of non-harmonic scales (p 518, "On the Sensations of
Tone", Hermann Helmoltz, translated by Alexander Ellis, Dover
1954). So I think we can scrub that one.

Maybe 5.5-EDO would sound good, but you'd have to be nimbler
than I am to play melodies in it by fingering every second fret
with the guitar tuned in 11-EDO. (I did try ...)

I'll probably revisit the 10-, 10.5- and 11-EDO cases again, with
retuned strings this time. But now, it's breakfast time!

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
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--- In MakeMicroMusic@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
> Dave,
>
> Has anyone named this scale yet?

It's called simply "phi 13" in tha Scala archive. It was suggested to
Erv Wilson way back, by a guy named Lorne Themes. I expect it has
occurred to many people.

You go ahead and call it what you like. :-)

-- Dave

--- In MakeMicroMusic@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
>
> Igliashon and all,
>
> Wrote the following Saturday morning, when I had plenty of time.
> Had meant to revisit the unfinished business - the relative tuning of
> the open strings - before sending this, but time flies when you work
> for a living. So rather than wait till the thread grows cold, I'm
sending
> this preliminary report to you now. YA
...

That's brilliant, Ya! And thanks for a good laugh too.

Yes. You'll get the most notes farthest from 12-equal if your stretch
or compression factor is a noble number, of the form

1 + (n+1)*phi
-------------
1 + n*phi

Where n is a whole number and phi is (sqrt(5)+1)/2.

-- Dave

> See
> http://dkeenan.com/Music/MicroGuitar.pdf

Fascinating. I'm going to have to spend some time with this one, but
it is DEFINITELY going to be useful down the line, when I start
thinking about non-equal frettings.

> I trust your ears. But I've got bad news for you. ;-) I don't think
>it gets any badder unless you either:
> (a) have fewer notes per octave, or
> (b) give up octaves,
> or both.

Indeed, it seems that way, but then you have to realize that there is
a question of "priority" here: my highest priority is to avoid
intervals that sound like fourth/fifths. My second highest priority
is to avoid "normal-sounding" major and minor thirds, since I've
already got 3 guitars to give me some unusual thirds. After that,
it's all just gravy. And most of these suggested tunings don't seem
to miss the 4th/5th by as much as I'd like. In fact, even giving up
octaves doesn't seem to help much in that respect, since for
stretched/compressed tunings the 4:3's and 3:2's are not mistuned by
the same amount, leaving one of them nice and wolfy while the other
is recognizable.

The more I look at it, the more it seems like 13-EDO is the way to go
for this current project. It misses 4:3 and 3:2 by a huge amount...I
think the only temperament that misses them farther is 11-EDO, and I
already have access to that temperament. 13 also misses 6:5 by ~40
cents, its nearest approx. to 5:4 is closer to 16:13, and its
wholetones are almost Just 10:9's. In a sense, I think 13 is the
yard-stick I'm using to measure everything else, since that was my
original impulse. However, all these suggestions are absolutely
fascinating, and there are more than a few I hope to explore in the
future. I really need to get something more readily retunable than a
guitar....

In the meantime, I'd really like to hear more composers try their
hands at these theoretically 'maximum dissonance' tunings...and not
just for dissonant music, either!

-Igs

> The more notes you have per octave the easier it gets to find
> something vaguely diatonic. If you stopped at 10 notes of this
> phi-generated scale, then it would be better.
>
> But I think you're not only trying to avoid 5-limit JI
approximations,
> but also approximations to intervals of 12-equal. And these scales
do
> have 100 cent steps, and sometimes two in a row.
>
> > In fact it's very very close to subset of 36-equal, which
> > doesn't seem like the most dissonant tuning out there.
>
> Yes it is essentially a subset of 36-equal (and so of 72-equal), but
> as Erv Wilson notes, on the last page of
> http://www.anaphoria.com/tres.PDF
> 72-ET contains both the best and the worst.
>
> The golden generator 50/72 oct (or equivalently 22/72 oct) just
> happens to be the one whose multiples take the longest to hit a
> consonance and that hits the fewest consonances per note. George
> Secor's miracle generator of 7/72 is the opposite.
>
> > Hmm...I dunno about this one. It's a cool idea, to be sure, but
> > maybe this recursive combinational tone idea is a little too
> > impractical for this current project of mine.
> >
> > However, I did just recently notice that 23-EDO contains very
good
> > approximations to both Phi and Pi, as well as a host of other
bizarre
> > intervals. However, it violates my first criterion (less than 19
> > steps per approximate 2:1), so maybe 11.5-EDO? i'll have to play
> > around a bit.
>
> You could try using 7/23 oct as the generator (~365 cents) and stop
at
> 10 notes. But this gives you better 8:9s than the 11/36 oct
generator.
>
> If you wanted primarily to get as far away from any 12-equal
intervals
> as possible then I'd advise scrapping the octave and using a
generator
> which is some noble multiple of the 12-equal semitone. The simplest
> are 100*phi = 161.8-cET (7.4-ET) and 100/phi = 61.8-cET (19.4-ET)
but
> these probably have too few and too many notes respectively.
>
> This is interesting
> http://mathworld.wolfram.com/SilverRatio.html
> but it gives us noble ratios _further_ from 1, whereas we need them
> closer to 1.
>
> Please read
> "The Noble Mediant: Complex ratios and metastable musical
intervals"
> by Margo Schulter and myself. See
> http://dkeenan.com/Music/NobleMediant.txt
>
> The next noblest number closer to 1 is probably the noble mediant of
> 1/1 and 3/2 = (1+3phi)/(1+2*phi) ~= 1.382
> so you could try 138.2-cET (8.7-ET) and 72.4-cET (16.6-ET)
>
> I note that Graham Breed also suggested 16.6-ET, based on a
completely
> different method, in
> /makemicromusic/topicId_9723.html#9741
>
> -- Dave Keenan

--- In MakeMicroMusic@yahoogroups.com, "daniel_anthony_stearns"
<daniel_anthony_stearns@y...> wrote:
> Yes, the site was down last night for some reason, but it's back now:
>
> http://meowing.memh.uc.edu/~chris/micromp3s/suite.html

Meester Stearns: I must say, a year ago I probably would have hated
this. But now I find it to be one of the freshest, most ear-opening
pieces I've ever heard! It sounds almost algorithmic. The tuning is
very appropriate; it's very pensive and slightly agitated, a mood that
is probably expressed better in 13 than most other tunings. Jolly
good. Schoenberg would likely have pooped himself.

-Igs

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
> Indeed, it seems that way, but then you have to realize that there
> is a question of "priority" here: my highest priority is to avoid
> intervals that sound like fourth/fifths. My second highest priority
> is to avoid "normal-sounding" major and minor thirds, since I've
> already got 3 guitars to give me some unusual thirds. After that,
> it's all just gravy. And most of these suggested tunings don't seem
> to miss the 4th/5th by as much as I'd like. In fact, even giving up
> octaves doesn't seem to help much in that respect, since for
> stretched/compressed tunings the 4:3's and 3:2's are not mistuned by
> the same amount, leaving one of them nice and wolfy while the other
> is recognizable.
>
> The more I look at it, the more it seems like 13-EDO is the way to
> go for this current project. It misses 4:3 and 3:2 by a huge
> amount...I
> think the only temperament that misses them farther is 11-EDO, and I
> already have access to that temperament. 13 also misses 6:5 by ~40
> cents, its nearest approx. to 5:4 is closer to 16:13, and its
> wholetones are almost Just 10:9's. In a sense, I think 13 is the
> yard-stick I'm using to measure everything else, since that was my
> original impulse.

When you put it like that, then yes, you can't do worse than 13-EDO
(even including fractional EDOs, from 7 to 19). Go for it. 14-EDO is a
close second. 11-EDO is not as bad as those. To get worse you have to
go down to 6.625-EDO.

-- Dave Keenan

nice post Yahya,thanks. For a long time I had an old Kay jumbo with a
moveable bridge, and I used it for the thing. But it's really only
quick and dirty approximations of the theoretically implied equal
tunings and the errors grow quite large the further you move the
bridge...still,it's a beautiful resource , and I've been looking to
replace this guitar with a new, and more functionally sound one for
some time.
daniel

--- In MakeMicroMusic@yahoogroups.com, "Yahya Abdal-Aziz"
<yahya@m...> wrote:
>
> Igliashon and all,
>
> Wrote the following Saturday morning, when I had plenty of time.
> Had meant to revisit the unfinished business - the relative tuning
of
> the open strings - before sending this, but time flies when you
work
> for a living. So rather than wait till the thread grows cold, I'm
sending
> this preliminary report to you now. YA
>
> -------------
>
> Here's my report on some simple experiments on a guitar. (If
you're
> past this stage, there's probably not much new in it for you.) I'm
> pleased to say they were non-destructive esperiments, and can easily
> be repeated without any real expense.
>
> I wrote, earlier:
> > ... the fewer notes the EDO scale has to the octave (or
> > double octave, per Graham), the weirder it may sound.
Particularly
> > when the period doesn't divide 12, such as 2, 3, 4 and 6-EDO, we
get
> > some nice (easily-fingered) scales that don't remind us too
inexorably
> > of common repertoire music.
> >
> > For instance, 7-EDO is the theoretical Thai court music scale,
and
> > that sounds particularly rootless or restless (to me at least).
> >
> > I've long believed that any artist worthy of the name needs to be
> > able to make the most of limited resources - as they are often a
> > fact of life - and besides, the mental discipline of doing so is
also
> > a real challenge to creativity. In any scale whatsoever, it's a
> > wonderful exercise to try to make engaging music with, say, only
> > three or four distinct pitches.
> >
> > For some nicely weird 11-EDO music, I recommend a listen to Bill
> > Sethares' "The Turquoise Dabo Girl", from Xentonality.
> >
> > 6-EDO is of course the whole-tone scale that Debussy exploited so
> > brilliantly. But how about 5-EDO? With its very limited gamut,
and
> > no point of harmonic repose, that should stretch musicianship and
> > inventiveness pretty far ...
>
> Lying in bed this morning, I had a silly idea. What if the
fingerboard
> of my guitar were made in rubber, and I stretched it, uniformly, by
> say 10%? - but left the bridge in the same place. The frets would
> become 10% further apart, and the notes 10% higher (in relative
> frequencies 100 cents --> 110 cents). Or equivalently, and much
more
> usefully, move the bridge one-eleventh of the length closer to the
nut?
>
> The numbers seemed quite good - to avoid any GOOD consonances
> (which are bad in this context), you'd want the minor and major
third
> and sixth, as well as the fourth, fifth and octave, to be
significantly
> off. That is, the notes from frets 3, 4, 5, 7, 9, 10 and 12
shouldn't be
> too consonant. With a 10% stretch, these turn into 330, 400, 550,
> 770, 990, 1100 and 1320 respectively. Only the last three are near
to
> whole 12-EDO semitones, and maybe fiddling with the stretch factor
> could fix that.
>
> But how easily could I adapt the guitar, perhaps by adding another
> bridge nearer the sound hole?
>
> So I got out of bed, got a glass of water and went to take my
tablets
> before grabbing the guitar. Unfortunately, I slipped in a puddle
on
> the tile floor in the kitchen (dog getting old, weak bladder) which
> sort of delayed proceedings while we cleaned up broken glass ... :-(
>
> When I got the guitar, I saw I could easily slip a pen underneath
the
> strings to act as a movable bridge, and tension it by jamming a thin
> bit of plastic (actually, the handle of a pink feather duster!)
under
> one end. The pen was a cheap gel one, with an almost straight
plastic
> barrel, from K-Mart. A ball-point would do just as well.
>
> Success! In no time, I was picking out "tunes" in my approximate
> 11-EDO.
>
> With the bridge almost at the edge of the soundhole (this is a flat-
> top Spanish guitar), I had very close to 10-EDO, and I'm sorry to
> say, this was very tuneful indeed. Only a couple of the intervals
> were bad enough for Igliashon's purpose, of having no readily
> available consonances. I jotted down one of the tunes I picked, in
> what I can only think of as 10-EDO E minor; only the thirds are not
> really melodic. I may convert that to a Scala .seq file to post to
> tuning-files to show what I mean about the thirds.
>
> With a 20% stretch, the consonances turn into 360, 480, 600, 840,
> 1080, 1200 and 1440 respectively. But this (10-EDO) scale also
> includes 720, which is a passable fifth, and does give a perfect
> octave.
>
> The 11-EDO setting (10% stretch) was more dissonant than the
> 10-EDO, by far. The fourths and fifths are really BAD, and that's
> good :-) Fact, the only thing that sounds halfway good is playing
> diminished chord positions on the top four strings; they sound
> rather like ordinary diminished chords, but stretched (I can't call
> them "augmented", can I?). All the major, minor and dom seventh
> chord (fingerings, that is) that I tried do sound quite bad. The
> minor chords were much better (therefore worse) than the others.
> (But see Caveat below.)
>
> To avoid the octave completely, I tried 10.5-EDO. This was even
> more successful than 11-EDO. The minor chords were still
> acceptable, particularly in speed, but everything else was "off".
> (Also see Caveat below.)
>
> With 9-EDO, the resonance is weakened greatly with the bridge
> moved so far over the sound-hole. However, I found it made for
> interesting arpeggio figures when shifting a diminished chord
> position or a barre chromatically up and down a few frets.
> (Also see Caveat below.)
>
> One thing I did notice in all tunings was that fast melodic passages
> sound pretty good; again, especially in "minor keys". I suspect
> that only slow music, that emphasises the most discordant intervals,
> will sound "weird" enough for Igliashon's purposes.
>
> Caveat:
> I should mention that in all of the above, I did not alter the basic
> tuning of the guitar, which would have become stretched fourths
> and a stretched third. For example, in 9-EDO, the strings are
> tuned, not 5, 5, 5, 4 and 5 steps apart, but about 4, 4, 4, 3.5 and
> 4 steps apart. Unless the strings are an exact number of steps
> of your gamut apart, the chords are not drawn from the same gamut,
> since each string has its own tuning. So the results above don't
> accurately represent 10-EDO, 10.5-EDO or 11-EDO, do they?
> Except on one string at a time. Although, by chance, the strings
> happened to be almost in tune in 9-EDO. (Actually, it's not
chance,
> but maths - 4 steps of 9 is 444 cents, while 5 steps of 12 is 417.)
>
> Therefore the conclusions I drew above about chords are
> probably wrong. I need to retune the strings to notes of the
> n-EDO gamut I'm using to be able to hear what chords really
> sound like in it.
>
> -------------------
>
> The point of the standard guitar tuning is to maximise playability;
> without shifting hand position, you can use up all your fingers (and
> no more!) to get different notes from one string, then move to
> the next string to continue the sequence of notes.
>
> So I tried tuning the strings to the standard number of steps apart,
> in 9-EDO, by leaving the top E string fixed and dropping the others.
> All the strings were quite playable. The chords I played then were
> at least all drawn from the same gamut.
>
> (Side note: after this retuning, by removing the movable bridge, I
> found that I had a usable non-harmonic tuning, with the notes tuned
> about 6.25, 6.25, 6.25, 5 and 6.25 steps apart. This produces no
> recognisable harmonies, no matter how I finger chords, and the
> consonances you get in melodies played in close positions of the
hand
> are limited to those of one tetrachord (fourth) at a time. We might
> call this chance discovery an accidentally non-harmonic tuning - if
> that didn't cause any confusion.)
>
> And yes, 9-EDO IS weird - especially melodically. I hardly know how
> to make a tune in it. Of course, almost every chord is wrong: the
> gamut consists of 0, 133, 267, 400, 533, 677, 800, 933, 1067, 1200
> cents above Doh. So you get an equally-tempered augmented chord
> as your most consonant triad. An interesting scale pattern is -
> l s l s l s
> Its inverse is -
> s l s l s l
> In 9-EDO, s=1 and l=2.
>
> Translating these into 12-EDO, with s=1 and l=3 we get the scales:
> C D# E G Ab B C and C Db E F G# A C. (The second scale
is
> actually just a transposition of a mode of the first.) Both have a
> number of harmonic chords, as follows -
>
> The first has major triads - C E G and Ab C D# and E G# B,
> minor triads - C D# G and E G B and Ab B D#,
> and augmented triads - C E Ab and G B D#.
>
> The second has major triads - Db F G# and F A C and A Db E,
> minor triads - Db E G# and F G# C and A C E,
> and augmented triads - C E G# and Db F A.
> (As expected, these are transpositions of those of the first.)
>
> The relevance of this to 9-EDO is that its third step, 400 cents,
> is exactly the fourth step of 12-EDO, a (rough) minor third. So
> this interval is always available in 9-EDO, which makes it rather
> (sub-)harmonic. This doesn't meet Igliashon's desire to eliminate
> all the usual consonances, as it leaves minor thirds and major
> sixths.
>
> Back in 9-EDO, with s=1 and l=2, those two scales become:
> C D#- E G- Ab B- C and C Db+ E F+ G# A+ C,
> where by + and - I mean a pitch shift of 33 cents up or down.
> (With this convention we could notate the entire 9-EDO gamut as -
> C Db+ D#- E F+ G- G#/Ab A+ B-
C
> 0 133 267 400 533 677 800 933
1067 1200.)
> The harmonic chords of these scales in 12-EDO become -
>
> First scale - major triads - C E G- and Ab C D#- and E G# B-,
> minor triads - C D#- G- and E G- B- and Ab B- D#-,
> and augmented triads - C E Ab and G- B- D#-.
>
> Second scale - major triads - Db+ F+ G# and F+ A+ C and A+ Db+ E ,
> minor triads - Db+ E G# and F+ G# C and A+ C E,
> and augmented triads - C E G# and Db+ F+ A+.
>
> The major and minor triads all have -
> o a fairly good major third of 400 cents
> (well, we're used to hearing it as good!),
> o a bad minor third of 267 cents, and
> o a bad fifth of 677 cents.
>
> The augmented triads are exactly as good as in 12-EDO.
>
> It seems that 9-EDO is harmonically bad, compared with
> 12-EDO. For Igliashon's purposes, it's better than 12-EDO,
> but still probably not weird enough.
>
> Now, I know that some people may say "There are easier
> ways of showing this", and they may be right. But for me,
> musically, there's no better way of finding something out
> than by trying it.
>
> ---
>
> What about 5-EDO? It's just every second note from 10-EDO;
> however, having the guitar tuned in 10-EDO doesn't necessarily
> make it easy to finger 5-EDO. The notes are 240, 480, 720, 960
> and 1200. The second note is a flattish fourth and the third is a
> sharpish fifth; altogether too consonant, I think. This is, of
course,
> the "Javese Salendro, assumed" scale reported by Ellis as number
> 95 in his table of non-harmonic scales (p 518, "On the Sensations
of
> Tone", Hermann Helmoltz, translated by Alexander Ellis, Dover
> 1954). So I think we can scrub that one.
>
> Maybe 5.5-EDO would sound good, but you'd have to be nimbler
> than I am to play melodies in it by fingering every second fret
> with the guitar tuned in 11-EDO. (I did try ...)
>
> I'll probably revisit the 10-, 10.5- and 11-EDO cases again, with
> retuned strings this time. But now, it's breakfast time!
>
> Regards,
> Yahya
>
> --
> No virus found in this outgoing message.
> Checked by AVG Anti-Virus.
> Version: 7.0.308 / Virus Database: 266.11.8 - Release Date: 10/5/05

thanks Igliashon! A few things I thought I'd mention are that only the
first section is in 13Edo, and I really thought of that section in much
more of a Bartok type of "extended folk tonality" context than
an "atonal" one....anyway, thanks , and I'll post some of the 13Edo
Ukulele soon.....it's very sweet,well aliensweet, but definitely not
harsh at all. good luck......and welcome to the dark side,ha!!!

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@s...> wrote:
> --- In MakeMicroMusic@yahoogroups.com, "daniel_anthony_stearns"
> <daniel_anthony_stearns@y...> wrote:
> > Yes, the site was down last night for some reason, but it's back
now:
> >
> > http://meowing.memh.uc.edu/~chris/micromp3s/suite.html
>
> Meester Stearns: I must say, a year ago I probably would have hated
> this. But now I find it to be one of the freshest, most ear-opening
> pieces I've ever heard! It sounds almost algorithmic. The tuning is
> very appropriate; it's very pensive and slightly agitated, a mood
that
> is probably expressed better in 13 than most other tunings. Jolly
> good. Schoenberg would likely have pooped himself.
>
> -Igs

--- In MakeMicroMusic@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
>
> Here's my report on some simple experiments on a guitar. (If
> you're
> past this stage, there's probably not much new in it for you.) I'm
> pleased to say they were non-destructive esperiments, and can easily
> be repeated without any real expense.
>

> When I got the guitar, I saw I could easily slip a pen underneath
> the strings to act as a movable bridge, and tension it by jamming a
> thin bit of plastic (actually, the handle of a pink feather duster!)
> under one end. The pen was a cheap gel one, with an almost straight
> plastic barrel, from K-Mart. A ball-point would do just as well.
>
> Success! In no time, I was picking out "tunes" in my approximate
> 11-EDO.
>

I tried this out yesterday - and it worked! He he, excellent idea -
and very convenient to my current interest in 5-TET and 10-TET!

Interesting detail: the canonical adaptation of standard guitar tuning
to 10-TET yields a completely regular result, i.e. all strings are one
fourth apart (the irregularity of the major third between the g and
the b string vanishes).

But the fingerings in 5-TET are indeed a little boring... Any good
ideas around for alternative tunings of the strings of a 10-TET
guitar?

Maybe I will try dadgad next time...
--
Hans Straub

--- In MakeMicroMusic@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:
> --- In MakeMicroMusic@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
> wrote:
> >
> > Here's my report on some simple experiments on a guitar. (If
> > you're
> > past this stage, there's probably not much new in it for you.) I'm
> > pleased to say they were non-destructive esperiments, and can easily
> > be repeated without any real expense.
> >
>
>
> > When I got the guitar, I saw I could easily slip a pen underneath
> > the strings to act as a movable bridge, and tension it by jamming a
> > thin bit of plastic (actually, the handle of a pink feather duster!)
> > under one end. The pen was a cheap gel one, with an almost straight
> > plastic barrel, from K-Mart. A ball-point would do just as well.
> >
> > Success! In no time, I was picking out "tunes" in my approximate
> > 11-EDO.
> >
>
> I tried this out yesterday - and it worked! He he, excellent idea -
> and very convenient to my current interest in 5-TET and 10-TET!

As Dan Stearns pointed out, this gives fairly inaccurate results,
getting worse the further you get from 12-EDO. But hey, for zero cost,
who's complaining.

That is, it is very inaccurate if you determine the new bridge
position for N-EDO by moving it until the Nth fret gives an octave.
For example, in the case of 10-EDO, if you place it so the 10th fret
gives an octave you will have errors of 10 to 17 cents on frets 2 thru 8.

If instead you move the bridge so that the 5th fret gives a
half-octave then the errors for frets 1 thru 6 will be less than 5
cents, although frets 8 and beyond will be unusable.

-- Dave Keenan