Favorite Xenharmonic Chords/Intervals?

Well, after all this arguing that's been going on lately, I thought it might be nice to bring the focus back to music a bit by asking y'all to post your favorite "xenharmonic" intervals or chords. I rarely get the opportunity to talk about such things with people in real life.

For me, I find that neutral sevenths really "pop", especially if they're near 11/6, and even more if you use them on top of a minor triad of some sort. Neutral ninths around 24/11 are really cool, too, especially on top of a near-4:5:7 triad. Hands down, though, for a triad that just screams "recognizable-but-alien tonality" to me, I gotta go with the 16:18:21 I just suggested in Kite's post.

What do you lot fancy?

-Igs

Favorite intervals
11/6
12/11
11/7
9/7
18/11
5/3 (only because 12TET does this one so BADLY)
7/4
19/10
22/15
50/33

Favorite chords
Too many to list...but mainly those reaching past the 10th harmonic and with
fairly even critical band spacing IE 11:12:14:16 or 12:15:17:20.

Igs>"For me, I find that neutral sevenths really "pop", especially if they're
near 11/6, and even more if you use them on top of a minor triad of some sort."
Agreed, I love the 11/6, especially compared to 15/8. Use them in virtually
all my new scales...they just sound like more relaxed/serene/spread-out 7ths to
me.

>"Neutral ninths around 24/11 are really cool, too"
Funny because they are, of course, an octave over neutral seconds, which are
among my favorite intervals and both of those intervals appear a whole lot in
scales I've created.

>"Hands down, though, for a triad that just screams "recognizable-but-alien
>tonality" to me, I gotta go with the 16:18:21"
The numbers call out to me 8:9:10 chord but with the 10 suspended to form a
7/6 dyad between the 18 and 21. There is only dyad here that is "weird" out of
the three possible, the 21/16...and it just happens to have the widest space and
least root-tone critical band issues...which is why I assume explains much of
how the chord sounds "recognizable".

--- On Sun, 7/18/10, cityoftheasleep <igliashon@...> wrote:

> Well, after all this arguing that's
> been going on lately, I thought it might be nice to bring
> the focus back to music a bit by asking y'all to post your
> favorite "xenharmonic" intervals or chords.  I rarely
> get the opportunity to talk about such things with people in
> real life.

This is more an issue with timbre blending than chords, but I randomly discovered a sound combination I like: a clarinet and trombone (or maybe bassoon) playing 16/11 apart.

As far as chords go, my favorite when I was a beginner in xenharmony was the neutral seventh (or "Rast seventh") chord, 18:22:27:33. I've also used the septimal minor seventh, 12:14:18:21, plenty of times. I also like the "wolf fourth", especially if necessary to avoid a wolf *fifth*, so I'm almost as likely to voice a major sixth chord as 16:20:24:27 as I am 12:15:18:20 (the major ninth will have to have a wolf fourth anyway).

And then there's the huge Fibonacci chord I have at the beginning of an unfinished piece: 1:2:3:5:8:13:22:34:55:89:144, built on G0 = 24.5 Hz. It's very well-approximated in 72-edo in fact; all intervals from 13/8 on up are 833.33 cents.

~D.

I really want to reply to this, but I'm a by-ear composer when it comes to microtonality. At least for the most part.

What I can say is that recently I've been loving the sound of 11/8. These days I'm trying to use that instead of plain fourths. Because I likes it.

Sean

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Well, after all this arguing that's been going on lately, I thought it might be nice to bring the focus back to music a bit by asking y'all to post your favorite "xenharmonic" intervals or chords. I rarely get the opportunity to talk about such things with people in real life.
>
> For me, I find that neutral sevenths really "pop", especially if they're near 11/6, and even more if you use them on top of a minor triad of some sort. Neutral ninths around 24/11 are really cool, too, especially on top of a near-4:5:7 triad. Hands down, though, for a triad that just screams "recognizable-but-alien tonality" to me, I gotta go with the 16:18:21 I just suggested in Kite's post.
>
> What do you lot fancy?
>
> -Igs
>

Sevish>"What I can say is that recently I've been loving the sound of 11/8.
These days I'm trying to use that instead of plain fourths. Because I likes
it."

Between you and Igs it seems you guys are naming a BUNCH of the intervals
used in my Infinity scale system (not to mention Ptolemy's scale system), and a
lot of the intervals you prefer seem to mirror exotic ones I "guessed"
12-TET-accustomed people would prefer.

Well here is a counter question
My new interval system uses the following higher-limit intervals assuming the
average brain likes them nearly so much as lower limit ones

11/6
11/8
11/9
12/11
15/11
18/11
22/15

Do any of you NOT like any of the above intervals? (and, if possible, how
come?)
And, a side question, is it fair to say the brain (when dealing with
11-or-higher limit intervals) gravitates toward those with either the numerator
or denominator is directly divisible by either 2 or 3?

Not to be a nudge but what's been said on this thread seems to hint there's an
awful lot of 11-limit intervals that work on par, in many cases, with their
lower-limit counterparts...I'm wondering if there is a loophole in this idea or
if the patterns I believe I'm seeing really hold their ground.

I echo Sean in being a mostly by ear microcomposer at this point.

With that being said I have found several intervals (for which I don't know
the name) when playing my fretless guitar - all of which fit into blues
nicely

(the one I know) 5/4 - the true third
a ~quarter flat tritone
a ~quarter flat minor 7th
a ~quarter flat minor 3rd
a ~quarter flat major 7th

Also - what is also interesting is how the dependencies of these intervals
play with each other .

Chris

On Mon, Jul 19, 2010 at 3:36 PM, sevishmusic <sevish@...> wrote:

>
>
> I really want to reply to this, but I'm a by-ear composer when it comes to
> microtonality. At least for the most part.
>
> What I can say is that recently I've been loving the sound of 11/8. These
> days I'm trying to use that instead of plain fourths. Because I likes it.
>
> Sean
>
>
> -
>

I used to really like 9 and chords like 5:6:7:9 and 4:5:6:9.
These days I don't play favorites.

-Carl

--- In tuning@yahoogroups.com, "sevishmusic" <sevish@...> wrote:
>
> I really want to reply to this, but I'm a by-ear composer when
> it comes to microtonality. At least for the most part.
>
> What I can say is that recently I've been loving the sound
> of 11/8. These days I'm trying to use that instead of plain
> fourths. Because I likes it.
>
> Sean
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >
> > Well, after all this arguing that's been going on lately,
> > I thought it might be nice to bring the focus back to music
> > a bit by asking y'all to post your favorite "xenharmonic"
> > intervals or chords. I rarely get the opportunity to talk
> > about such things with people in real life.
>

Just subminor triads are awesome, especially when you turn them into
subminor 9 chords. It works really well when you put it in 2nd inversion. So
for Fsm9 you'd end up with

C Eb< F G Ab< C

That G-Ab< dyad is kind of hard to swallow, but once you get around it it
works pretty well. Neil Haverstick used a bunch of chords like this in
"Beautiful Springtime" off of his latest album "Spider."

-Mike

On Sun, Jul 18, 2010 at 9:23 PM, cityoftheasleep <igliashon@...>wrote:

>
>
> Well, after all this arguing that's been going on lately, I thought it
> might be nice to bring the focus back to music a bit by asking y'all to post
> your favorite "xenharmonic" intervals or chords. I rarely get the
> opportunity to talk about such things with people in real life.
>
> For me, I find that neutral sevenths really "pop", especially if they're
> near 11/6, and even more if you use them on top of a minor triad of some
> sort. Neutral ninths around 24/11 are really cool, too, especially on top of
> a near-4:5:7 triad. Hands down, though, for a triad that just screams
> "recognizable-but-alien tonality" to me, I gotta go with the 16:18:21 I just
> suggested in Kite's post.
>
> What do you lot fancy?
>
> -Igs
>
>
>

22/15 is nasty if naked.

You are going to 22/15? Do no forget your whip!

-c

On Jul 19, 2010, at 4:03 PM, Michael wrote:

>
> Sevish>"What I can say is that recently I've been loving the sound of 11/8. These days I'm trying to use that instead of plain fourths. Because I likes it."
>
> Between you and Igs it seems you guys are naming a BUNCH of the intervals used in my Infinity scale system (not to mention Ptolemy's scale system), and a lot of the intervals you prefer seem to mirror exotic ones I "guessed" 12-TET-accustomed people would prefer.
>
>
> Well here is a counter question
> My new interval system uses the following higher-limit intervals assuming the average brain likes them nearly so much as lower limit ones
>
> 11/6
> 11/8
> 11/9
> 12/11
> 15/11
> 18/11
> 22/15
>
>
> Do any of you NOT like any of the above intervals? (and, if possible, how come?)
> And, a side question, is it fair to say the brain (when dealing with 11-or-higher limit intervals) gravitates toward those with either the numerator or denominator is directly divisible by either 2 or 3?
>
>
> Not to be a nudge but what's been said on this thread seems to hint there's an awful lot of 11-limit intervals that work on par, in many cases, with their lower-limit counterparts...I'm wondering if there is a loophole in this idea or if the patterns I believe I'm seeing really hold their ground.
>
>

On Mon, Jul 19, 2010 at 4:09 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I echo Sean in being a mostly by ear microcomposer at this point.
>
> With that being said I have found several intervals (for which I don't know the name) when playing my fretless guitar - all of which fit into blues nicely
>
> (the one I know) 5/4 - the true third
> a ~quarter flat tritone

That would be 11/8.

> a ~quarter flat minor 7th

That's about 7/4.

> a ~quarter flat minor 3rd

That's about 7/6.

> a ~quarter flat major 7th

I usually hear that as a compound interval representing a neutral
third + a fifth, or a fifth + a neutral third, or however you want to
think of it. Kind of like how a major 7th is 5/4 * 3/2, I think of it
as 11/9 * 3/2, or whatever neutral third you prefer.

-Mike

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I echo Sean in being a mostly by ear microcomposer at this point.
>
> With that being said I have found several intervals (for which I don't know
> the name) when playing my fretless guitar - all of which fit into blues
> nicely
>
> (the one I know) 5/4 - the true third
> a ~quarter flat tritone
> a ~quarter flat minor 7th
> a ~quarter flat minor 3rd
> a ~quarter flat major 7th

What does "quarter flat" mean and what are they flat from?
You can't just say "tritone" or "minor third" when talking about tuning.

Hi Gene,

I hope you are having a very pleasant evening.
(Or whatever is appropriate for your timezone when you receive this reply).

With all due respect I think I can describe intervals as I did when it is in
the context of 12 edo.

I described in some detail on this email list the $60 12 edo electric guitar
I bought, made fretless, and the nice lines the plastic wood made where the
frets used to be.

Thanks,

Chris

On Mon, Jul 19, 2010 at 5:52 PM, genewardsmith
<genewardsmith@...>wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > I echo Sean in being a mostly by ear microcomposer at this point.
> >
> > With that being said I have found several intervals (for which I don't
> know
> > the name) when playing my fretless guitar - all of which fit into blues
> > nicely
> >
> > (the one I know) 5/4 - the true third
> > a ~quarter flat tritone
> > a ~quarter flat minor 7th
> > a ~quarter flat minor 3rd
> > a ~quarter flat major 7th
>
> What does "quarter flat" mean and what are they flat from?
> You can't just say "tritone" or "minor third" when talking about tuning.
>
>
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:

> With all due respect I think I can describe intervals as I did when it is in
> the context of 12 edo.

This still leaves me guessing, but narrows it down some. My guesses are the same ones as before:

> > > a ~quarter flat tritone
550 cents? An 11/8

> > > a ~quarter flat minor 7th
950 cents? Possibly 7/4

> > > a ~quarter flat minor 3rd
250 cents? Possibly 7/6, or could be 15/13

> > > a ~quarter flat major 7th
1050 cents? An 11/6

cityoftheasleep wrote:
> Well, after all this arguing that's been going on lately, I thought
> it might be nice to bring the focus back to music a bit by asking
> y'all to post your favorite "xenharmonic" intervals or chords. I
> rarely get the opportunity to talk about such things with people in
> real life.
> > For me, I find that neutral sevenths really "pop", especially if
> they're near 11/6, and even more if you use them on top of a minor
> triad of some sort. Neutral ninths around 24/11 are really cool,
> too, especially on top of a near-4:5:7 triad. Hands down, though,
> for a triad that just screams "recognizable-but-alien tonality" to
> me, I gotta go with the 16:18:21 I just suggested in Kite's post.
> > What do you lot fancy?
> Melodic or harmonic? For melodic use I like the neutral second that you get by dividing the minor third into equal parts. As far as harmony, I guess the 7/6 is one that stands out, especially in something like a 1/(7:6:5:4) chord, or a 6:7:9.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> Well here is a counter question
> My new interval system uses the following higher-limit intervals assuming the
> average brain likes them nearly so much as lower limit ones
>
> 11/6
> 11/8
> 11/9
> 12/11
> 15/11
> 18/11
> 22/15
>
> Do any of you NOT like any of the above intervals? (and, if possible, how
> come?)

I love all of these, but I have to say that 11/8 has pretty much never sounded like a "consonance" to me and I actually find the near-by 15/11 to sound quite a bit nicer. But I think that's a good thing, I think 11/8 is a really great dissonance and is actually quite a bit more unsettling than most of the various "augmented fourths" (7/5, 18/13, 27/20) I've encountered. However, I probably wouldn't want to use a scale composed of only these intervals; they are very xenharmonic, but I'm a firm believer in the idea that one needs some familiar intervals for contrast, or else the xenharmony gets obscured by its own "shock value".

> Not to be a nudge but what's been said on this thread seems to hint there's an
> awful lot of 11-limit intervals that work on par, in many cases, with their
> lower-limit counterparts...I'm wondering if there is a loophole in this idea or
> if the patterns I believe I'm seeing really hold their ground.

I think it's more the case that 11-limit intervals are probably some of the most xenharmonic, and a lot of people on this list might have a bias toward favoring them. My predilection for them is certainly not something I'd expect to appear in the "general population". I mean, people on the tuning list aren't a good sample population for determining whether intervals "sound good", because we tend to either a much broader sense of what is harmonically acceptable, or else a much narrower one.

OTOH, when you consider that 12-tET is full of near-Just 17 and 19-limit intervals (18/17, 19/16, 24/19, 17/12, 19/12, 32/19, and 17/9), and average people find 12-tET perfectly acceptable, near-Just 11-limit intervals might very well be accepted. I bet 13/11 and 14/11 could pass unnoticed by the average listener, and most would probably hear 11/8 as a variety of tritone (i.e. dissonant but still "normal")...even 11/10 could probably pass as a whole-tone in the right context (like in 22-tET's Pajara/decatonic scale). Maaaaybe you could sneak 15/11 by as a fourth. Maaaaaybe. It works in 9-EDO, more or less, but it really needs the help of the familiar 400¢ major third. But isn't the point of making xenharmonic music to get the audience to go "WTF! That sounds WEIRD!!" (but in a good way)?

>"OTOH, when you consider that 12-tET is full of near-Just 17 and 19-limit
>intervals (18/17, 19/16, 24/19, 17/12, 19/12, 32/19, and 17/9), and average
>people find 12-tET perfectly acceptable, near-Just 11-limit intervals might
>very well be accepted."
True enough...I guess the question is what makes a smooth-sounding higher
limit interval work? I think my chief example would be something like 16/11 vs.
22/15...numerically you'd think 16/11 would have to sound smoother yet in
reality I swear it sounds much rougher (at least to me).

>"I mean, people on the tuning list aren't a good sample population for
>determining whether intervals "sound good", because we tend to either a much
>broader sense of what is harmonically acceptable, or else a much narrower one."
But is there any reason not to open the can of worms of "maybe the limit
system does not always cover how easy to listen to an interval is". One example
again of 16/11 vs. 22/15. Another, I believe, is 50/33 vs. 17/11 (with 50/33
sounding better). Or 27/20 vs. 15/11 (27/20 seems to be to sound better). Of
course I think higher limit intervals are harder to listen to "on the
average"...but some higher limit intervals seem much more consonant than others
which look similar or simpler as fractions.

Igs>"It works in 9-EDO, more or less, but it really needs the help of the
familiar 400¢ major third. But isn't the point of making xenharmonic music to
get the audience to go "WTF! That sounds WEIRD!!" (but in a good way)?"
Heh, kind of. :-) My opinion is that is sound be weird enough to sound fresh
but interpretable enough that it leaves no doubt the musician knows what he/she
is doing and isn't going 'off-tone' due to inexperience or lack of talent. Not
to drag on but virtually every common musician I've shown micro-tonal music to
has responded "either it's avant-garde or the guy needs to learn music
theory"...minus those I've shown things like Erv Wilson's Hexanies and 6-tone
MOS scales in which case they simply called it very innovative.

One very obvious example of "weird but very coherent" in my mind would have
to be the Beatles "Sgt. Pepper's" album or the Chemical Brother's Surrender
album (both which, oddly enough, sound alike in quite a few ways to me).

>"I love all of these, but I have to say that 11/8 has pretty much never sounded
>like a "consonance" to me and I actually find the near-by 15/11 to sound quite
>a bit nicer."
I learn toward 15/11 as well, but don't find 11/8 grating by any means and
love tempering between 15/11 and 11/8. 11/8 sounds confident enough to me
though (in the same way something like 50/33 sounds passable to me as a
special/sharper/brighter version of a 5th)...though maybe it's just my ears and
their bias (and not applicable to many people).

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> True enough...I guess the question is what makes a smooth-sounding higher
> limit interval work? I think my chief example would be something like 16/11 vs.
> 22/15...numerically you'd think 16/11 would have to sound smoother yet in
> reality I swear it sounds much rougher (at least to me).

22/15 is in the outer fringes of the fifth orbit, and 16/11 isn't. 16/11 is out there on its own. 11/8 and 16/11 are core xenharmonic intervals partly for this reason, and I find them quite interesting.

Another, I believe, is 50/33 vs. 17/11 (with 50/33
> sounding better).

50/33 is much closer to 3/2; it's basically the screwed-up fifth of 5edo, which is neither fish nor fowl. But having many expectations of a 17-interval is overly optimistic anyway.

Or 27/20 vs. 15/11 (27/20 seems to be to sound better).

27/20 is in the outer reaches of the fourth orbit, being a comma sharper.

> I learn toward 15/11 as well, but don't find 11/8 grating by any means and
> love tempering between 15/11 and 11/8.

121/120 tempering. Equates the 12/11 and 11/10 3/4-2/3 tone intervals also.

Gene>"22/15 is in the outer fringes of the fifth orbit"
I figure it's way over 13 cents off a fifth and more like 17 cents off. And
even if that were "close enough"...meanwhile 1.485, which is the same
logarithmic difference, sounds much worse to me. I have my doubts as to if
"being in orbit around 3/2" is enough to explain how it avoids sounding fowl
even if it (like virtually any other interval minus the octave) doesn't sound
near as relaxed as a pure fifth.

>"16/11 is out there on its own. 11/8 and 16/11 are core xenharmonic intervals
>partly for this reason"
Figures...as 11/8 is over 20 cents from 7/5 and 16/11 is very far from both
7/5 and 3/2.

Me>>Another, I believe, is 50/33 vs. 17/11 (with 50/33
>> sounding better).
Gene>"50/33 is much closer to 3/2;"
...though it's still about 17 cents away and I have tough time believing
17-cents difference doesn't make it "it's own tone" enough to be compared with
other tones far from any obvious low-limit ratios.

>"27/20 is in the outer reaches of the fourth orbit, being a comma sharper."
So who or what defines how far an "orbit" can reach? One major example I've
found: compare the sound of 13/8 vs. that of 18/11 (both very close to each
other, about 11 cents apart)...they certainly sound very different to me,
especially surprising given your "orbit" examples above with reach, as you said,
to "commatic" proportions (much larger differences).

Me>> I learn toward 15/11 as well, but don't find 11/8 grating by any means
and

>> love tempering between 15/11 and 11/8.
Gene> 121/120 tempering. Equates the 12/11 and 11/10 3/4-2/3 tone intervals
also.
Seems like a good way to do things even judging by pure mathematical error and
not just aesthetics. Put a note in-between a 121/120 gap between two tones and
you get about 7 cents gap on each side...I figure; no wonder the ear seems to
have so little problem interpreting that kind of tempering. Side
question...what are your favorite scales using 121/120 tempering?

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> 22/15 is nasty if naked.
>
> You are going to 22/15? Do no forget your whip!
>
> -c

Just caught this, Caleb...are you intentionally referencing Nietzsche's infamous comment about women, or is it a coincidence?

-Igs

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Gene>"22/15 is in the outer fringes of the fifth orbit"
> I figure it's way over 13 cents off a fifth and more like 17 cents off. And
> even if that were "close enough"...meanwhile 1.485, which is the same
> logarithmic difference, sounds much worse to me. I have my doubts as to if
> "being in orbit around 3/2" is enough to explain how it avoids sounding fowl
> even if it (like virtually any other interval minus the octave) doesn't sound
> near as relaxed as a pure fifth.
//
> Gene>"50/33 is much closer to 3/2;"
> ...though it's still about 17 cents away and I have tough time believing
> 17-cents difference doesn't make it "it's own tone" enough to be compared with
> other tones far from any obvious low-limit ratios.
>
> >"27/20 is in the outer reaches of the fourth orbit, being a comma sharper."
> So who or what defines how far an "orbit" can reach? One major example I've
> found: compare the sound of 13/8 vs. that of 18/11 (both very close to each
> other, about 11 cents apart)...they certainly sound very different to me,
> especially surprising given your "orbit" examples above with reach, as you said,
> to "commatic" proportions (much larger differences).

I wonder if it's enough to look at the fifth as a dyad, as opposed to the boundary of a triad? I mean, 22/15 can work as a fifth in certain contexts, and it's almost 40 cents flat of 3/2. Its functionality as a fifth may not be apparent when looking at it "bare", as Caleb pointed out, but stick a well-tuned third in between the root and the 22/15 and suddenly you have a recognizable "root-third-fifth" triad.

I have a suspicion that using ratios to define interval classes may prove to be a dead end, since there's a lot of leeway in terms of how a ratio can function in a scale. If you look at a major scale generated by a 690¢ fifth, the major third comes out to a near-perfect 16/13, and the major seventh comes out to a near-perfect 11/6. Neither of these are close enough to the 5/4 or 15/8 that define the "normal" versions of those interval classes, and yet you can play diatonic music in that scale and the identity of those interval classes comes through despite being distant from the traditional ratios. So then the question arises: where does the identity of an interval class actually come from?

-Igs

Heh. My rhetoric about 22/15 seemed so lurid that I couldn't help but make fun of myself.

It might be of interest that Nietzsche put that line in the mouth of an old woman, or perhaps it was
a reference to his brief affair with his half-a-girlfriend, Lou Soleme, who he posed with his friend Paul Ree holding a whip.
In any case, it comes from his worst book, imo, when he was on manic rebound after Lou rejected him. Strangely, he always maintained it was his best. But enough about Fred.

My feeling about the interval is that it's so close to the pull or orbit of 3/2 that, by itself, it sounds like a very out-of-tune 3/2. 39 cents is too close for comfort, too far for euphony.

But I'm learning that intelligent people disagree.

For me, for example, 11/8, and more so 11/4, 11/2, 11/1 sound like consonances, if in high enough register.

-c

On Jul 20, 2010, at 6:27 PM, cityoftheasleep wrote:

> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
> >
> > 22/15 is nasty if naked.
> >
> > You are going to 22/15? Do no forget your whip!
> >
> > -c
>
> Just caught this, Caleb...are you intentionally referencing Nietzsche's infamous comment about women, or is it a coincidence?
>
> -Igs
>
>

On Tue, Jul 20, 2010 at 7:28 PM, cityoftheasleep
<igliashon@...> wrote:
>
> I wonder if it's enough to look at the fifth as a dyad, as opposed to the boundary of a triad? I mean, 22/15 can work as a fifth in certain contexts, and it's almost 40 cents flat of 3/2. Its functionality as a fifth may not be apparent when looking at it "bare", as Caleb pointed out, but stick a well-tuned third in between the root and the 22/15 and suddenly you have a recognizable "root-third-fifth" triad.
>
> I have a suspicion that using ratios to define interval classes may prove to be a dead end, since there's a lot of leeway in terms of how a ratio can function in a scale. If you look at a major scale generated by a 690¢ fifth, the major third comes out to a near-perfect 16/13, and the major seventh comes out to a near-perfect 11/6. Neither of these are close enough to the 5/4 or 15/8 that define the "normal" versions of those interval classes, and yet you can play diatonic music in that scale and the identity of those interval classes comes through despite being distant from the traditional ratios. So then the question arises: where does the identity of an interval class actually come from?
>
> -Igs

Yeah. That. What the hell? And if you play in 27-et, where the fifths
are so flat that the minor triads are ~6:7:9, they sound no different
than 19-tet, where the minor triads are closer to 10:12:15.

I just messaged Carl about this offlist, but let's bring it here
instead. How does it work? The question really is - does JI even
matter at all?

-Mike

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> Side
> question...what are your favorite scales using 121/120 tempering?
>

I'm afraid I haven't given much thought to rank 4 temperaments, but I am a fan of 46et, and have used it there. It's also a highly characteristic feature of valentine/Carlos Alpha, where 11/10-12/11 is only two generator steps. Hence the 15 and 16 note MOS of valentine already have goodly quantities of 11-limit harmony, and could be a good place to start if you want to explore this further. 46 or 77 are good tuning choices for valentine.

If you are really wild&crazy and want to try a rank 3 temperament, there's zeus, which also tempers out 176/175. It's the 22&31&46 temperament, and a 22 note scale would be one obvious thing to try.

Mike B>"I just messaged Carl about this offlist, but let's bring it here

instead. How does it work? The question really is - does JI even
matter at all?"

Here's an odd thing. I do believe in classes and JI, but also that the both
are a lot more flexible than people think in some cases (though really are picky
in others, depending on the register).
IE I find from about 22/15 to 50/33 work fine as 5ths in the context of
larger-than-dyadic chord. So I think JI matters, but is a lot less picky than
many people think it is, especially in certain areas IE between 22/15 and 50/33
along with between 18/11 and 14/8. But it does seem very picky between 13/8 vs.
18/11.

In other words JI certainly doesn't seem to work the same way for every
interval area (it seems a lot more picky about some than others). And the limit
system certainly seems to have some gaping holes in how accurately it judges a
dyad's consonance, especially in larger chords (it seems, the larger the chord
and more the note clustering the less extent in which error effects/hurts the
chord).

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

> Yeah. That. What the hell? And if you play in 27-et, where the fifths
> are so flat that the minor triads are ~6:7:9, they sound no different
> than 19-tet, where the minor triads are closer to 10:12:15.
>
> I just messaged Carl about this offlist, but let's bring it here
> instead. How does it work? The question really is - does JI even
> matter at all?
>
> -Mike
>

The fifth of 27 is sharp, not flat, at 711.111... cents. Sharp fifths push major and minor thirds further apart (until they merge with fourths and seconds, respectively, at 5-EDO), whereas flat fifths push major and minor thirds closer together (until they become one neutral third in 7-EDO). But regardless....

Yeah, see, I have this hypothesis that I'm trying to find a way to test, and it's that the structure of a scale does more to define harmonic perception than the JI ratios approximated by the triads. A 7/6 might sound very different from a 6/5 when you put them side-by-side, but playing diatonic music using 6:7:9 triads doesn't necessarily "feel" different than if you use 10:12:15. I mean, it will of course sound somewhat different, but the ear doesn't seem to interpret the triads as having different "meanings"--a 7/6 can still *mean the same thing* as a 6/5 or a 13/11 or a 19/16 or a 23/19 if it's put in the same context. This is just one of the many reasons I think Marcel is mistaken--there's nothing essentially 5-limit about diatonic music, especially in 12-tET where a minor triad is basically a 16:19:24.

So this begs the question: how do you find the limits to the range of possible tunings of triads that allows their musical meaning to be preserved? I'm actually starting to suspect that, at least to our 12-tET-saturated ear, the range of interval classes might actually be +/- 50 cents from 12-tET intervals. But then, I've heard an approximate 7/6 function as a pretty good whole-tone before, so maybe it's not so simple.

-Igs

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> Here's an odd thing. I do believe in classes and JI, but also that the both
> are a lot more flexible than people think in some cases (though really are picky
> in others, depending on the register).
> IE I find from about 22/15 to 50/33 work fine as 5ths in the context of
> larger-than-dyadic chord. So I think JI matters, but is a lot less picky than
> many people think it is, especially in certain areas IE between 22/15 and 50/33
> along with between 18/11 and 14/8. But it does seem very picky between 13/8 vs.
> 18/11.

Does it? I think that depends on what function you're giving to 13/8 and 18/11. Using a generator of 140 cents, you get a 9-note scale of LLLLLLLLs that has an almost-perfect 13/8. Shift the generator slightly to about 142.1 cents and you get a scale of the same MOS configuration, with an almost-perfect 18/11 in the same spot the 13/8 was in. Are you convinced you could tell music made with these two scales apart?

> In other words JI certainly doesn't seem to work the same way for every
> interval area (it seems a lot more picky about some than others). And the limit
> system certainly seems to have some gaping holes in how accurately it judges a
> dyad's consonance, especially in larger chords (it seems, the larger the chord
> and more the note clustering the less extent in which error effects/hurts the
> chord).

The "limit" system is well-known to be a rough and incomplete gauge of consonance. As Gene has pointed out, you can stay within a given limit and yet get arbitrarily close to the next-highest limit using ratios of greater complexity. Likewise, a ratio at a higher limit may approximate more simply a ratio of great complexity at a lower limit. I still swear that as long as an interval falls within the range of recognizability for an interval class (i.e. as long as it's "clearly" some type of major/minor third, or fifth, or whatever), it can sound consonant if it's placed in the context of a familiar scale structure. How else can you account for the long-term success of 12-tET?

-Igs

Igs>The "limit" system is well-known to be a rough and incomplete gauge of
consonance."
Agreed...in both the cases of odd and prime limit (though prime seems
considerably more faulty).

>"As Gene has pointed out, you can stay within a given limit and yet get
>arbitrarily close to the next-highest limit using ratios of greater
>complexity. Likewise, a ratio at a higher limit may approximate more simply a
>ratio of great complexity at a lower limit."
>
But (and Gene feel free to "butt in")...didn't Gene refer to prime and not
odd limit? I'm talking about odd limit...which, as a recall, is considered the
golden standard for rating JI chords...and saying even that "more strict" limit
system often falls flat.

>"(i.e. as long as it's "clearly" some type of major/minor third, or fifth, or
>whatever), it can sound consonant if it's placed in the context of a familiar
>scale structure."
Right, but I guess you could say my point is often you can even go so far as
to substitute a neutral vs. minor third, for example, and still be "ok" so far
as preserving the feel. You could say my main concern, in a way, is how
neutral-like intervals function in chords IE intervals like 13/8 and 18/11 or
11/9...which often seem to defy the odd-limit system in what their numbers look
like vs. how resolved they sound within chords (especially larger ones that seem
to help shift their context toward matching/resolved-ness). Not to mention the
fact they are not very close to any low limit intervals so the "tempered version
of" argument doesn't seem valid for them when they are often 15+ cents off.

>"How else can you account for the long-term success of 12-tET?"
Sorry, but I think 12TET is a lousy example since all of its intervals (at
least within diatonic scales) are within 13 cents of JI diatonic. A better
example, I believe, is those triads based on "nasty" 16/11 dyad in your first
song on "Map of an Internal Landscape", which sound oddly relaxed.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> But (and Gene feel free to "butt in")...didn't Gene refer to prime and not
> odd limit? I'm talking about odd limit...which, as a recall, is considered the
> golden standard for rating JI chords...and saying even that "more strict" limit
> system often falls flat.

The odd-limit has worse problems in some ways. Odd limit cannot be based on "odd factors" or else it would have the same problem as prime limit (getting arbitrarily close to the next odd limit using more complex ratios), and it would also be a clumsier way of saying basically the same thing that prime-limit says, since odd factors can reduce to prime factors. But then you have the question: is 16/9 a 9-odd-limit ratio, because 16 is even? Or is it a 15-odd-limit ratio because 15 is the highest odd number before 16? What of intervals in a 2nd octave? 13/11, a 13-odd-limit ratio, becomes 26/11 in the 2nd octave--is it then an 11-odd-limit ratio, or a 25-odd-limit ratio? Also, look at what happens between two 9-odd-limit ratios like 9/8 and 5/3: you have 40/27, a 27-odd-limit ratio that pops up in a scale of 5-odd-limit ratios. The odd-limit is just a terribly inconsistent way of looking at scales. Much worse than prime-limit, IMHO.

> >"(i.e. as long as it's "clearly" some type of major/minor third, or fifth, or
> >whatever), it can sound consonant if it's placed in the context of a familiar
> >scale structure."
> Right, but I guess you could say my point is often you can even go so far as
> to substitute a neutral vs. minor third, for example, and still be "ok" so far
> as preserving the feel. You could say my main concern, in a way, is how
> neutral-like intervals function in chords IE intervals like 13/8 and 18/11 or
> 11/9...which often seem to defy the odd-limit system in what their numbers look
> like vs. how resolved they sound within chords (especially larger ones that seem
> to help shift their context toward matching/resolved-ness). Not to mention the
> fact they are not very close to any low limit intervals so the "tempered version
> of" argument doesn't seem valid for them when they are often 15+ cents off.

Well, I guess I shouldn't have said "clearly" but rather its antonym, "vaguely". My mistake. But yes, you are right, a neutral third can certainly function as either major or minor, depending on what context it is given. It actually seems more of a challenge to really get it to function as a "neutral third" to me. Playing in Mohajira, since there's still something of a chain of fifths, I'm often surprised that the I, IV, and V chords sound major but the vi and iii chords still sound minor, even though they have the same size thirds. Most bizarre.

> >"How else can you account for the long-term success of 12-tET?"
> Sorry, but I think 12TET is a lousy example since all of its intervals (at
> least within diatonic scales) are within 13 cents of JI diatonic. A better
> example, I believe, is those triads based on "nasty" 16/11 dyad in your first
> song on "Map of an Internal Landscape", which sound oddly relaxed.
>

Well, those triads weren't 16/11 but 40/27, actually between 40/27 and 22/15 at 675¢, with thirds at 300¢ and 375¢. But regardless--12-tET may hit within 13-15 cents of 5-limit JI diatonic, but it hits even closer to JI 19-limit diatonic. I'm not sure it's fair to say 19/16 gets "absorbed" by 6/5 or that 24/19 gets absorbed by 5/4; these are (theoretically) distinct intervals, and I think it's more likely that both 6/5 and 19/16 (as well as 5/4 and 24/19) get absorbed into a much more vaguely-defined concept of interval class. 5/4 and 6/5 may be easier to tune to by ear, but that doesn't give them be-all end-all status as the definition of the major and minor third interval classes. Or does it, and if so, why?

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>But then you have the question: is 16/9 a 9-odd-limit ratio, because 16 is even? Or is it a 15-odd-limit ratio because 15 is the highest odd number before 16?

It's 9-odd-limit,

What of intervals in a 2nd octave? 13/11, a 13-odd-limit ratio, becomes 26/11 in the 2nd octave--is it then an 11-odd-limit ratio, or a 25-odd-limit ratio?

It's still 13.

Also, look at what happens between two 9-odd-limit ratios like 9/8 and 5/3: you have 40/27, a 27-odd-limit ratio that pops up in a scale of 5-odd-limit ratios. The odd-limit is just a terribly inconsistent way of looking at scales. Much worse than prime-limit, IMHO.

It's not inconsistent, it's just not what you seem to want or expect it to be. Tenney height could be what you are looking for.

On Wed, Jul 21, 2010 at 1:12 AM, cityoftheasleep
<igliashon@...> wrote:
>
> The fifth of 27 is sharp, not flat, at 711.111... cents. Sharp fifths push major and minor thirds further apart (until they merge with fourths and seconds, respectively, at 5-EDO), whereas flat fifths push major and minor thirds closer together (until they become one neutral third in 7-EDO). But regardless....

Sorry, I meant the fourth was flat. So that 3 of them makes an
approximate 7/3 instead of 12/5.

> Yeah, see, I have this hypothesis that I'm trying to find a way to test, and it's that the structure of a scale does more to define harmonic perception than the JI ratios approximated by the triads. A 7/6 might sound very different from a 6/5 when you put them side-by-side, but playing diatonic music using 6:7:9 triads doesn't necessarily "feel" different than if you use 10:12:15. I mean, it will of course sound somewhat different, but the ear doesn't seem to interpret the triads as having different "meanings"--a 7/6 can still *mean the same thing* as a 6/5 or a 13/11 or a 19/16 or a 23/19 if it's put in the same context. This is just one of the many reasons I think Marcel is mistaken--there's nothing essentially 5-limit about diatonic music, especially in 12-tET where a minor triad is basically a 16:19:24.

I think you're right on and I have come to the same conclusion. I just
finished reading Rothenberg's stuff and I think he's in agreement as
well.

I saw a study a while ago where brain activity in response to a chord
was observed via MRI. The finding was that at first, there was
activity in one part of the brain (possibly the temporal lobe) which
deals with general sound processing, and then a delayed response in a
completely different part of the brain (the cortex, was it?) that
deals with logic and that sort of thing. They hypothesized that the
first one was just a general auditory response to the incoming sound,
and the second was a learned response that deals with "tonality".

So perhaps, perceptually, the first response reflects the brain trying
to fuse 4:5:6 as a subset of 1:2:3:4:5:6:7.... and the second response
reflects one "imagining" the major triad as part of the major scale.

> So this begs the question: how do you find the limits to the range of possible tunings of triads that allows their musical meaning to be preserved? I'm actually starting to suspect that, at least to our 12-tET-saturated ear, the range of interval classes might actually be +/- 50 cents from 12-tET intervals. But then, I've heard an approximate 7/6 function as a pretty good whole-tone before, so maybe it's not so simple.

Rothenberg has a concept called the "equivalence class" that deals
with this, but I didn't quite understand all of it.

-Mike

>"Also, look at what happens between two 9-odd-limit ratios like 9/8 and 5/3:
>you have 40/27, a 27-odd-limit ratio that pops up in a scale of 5-odd-limit
>ratios. The odd-limit is just a terribly inconsistent way of looking at
>scales. Much worse than prime-limit, IMHO."

The way I've learned to look at scales' flexibility has a lot to do with a
way Carl mentioned to me ages ago. Which is to see how many dyads I can form at
as low of an odd limit as possible. And, supposedly, that's what leading
academics do when they create and submit JI scales. The part I don't agree with
is going beyond that and insisting triads and larger must be very low
limit...because not only does that strictness create much fewer chord
possibilities but, according to this thread, the brain often doesn't even
categorize the emotion of the "extra Just chords" any better.

The question IMVHO then becomes what happens when you are dealing with more
than dyads? How does the "second reaction in your brain" deal with triads and
larger chords and does how does it's ability to interpolate become greater (or
so I've heard based on past posts) based on how many intervals are involved in a
chord and how clustered the chord is (and not just how close the dyads are to
strict JI)?

What if you have consecutive dyads of 3/2 and 8/7 in a scale and they form a
triad where two of the three dyads possible in the triad are strictly just (and
varied other examples)... How can you determine which combinations the brain can
snap it into place "as if it were a 4:6:7 triad"?

I can only assume tons of listening tests of listeners trained both in common
practice theory and chords with neutral intervals (IE 12/11) coupled with MRI
tests could really pull this all together...but do any of the rest of you have
other ideas to figure this out?

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> The odd-limit has worse problems in some ways. Odd limit cannot
> be based on "odd factors" or else it would have the same problem
> as prime limit (getting arbitrarily close to the next odd limit
> using more complex ratios), and it would also be a clumsier way
> of saying basically the same thing that prime-limit says, since
> odd factors can reduce to prime factors. But then you have the
> question: is 16/9 a 9-odd-limit ratio, because 16 is even? Or
> is it a 15-odd-limit ratio because 15 is the highest odd number
> before 16?

Before declaring a problem with odd limits it would be good
to understand how they work. 16/9 is in the 9-limit, and is
a "ratio of 9", which means it's not in the 7-limit.

A ratio R, in lowest terms, is a "ratio of N" when N is the
larger of the numerator and denominator after all factors of
2 have been removed from them. R is then also in the N-limit.

> What of intervals in a 2nd octave? 13/11, a 13-odd-limit
> ratio, becomes 26/11 in the 2nd octave--is it then an
> 11-odd-limit ratio, or a 25-odd-limit ratio?

Odd limit is based on octave equivalence, and should only
be used where octave equivalence is desired.

> Also, look at what happens between two 9-odd-limit ratios
> like 9/8 and 5/3: you have 40/27, a 27-odd-limit ratio that
> pops up in a scale of 5-odd-limit ratios. The odd-limit is
> just a terribly inconsistent way of looking at scales.
> Much worse than prime-limit, IMHO.

Odd limit is not intended for use with scales. But in your
example, all three of those ratios are in the 5-prime-limit,
which is no better.

-Carl

I guess Yahoo doesn't put quotes in anymore or something, but
it makes these messages illegible. -C.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >But then you have the question: is 16/9 a 9-odd-limit ratio,
> because 16 is even? Or is it a 15-odd-limit ratio because 15 is
> the highest odd number before 16?
>
> It's 9-odd-limit,
>
> What of intervals in a 2nd octave? 13/11, a 13-odd-limit ratio,
> becomes 26/11 in the 2nd octave--is it then an 11-odd-limit
> ratio, or a 25-odd-limit ratio?
>
> It's still 13.
>
> Also, look at what happens between two 9-odd-limit ratios like
> 9/8 and 5/3: you have 40/27, a 27-odd-limit ratio that pops up in
> a scale of 5-odd-limit ratios. The odd-limit is just a terribly
> inconsistent way of looking at scales. Much worse than prime-
> limit, IMHO.
>
> It's not inconsistent, it's just not what you seem to want or
> expect it to be. Tenney height could be what you are looking for.
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I saw a study a while ago where brain activity in response to
> a chord was observed via MRI. The finding was that at first,
> there was activity in one part of the brain (possibly the
> temporal lobe) which deals with general sound processing, and
> then a delayed response in a completely different part of the
> brain (the cortex, was it?) that deals with logic and that sort
> of thing. They hypothesized that the first one was just a
> general auditory response to the incoming sound, and the
> second was a learned response that deals with "tonality".

The conclusions that can be drawn from fMRI studies are very
much more limited than those routinely drawn by investigators.
So be careful.

> Rothenberg has a concept called the "equivalence class" that
> deals with this, but I didn't quite understand all of it.

Try the tuning faq draft
http://lumma.org/music/theory/TuningFAQ.txt

Find for "rank-order" to get the relevant question.

-Carl

Carl>"A ratio R, in lowest terms, is a "ratio of N" when N is the
larger of the numerator and denominator after all factors of
2 have been removed from them. R is then also in the N-limit."
....
...."Odd limit is based on octave equivalence, and should only
be used where octave equivalence is desired."

So I am guessing if someone wanted to make a special version of the concept
of odd limit work for tri-taves they would remove factors of 3 instead of 2 from
the numerator and denominator?

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

> Rothenberg has a concept called the "equivalence class" that deals
> with this, but I didn't quite understand all of it.

That's simply math terminology:

http://en.wikipedia.org/wiki/Equivalence_class

We are all familiar with the idea in the case of pitch classes.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> So I am guessing if someone wanted to make a special version of the concept
> of odd limit work for tri-taves they would remove factors of 3 instead of 2 from
> the numerator and denominator?

That would be logical, but no one can actually make it work. It won't.

On 21 July 2010 17:08, cityoftheasleep <igliashon@...> wrote:

> Well, those triads weren't 16/11 but 40/27, actually between 40/27 and 22/15 at 675¢, with thirds at 300¢ and 375¢.  But regardless--12-tET may hit within 13-15 cents of 5-limit JI diatonic,
<snip>

12-tET intervals don't, in fact, get within 15 cents of the JI
diatonic. Yes, the worst 5-limit interval is the minor third, which
is only a tad more than 15 cents away from 6/5. But the diatonic will
also have intervals of 10/9 and a wolf fifth.

Graham

[This article is best read with Yahoo's "Use Fixed Width Font"
option.]

Hello, all.

While Ivor Darreg's term "xenharmonic" is very encompassing,
including for example Zalzalian or neutral intervals routine in
some world musical traditions but less familiar in others, I'll
take this survey to be seeking especially sonorities which from a
given person's perspective seem strange, wonderful, and
beautiful.

Completeness calls for me not to omit two types of sonorities --
or, more precisely, the forms they take in my part of the musical
universe -- recently discussed here: the Tristan chord, and the
square root of two (i.e. 600 cents).

A bit of context may help, since my styles often mixing or
juxtapositing medieval European and Near Eastern elements, with
fifths and fourths as the most complex stable sonorities but many
degrees of blend/tension relished in polyphony, may lead to a
radically different perspective than other styles ranging from
18th-19th century European tonality to jazz, and from traditional
Balinese or Javanese gamelan to the types of maqam-based or
dastgah-based polyphony practiced by musicians such as Ozan
Yarman or Shaahin Mohajeri, or the delectable textures of Naomi
O'Sullivan.

My longstanding love affair with 21:16 may be one place to
begin. Certainly 16:21:24:28 is for me impressively, spaciously,
energetically, and transcendently concordant. What I might call
the "guide-intervals" of the outer 7:4 minor seventh and upper
7:6 minor third typically resolve by stepwise contrary motion to
the fifth and unison respectively, although other people might
treat this sonority pleasingly in many other ways. The effect for
me is like an encompassing interstellar medium illuminated by the
vibrant energy of the 21:16.

Especially smooth is 12:14:16:18:21, a type of septimal slendro
and also, as George Secor has observed in the context of his
17-tone well-temperament (17-WT) where the 64:63 is tempered out
so that all fifths and fourths are close to 3:2 or 4:3, a fine
closing sonority in many modern styles. While five-voice
sonorities are a bit rich in many of my stylistic settings, I
find that in a JI or near-JI version keeping the 21:16 between
two upper voices, also, is very suave, and could well conclude a
piece in the types of styles he addresses. In a neomedieval
setting, I would typically resolve it much like 12:14:18:21, with
the outer seventh contracting to a fifth and the minor thirds to
unisons.

George Secor has more generally instilled in me an interest in
isoharmonic chords or sonorities, with identical or similar
differences between tones, with 11:16:21:26 as one of my early
"finds." Here the "guide-interval" is the outer minor tenth, at
the quotidian ratio (as Ozan might say) of 13:11-plus-octave which seeks resolution to an octave. On a 24-note keyboard, with
an asterisk showing a note on the upper 12-note manual raised by
an interval of about 55-60 cents, a typical resolution might be:

F* E
C#* B
A B
D* E

Here the lowest voice ascends by about a 12:11, while the highest
descends by about a 13:12, thus resolving from 26:11 to 2:1.
I realize that sooner or later I'll need to produce some mp3
examples.

Conjuring up for me the image of a Big Band conductor at large in
mostly unsuspected regions of the universe, George Secor has
introduced me to such delights as 8:9:11:13 and 7:9:11:13, the
latter having an especially compelling resolution with the outer
voices of the large Zalzalian or neutral seventh at 13:7
expanding to the octave as each moves by a semitone or thirdtone:

Eb* E
C* B
A B
E* E

Typically E*-E is a descent of about 55-60 cents, and Eb*-E an
ascent of around 70-80 cents, the two adding up to something like
the 128 cents of 13:7, varying a bit, of course, in tempered
systems.

An interesting idea I got from a discussion of an "Infinity scale
system" was a sonority of 11:9:8:6 which is nicely approximated
in a version of Maqam Penchgah that I use based on Mustaqim, a
maqam like Rast but favoring a lower neutral third step which Ibn
Sina places at 39:32 and in some systems I use is closer to 11:9.
In Penchgah, the 4/3 of Rast or Mustaqim is replaced with a
smallish "tritone" equal to the neutral third plus a tone at
typically around 9:8, Here I give just ratios, often slightly
tempered in practice, with the 11:6 neutral seventh and 11:9
neutral third as "guide-intervals" resolving to fifth and unison:

11/6 D* 12:13 C# 22/13
11/8 A* 16:13 C# 22/13
11/9 G* 12:13 F# 44/39
1/1 E 44:39 F# 44/39

While the added 11:8 might be a bit rich for the style of
polyphony I often seek in a maqam or dastgan setting, where the
counterpoint should support and adorn rather than overwhelm the
_sayr_ or melodic "path" or procedure, I do find the effect
striking. Note how the classic medieval steps of 9/8 and 27/16
are often altered in the modern tuning systems I use to 44/39 and
22/13 or the like.

The 11:8 is indeed, as Gene has said, a xenharmonic interval par
excellence, and it may occur routinely in maqam-based polyphony
or take on various more "unusual" effects. The 8:11:13 sonority
has an intriguing quality, often inviting for me a resolution by
way of another sonority introduced to me by George Secor, a
suspended 16:21:24.

13/8 F* 3/2 E 14/9 E*
11/8 D* 21/16 C#* 7/6 B* 28/27 A*
1/1 A 28/27 A*

In my kind of typical stylistic context -- quite unlike in jazz
based on some 12-note circulating concept -- a "tritone
substitution" simply means a sonority where some kind of
"tritonic" interval is used in place of a regular fifth or
fourth. Here is one beautiful example using notes from a usual
set for a septimal version of Dastgah-e Shur:

7/4 F#* 13:14 F* 13/8
13/9 Eb* 9:8 F* 13/8
7/6 B* 13:14 Bb* 13/12
1/1 A 13:12 Bb* 13/12

Generally I would consider this four-voice version with 13/9,
present in Shur as a _moteqayyer_ or variable step acting as a
lowered version of the fifth often favored in descending passages
and in certain of the gusheh-s or modal themes that make up the
Shur family, as a bit ostentatious for a usual dastgah-based
style, by comparison to a more routine 1/1-7/6-7/4 or 12:14:21. However, when colorful xenharmonicism is the goal, this version
with the 13/9, ~36:42:52:63, is certainly ear-catching!

Now for the Tristan chord, or rather the kind of sonority I
obtain with the same spelling of Eb-A-C#-F#. Here I find that
attractive "guide-intervals" are the augmented sixth Eb-C#, a
small neutral seventh at around 1038 cents or 51/28 in my current
temperament, and the augmented ninth Eb-F#, likewise a small
neutral tenth at around 1534 cents, an octave plus something like
63/52, 40/33, or 17/14. As usual, the seventh tends to contract
to a fifth, with the tenth accordingly contracting to an octave:

1534 F# -207 E 1327
1038 C# -207 B 831
623 A +207 B 831
0 Eb +127 E 127

In this treatment, the regular augmented fourth or tritone A at
623 cents, or somewhere between 63/44 and 56/39, adds color and
tension while the small neutral seventh and tenth guide the
progression, much as they would do in a familiar context like
the following, shown in the close voicing I most often follow,
although a wider spacing using a neutral tenth, I now realize,
offers a pleasant variation:

1038 C# -207 B 831
704 Bb +127 B 831
334 F# -207 B 127
0 Eb +127 E 127

In this kind of stylistic and intonational environment where the
"Tristan chord" -- or its parallel universe counterpart! --
brings into play small neutral sevenths and tenths and resolving
steps around 14:13, the explanation of this colorful sonority
may be rather less complicated than in the theoretical milieus
where it has brought forth such an effusion of ink over the past
century and more. However, one point does bear mention.

If the voices are transposed or inverted to produce the form
A-C#-Eb-F#, then we have a different resolution that stands out,
one guided by the intervals of the regular major third A-C# and
major sixth A-F# expanding respectively to the fifth and
octave. with the tritone (or here more precisely a diminished
fifth A-Eb at around 39/28 or 88/63) again as an element of added
tension or color:

911 F# +81 G +992 (1200)
577 Eb -81 D +496 (704)
415 C# +81 D +496 (704)
0 A -208 G -208 (0)

Here the three-voice progressive of A-C#-F# to G-D-G is a
14th-century European classic here realized in a modern
temperament not too far from Pythagorean for the regular diatonic
intervals (e.g. something around 33/26 or 14/11 for 81/64; 22/13
for 27/16; and 22/21 for the classic limma at 256/243). with the
tritone as a coloristic addition. This addition results in a
curious quirk: a diminished third of 162 cents, fairly close to
11:10, between the middle voices, resolves to a unison with each
voice moving by a usual near-22:21 semitone!

This maybe not-so-routine resolution of a vertical diminished
third, a touch worthy of the exotic expectations that might be
aroused by an inversion of the "Tristan chord" (albeit hardly in
a tuning intended by Wagner, whatever the parameters of that
category might be), could be taken as a variation on the familiar
idiom where an outer augmented sixth expands to an octave, with
each voice moving by a regular semitone or limma:

1038 C# +81 D +1119 (1200)
415 G +208 A +623 (704)
0 Eb -81 D -81 (0)

Thus the "Tristan chord" and some of its associations as tuned
in my neck of the xenharmonic woods.

This leaves us for the moment with one vital xenharmonic theme:
the square root of two, of course, or 600 cents. While this
interval is a very bulwark of some harmonic structures and styles
including Paul Erlich's decatonic, and notably greeted with less
than full enthusiasm in certain quarters, I find it simply one
possible size for an augmented fourth or diminished fifth, and in
one of my favorite tuning systems a rare interval thus worthy to
be cherished in a very special use.

In this system the square root of 2 wasn't a planned feature,
just a happy result of "nanotempering" on a device using binary
millioctaves (1024-EDO). While "nanotemperament" may have
different meanings for different people, for example a nuance so
small as to be aurally undetectible and thus inconsequential, for
me it is mainly a synonym for the pastime of reflectively
performing the equivalent of Scala's QUANTIZE 1024.

Anyway, the xenoharmonic sonority leading to this story is
12:14:17:21 or 1/1-7/6-17/12-7/4 (0-267-603-969 cents in JI), featuring the "rare" 17:12 tritone, only approximated at one
location in this temperament, Eb-G#*. This could be viewed as
another sonority where a tritone is substtituted for a usual
fifth or fourth, here 17/12 in place of the 3/2 in 12:14:18:21.

An inspiration was Helmholtz with his diminished seventh chord at
10:12:14:17, whose use of the 7:6 septimal minor third and 17:14
"supraminor" third suggested to me 12:14:17:21, a variation
combining two beloved sonorities: 12:14:21 (a common three-voice
relative of the complete 12:14:18:21); and 14:17:21.

As it happened, in my chosen nanotemperament Eb-G#* came to 512
binary millioctaves -- or, in other words, precisely 600 cents!
And the result sounds just fine. with the minor complication on
paper (or on screen) that steps at 57.4 and 761.7 cents round to
57 and 762 cents, but form a fifth that rounds to 704 rather than
705 cents:

969 C* -207 Bb* 762 (704)
600 G#* +162 Bb* 762 (704)
265 F* -207 Eb* 57 (0)
0 Eb +57 Eb* 57 (0)

Here the guide-intervals are the minor third and seventh above
the lowest voice at 265 and 969 cents, a slightly narrow 7/6 and
virtually just 7/4, which contract to the unison and fifth, with
the lowest voice ascending by a 57-cent step, the notably narrow
tempered equivalent of 28:27 at 63 cents, and actually much
closer to 91:88 at 58 cents. The upper three voices in themselves
would form one of the standard resolutions from a tempered
14:17:21 (here 0-335-704 cents) to a stable fifth, with the
middle voice of this trio ascending G#*-Bb* or 162 cents, a
virtually just 56:51.

At any rate, the 600-cent tempering of 17:12 sounds just fine,
and maybe this escapade of tempering out the 289:288, the
difference between 17:12 at 603 cents and 24:17 at 597 cents, may
recall some of Paul Erlich's writings about the 50:49.

With many thanks,

Margo Schulter
mschulter@...

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 21 July 2010 17:08, cityoftheasleep <igliashon@...> wrote:
>
> > Well, those triads weren't 16/11 but 40/27, actually between 40/27 and 22/15 at 675¢, with thirds at 300¢ and 375¢.  But regardless--12-tET may hit within 13-15 cents of 5-limit JI diatonic,
> <snip>
>
> 12-tET intervals don't, in fact, get within 15 cents of the JI
> diatonic. Yes, the worst 5-limit interval is the minor third, which
> is only a tad more than 15 cents away from 6/5. But the diatonic will
> also have intervals of 10/9 and a wolf fifth.
>
>
> Graham
>

Ah yes, point taken, Graham. Such is the peril of looking at ET's as approximations to JI scales. In one modal rotation, the approximation may seem decent, but rotate to another mode, and the approximation seems poor. If we look at the Dorian (ii) mode of the JI major scale 1/1-9/8-5/4-4/3-3/2-5/3-15/8-2/1, we have 1/1-10/9-32/27-4/3-40/27-5/3-16/9-2/1, two intervals of which are quite a ways from 12-tET. Quite a different scale from a JI Dorian scale, which would probably be 1/1-9/8-6/5-4/3-3/2-5/3-9/5-2/1. I guess that's the upshot of tempering: two JI ratios collapse into one tempered interval, so that interval's error really should take into account both of the collapsed intervals. So the 12-tET major scale isn't just approximating the 5-limit JI major scale, it's also approximating the Ionian mode of other JI modes. I never thought about this before. Interesting. So if you look at temperaments not just according to harmonics but as two JI scales merged by tempering out a comma, there's sort of a continuum where the approximations will lean closer to one scale than the other. For instance, while 12-tET hits closer to 9/8, 32/27, 81/64, 4/3, 3/2, 27/16, and, 243/128, 19-tET is (variably) closer to 10/9, 6/5, 5/4, 27/20, 40/27, 5/3, and 15/8...this is giving me something to think about.

-Igs

Igs> 12-tET intervals don't, in fact, get within 15 cents of the JI
> diatonic. Yes, the worst 5-limit interval is the minor third, which
> is only a tad more than 15 cents away from 6/5. But the diatonic will
> also have intervals of 10/9 and a wolf fifth.
Right, and the wolf fifth is an IMVHO terrible sounding 40/27 (13/9, 22/15,
50/33, 14/9...IMVHO all sound more relaxed to me despite also being
wolf/non-pure 5th-ish tones). Not to mention, as you stated, the minor third is
at about 13/11...quite far away from both 6/5 and 7/6. So I'd say that's two
very sour dyads and one slightly sour one (the 10/9).

I don't see why JI diatonic often gets credit for the "home field advantage"
so far as sounding resolved and can easily understand how many musicians who
have heard JI diatonic still lean toward things like 12TET for "feeling more
balanced". Especially considering the lack of balance in diatonic JI (IE almost
everything is tremendously pure and then you have a fifth and minor third that
are just about as sour as they can be).

The way I see it, if you're willing to settle for a wolf fifth, there are so
many other options than JI that get many more relaxed-sounding intervals as a
result of "compromising" that fifth and often at least have "wolf" fifths that
aren't nearly as disproportionately sour as diatonic JI's 40/27.
And if you give up two fifths or so ALA Ptolemy's scales I swear you get tons
of very sweet sounding not to mention fresh and exotic intervals.

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

Thank you, Margo. Much to digest in here!