What other temperaments support these 12-tet scales?

I was messing around with the following 12-tet pajara-ish scale:

C C# D# E F# G# A B C

You can think of it as C lydian aug #2, but with a #1 added in as well. It
really makes more sense if you think of it as not being an altered diatonic
scale. Or, you can think of it as the following two scales combined, by
equivocating enharmonic notes:

C Db Eb Fb Gb Ab A C
C D# E F# G# A B C

Turns out this scale can produce a bunch of really interesting chord
progressions, and ones I'd never heard before, like F#m7/C -> G#m7/C. I'll
get some listening examples up when I have a second. It also turns out to
have a lot of really interesting modes, like this one

C C# D E F G A Bb C

In which C-D-Bb-C# (or C-D-A-Bb-C#) is an especially interesting sonority,
possibly because it approximates 8:9:7:17.

Turns out one of the modes of this scale is the relatively less exciting
"major bebop" scale (C D E F G G# A B C), which is usually just treated as a
diatonic scale with an "unofficial extra note" that people like to play as a
passing tone. I've never heard of it being dissected modally, or used to
form harmonies, though. Are there any interesting rank-2 temperaments (or
other ET's) that support this scale, just like all of Paul's pajara scales
are supported in 22-equal just as are they in 12-equal? I thought it was
pajara, but it doesn't look like it works in 22-et. The fact that it seems
to be taking advantage of 128/125 and 81/80 as unison vectors is pretty
depressing in that perhaps it's confined to 12, but maybe there's some way
of seeing it that I'm missing (perhaps in the 7-limit)...

PS: this scale is, in 12-et, Rothenberg proper but not strictly proper.

-Mike

The essential shape of it is Magen Abot. I've got a lot to comment on this, but don't have time... maybe I'll get a chance to-nite.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I was messing around with the following 12-tet pajara-ish scale:
>
> C C# D# E F# G# A B C
>
> You can think of it as C lydian aug #2, but with a #1 added in as well. It
> really makes more sense if you think of it as not being an altered diatonic
> scale. Or, you can think of it as the following two scales combined, by
> equivocating enharmonic notes:
>
> C Db Eb Fb Gb Ab A C
> C D# E F# G# A B C
>
> Turns out this scale can produce a bunch of really interesting chord
> progressions, and ones I'd never heard before, like F#m7/C -> G#m7/C. I'll
> get some listening examples up when I have a second. It also turns out to
> have a lot of really interesting modes, like this one
>
> C C# D E F G A Bb C
>
> In which C-D-Bb-C# (or C-D-A-Bb-C#) is an especially interesting sonority,
> possibly because it approximates 8:9:7:17.
>
> Turns out one of the modes of this scale is the relatively less exciting
> "major bebop" scale (C D E F G G# A B C), which is usually just treated as a
> diatonic scale with an "unofficial extra note" that people like to play as a
> passing tone. I've never heard of it being dissected modally, or used to
> form harmonies, though. Are there any interesting rank-2 temperaments (or
> other ET's) that support this scale, just like all of Paul's pajara scales
> are supported in 22-equal just as are they in 12-equal? I thought it was
> pajara, but it doesn't look like it works in 22-et. The fact that it seems
> to be taking advantage of 128/125 and 81/80 as unison vectors is pretty
> depressing in that perhaps it's confined to 12, but maybe there's some way
> of seeing it that I'm missing (perhaps in the 7-limit)...
>
> PS: this scale is, in 12-et, Rothenberg proper but not strictly proper.
>
> -Mike
>

Wow that scale has a lot of diminished triads available (around 5): much
more so than major triads (I counted about 2) and minor (I counted about 3).
Interesting scale...seems to lend itself toward odd chords (diminished suspended
add2 etc.) much more than diatonic.

Far as your chord the (C-D-A-Bb-C#)...for some bizarre reason it sounds to me
like a C D A C# with an extra note thrown in...but C A# C# sounds OK to me by
itself. I wonder why that is....

The chord appears to map as

1/1 C
9/8 D
5/3 A
16/9 A#
17/8 C#

I noticed
A) In C A# C# the interval between A# and C# is about 6/5. Oddly enough
(other than that dyad) that chord seems a mix of an x/9 and x/8 dyad. I wonder
why this one works so well considering there seems to be no obvious root
implied...maybe the fact it follows an Ls pattern (ALA major chords) and the low
critical band dissonance between C5 and A#5 plus C5 and C#6 has something to do
with it.

B) In C D A C# the only interval not nearest round-able to an "x/8"
fraction is 5/3....the C# over A is a pretty bad 5/4 (think over 13 cents
off)...but for some odd reason the brain seems to forgive it. It seems to me
the tendency toward x/8 fractions probably makes it work so well (IE it's pretty
close to a straight x/8 harmonic series).

MOS-wise, it looks like it's just a diminished scale with two of the steps flipped: sLsLLsLs. It's like Paul's "pentachordal" scales, which are not strictly Pajara (they're not MOS) but are derived from it. If you're cool with enharmonics being equivalent, you can play this scale in any EDO divisible by 4...so 16, 20, 24, 28, 32, 36, 40, 44, etc. However, I'm not sure any of these will afford much harmonic advantage. 28-EDO gives basically a pure 5/4, but you also get stuck with the 7-EDO fifth of around 685 cents.

Hope that helps.

-Igs

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> The essential shape of it is Magen Abot. I've got a lot to comment on this, but don't have time... maybe I'll get a chance to-nite.
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > I was messing around with the following 12-tet pajara-ish scale:
> >
> > C C# D# E F# G# A B C
> >
> > You can think of it as C lydian aug #2, but with a #1 added in as well. It
> > really makes more sense if you think of it as not being an altered diatonic
> > scale. Or, you can think of it as the following two scales combined, by
> > equivocating enharmonic notes:
> >
> > C Db Eb Fb Gb Ab A C
> > C D# E F# G# A B C
> >
> > Turns out this scale can produce a bunch of really interesting chord
> > progressions, and ones I'd never heard before, like F#m7/C -> G#m7/C. I'll
> > get some listening examples up when I have a second. It also turns out to
> > have a lot of really interesting modes, like this one
> >
> > C C# D E F G A Bb C
> >
> > In which C-D-Bb-C# (or C-D-A-Bb-C#) is an especially interesting sonority,
> > possibly because it approximates 8:9:7:17.
> >
> > Turns out one of the modes of this scale is the relatively less exciting
> > "major bebop" scale (C D E F G G# A B C), which is usually just treated as a
> > diatonic scale with an "unofficial extra note" that people like to play as a
> > passing tone. I've never heard of it being dissected modally, or used to
> > form harmonies, though. Are there any interesting rank-2 temperaments (or
> > other ET's) that support this scale, just like all of Paul's pajara scales
> > are supported in 22-equal just as are they in 12-equal? I thought it was
> > pajara, but it doesn't look like it works in 22-et. The fact that it seems
> > to be taking advantage of 128/125 and 81/80 as unison vectors is pretty
> > depressing in that perhaps it's confined to 12, but maybe there's some way
> > of seeing it that I'm missing (perhaps in the 7-limit)...
> >
> > PS: this scale is, in 12-et, Rothenberg proper but not strictly proper.
> >
> > -Mike
> >
>

On Fri, Aug 27, 2010 at 11:13 AM, cityoftheasleep
<igliashon@...> wrote:
>
> MOS-wise, it looks like it's just a diminished scale with two of the steps flipped: sLsLLsLs. It's like Paul's "pentachordal" scales, which are not strictly Pajara (they're not MOS) but are derived from it. If you're cool with enharmonics being equivalent, you can play this scale in any EDO divisible by 4...so 16, 20, 24, 28, 32, 36, 40, 44, etc. However, I'm not sure any of these will afford much harmonic advantage. 28-EDO gives basically a pure 5/4, but you also get stuck with the 7-EDO fifth of around 685 cents.

Yeah, that's one of the questions that this has all brought up for me:
how do you determine whether a scale "comes from a temperament" or
not? Is the harmonic minor or melodic minor scale not from meantone,
because they're not MOS?

But as to your suggestion - what if we ended up in a temperament where
one or more of those minor thirds became a 7/6 instead of a 6/5? Then
in 12-tet they would "collapse" 4-tet to a diminished chord, because
36/35 is eliminated.

Also note that one of the modes is the bebop scale, C D E F G G# A B
C, which is a major scale with G# added, although it's a lot more
interesting if you use it to produce chords rather than just as a
major scale with this one chromatic tendency. But if you look at it
that way, it's also the composite of these two scales:

C D E F G Ab B C (harmonic major)
C D E F G# A B C (Ionian augmented)

Equating the two implies that G# = Ab, which implies 128/125. 128/125
and 81/80 being eliminated means this only works in 12-equal (and also
leads to 648/625 vanishing, as per your ideas). So perhaps if we allow
81/80 to not vanish, we'd get some kind of augmented temperament, and
combining this with some 7-limit commas that 12-tet supports would
lead to the (or one of the) rank 1 temperaments this supports.

I really wish I was comfortable enough with the math to work this all out.

-Mike

Also, keeping with your original train of thought, if we're going to
say that this is a type of diminished temperament, then that implies
648/625 vanishing. 648/625 = 128/125 * 81/80, so that would imply that
128/125 * 81/80 ~ 1/1. This means that either one of those intervals
is going to have to be reversed.

Reversing 128/125 would lead to a situation where 3 major thirds
exceeds the octave rather than falling short of it, which might
naturally lend to equating them with 14/11 or 9/7. Reversing 81/80
would lead to a situation where 4 perfect fifths would be LESS than a
major third (hence 28-et).

12-equal is the case where none of these intervals are reversed and
instead both reduced to 0 cents.

-Mike

On Fri, Aug 27, 2010 at 4:12 PM, Mike Battaglia <battaglia01@...> wrote:
> On Fri, Aug 27, 2010 at 11:13 AM, cityoftheasleep
> <igliashon@...> wrote:
>>
>> MOS-wise, it looks like it's just a diminished scale with two of the steps flipped: sLsLLsLs. It's like Paul's "pentachordal" scales, which are not strictly Pajara (they're not MOS) but are derived from it. If you're cool with enharmonics being equivalent, you can play this scale in any EDO divisible by 4...so 16, 20, 24, 28, 32, 36, 40, 44, etc. However, I'm not sure any of these will afford much harmonic advantage. 28-EDO gives basically a pure 5/4, but you also get stuck with the 7-EDO fifth of around 685 cents.
>
> Yeah, that's one of the questions that this has all brought up for me:
> how do you determine whether a scale "comes from a temperament" or
> not? Is the harmonic minor or melodic minor scale not from meantone,
> because they're not MOS?
>
> But as to your suggestion - what if we ended up in a temperament where
> one or more of those minor thirds became a 7/6 instead of a 6/5? Then
> in 12-tet they would "collapse" 4-tet to a diminished chord, because
> 36/35 is eliminated.
>
> Also note that one of the modes is the bebop scale, C D E F G G# A B
> C, which is a major scale with G# added, although it's a lot more
> interesting if you use it to produce chords rather than just as a
> major scale with this one chromatic tendency. But if you look at it
> that way, it's also the composite of these two scales:
>
> C D E F G Ab B C (harmonic major)
> C D E F G# A B C (Ionian augmented)
>
> Equating the two implies that G# = Ab, which implies 128/125. 128/125
> and 81/80 being eliminated means this only works in 12-equal (and also
> leads to 648/625 vanishing, as per your ideas). So perhaps if we allow
> 81/80 to not vanish, we'd get some kind of augmented temperament, and
> combining this with some 7-limit commas that 12-tet supports would
> lead to the (or one of the) rank 1 temperaments this supports.
>
> I really wish I was comfortable enough with the math to work this all out.
>
> -Mike

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

> Yeah, that's one of the questions that this has all brought up for me:
> how do you determine whether a scale "comes from a temperament" or
> not? Is the harmonic minor or melodic minor scale not from meantone,
> because they're not MOS?

That was a question I was actually faced with when I was classifying the 7-note strictly proper scales of 31et. 31 has so many linear temperaments it supports--each step size from 1 to 15 results in a new temperament if used as a generator. So where, if anywhere, should I stick one of these scales in terms of such a temperament? What I did was to find the span of the scale in terms of each generator. Often one was clearly less than the rest, and so classifying it as belonging to a particular temperament was easy. Sometimes it was closer, but going with the smallest seemed to make the most sense.

> Also note that one of the modes is the bebop scale, C D E F G G# A B
> C, which is a major scale with G# added, although it's a lot more
> interesting if you use it to produce chords rather than just as a
> major scale with this one chromatic tendency. But if you look at it
> that way, it's also the composite of these two scales:
>
> C D E F G Ab B C (harmonic major)
> C D E F G# A B C (Ionian augmented)
>
> Equating the two implies that G# = Ab, which implies 128/125. 128/125
> and 81/80 being eliminated means this only works in 12-equal (and also
> leads to 648/625 vanishing, as per your ideas).

I think analyzing scales makes a lot more sense in 31 than in 12. 12 leaves you scratching your head like this all too often.

> I really wish I was comfortable enough with the math to work this all out.

I'm not clear on what you want to work out.

Personally I'd be concerned with first choosing a Just ideal for the scale, then worrying about the commas, temperament and practical tuning. This scale is very, very nice in 41-edo for example. I got this by starting with the F# as 7/5, and keeping the proportions of s,L, roughly so rather than strictly MOS (roughly s,L, as they'd almost certainly be in a historical Magen Abot mode).

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Also, keeping with your original train of thought, if we're going to
> say that this is a type of diminished temperament, then that implies
> 648/625 vanishing. 648/625 = 128/125 * 81/80, so that would imply that
> 128/125 * 81/80 ~ 1/1. This means that either one of those intervals
> is going to have to be reversed.
>
> Reversing 128/125 would lead to a situation where 3 major thirds
> exceeds the octave rather than falling short of it, which might
> naturally lend to equating them with 14/11 or 9/7. Reversing 81/80
> would lead to a situation where 4 perfect fifths would be LESS than a
> major third (hence 28-et).
>
> 12-equal is the case where none of these intervals are reversed and
> instead both reduced to 0 cents.
>
> -Mike
>
> On Fri, Aug 27, 2010 at 4:12 PM, Mike Battaglia <battaglia01@...> wrote:
> > On Fri, Aug 27, 2010 at 11:13 AM, cityoftheasleep
> > <igliashon@...> wrote:
> >>
> >> MOS-wise, it looks like it's just a diminished scale with two of the steps flipped: sLsLLsLs. It's like Paul's "pentachordal" scales, which are not strictly Pajara (they're not MOS) but are derived from it. If you're cool with enharmonics being equivalent, you can play this scale in any EDO divisible by 4...so 16, 20, 24, 28, 32, 36, 40, 44, etc. However, I'm not sure any of these will afford much harmonic advantage. 28-EDO gives basically a pure 5/4, but you also get stuck with the 7-EDO fifth of around 685 cents.
> >
> > Yeah, that's one of the questions that this has all brought up for me:
> > how do you determine whether a scale "comes from a temperament" or
> > not? Is the harmonic minor or melodic minor scale not from meantone,
> > because they're not MOS?
> >
> > But as to your suggestion - what if we ended up in a temperament where
> > one or more of those minor thirds became a 7/6 instead of a 6/5? Then
> > in 12-tet they would "collapse" 4-tet to a diminished chord, because
> > 36/35 is eliminated.
> >
> > Also note that one of the modes is the bebop scale, C D E F G G# A B
> > C, which is a major scale with G# added, although it's a lot more
> > interesting if you use it to produce chords rather than just as a
> > major scale with this one chromatic tendency. But if you look at it
> > that way, it's also the composite of these two scales:
> >
> > C D E F G Ab B C (harmonic major)
> > C D E F G# A B C (Ionian augmented)
> >
> > Equating the two implies that G# = Ab, which implies 128/125. 128/125
> > and 81/80 being eliminated means this only works in 12-equal (and also
> > leads to 648/625 vanishing, as per your ideas). So perhaps if we allow
> > 81/80 to not vanish, we'd get some kind of augmented temperament, and
> > combining this with some 7-limit commas that 12-tet supports would
> > lead to the (or one of the) rank 1 temperaments this supports.
> >
> > I really wish I was comfortable enough with the math to work this all out.
> >
> > -Mike
>

Mike, I should add that an obvious concern with the scale is that without a specific modality, pedalling the C as you do in the x/C chord example you gave, in any tuning with more or less Just fifths, the thing is going to tend to drive, by virtue of historical habits vis a vis fifths as well as spectral tendencies, toward the C# as tonic. So, one approach to a Just ideal would be weight the spectrum toward C (17/16, 19/16, 20/16, 22/16... would do it). But I think the tension of keeping it floating over C is groovier, so I'd got for 7/5 for F#.

Notice that you've got the Viennese trichord on the tonic, LOL. 0,1,6.

-Cameron

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Personally I'd be concerned with first choosing a Just ideal for the scale, then worrying about the commas, temperament and practical tuning. This scale is very, very nice in 41-edo for example. I got this by starting with the F# as 7/5, and keeping the proportions of s,L, roughly so rather than strictly MOS (roughly s,L, as they'd almost certainly be in a historical Magen Abot mode).
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > Also, keeping with your original train of thought, if we're going to
> > say that this is a type of diminished temperament, then that implies
> > 648/625 vanishing. 648/625 = 128/125 * 81/80, so that would imply that
> > 128/125 * 81/80 ~ 1/1. This means that either one of those intervals
> > is going to have to be reversed.
> >
> > Reversing 128/125 would lead to a situation where 3 major thirds
> > exceeds the octave rather than falling short of it, which might
> > naturally lend to equating them with 14/11 or 9/7. Reversing 81/80
> > would lead to a situation where 4 perfect fifths would be LESS than a
> > major third (hence 28-et).
> >
> > 12-equal is the case where none of these intervals are reversed and
> > instead both reduced to 0 cents.
> >
> > -Mike
> >
> > On Fri, Aug 27, 2010 at 4:12 PM, Mike Battaglia <battaglia01@> wrote:
> > > On Fri, Aug 27, 2010 at 11:13 AM, cityoftheasleep
> > > <igliashon@> wrote:
> > >>
> > >> MOS-wise, it looks like it's just a diminished scale with two of the steps flipped: sLsLLsLs. It's like Paul's "pentachordal" scales, which are not strictly Pajara (they're not MOS) but are derived from it. If you're cool with enharmonics being equivalent, you can play this scale in any EDO divisible by 4...so 16, 20, 24, 28, 32, 36, 40, 44, etc. However, I'm not sure any of these will afford much harmonic advantage. 28-EDO gives basically a pure 5/4, but you also get stuck with the 7-EDO fifth of around 685 cents.
> > >
> > > Yeah, that's one of the questions that this has all brought up for me:
> > > how do you determine whether a scale "comes from a temperament" or
> > > not? Is the harmonic minor or melodic minor scale not from meantone,
> > > because they're not MOS?
> > >
> > > But as to your suggestion - what if we ended up in a temperament where
> > > one or more of those minor thirds became a 7/6 instead of a 6/5? Then
> > > in 12-tet they would "collapse" 4-tet to a diminished chord, because
> > > 36/35 is eliminated.
> > >
> > > Also note that one of the modes is the bebop scale, C D E F G G# A B
> > > C, which is a major scale with G# added, although it's a lot more
> > > interesting if you use it to produce chords rather than just as a
> > > major scale with this one chromatic tendency. But if you look at it
> > > that way, it's also the composite of these two scales:
> > >
> > > C D E F G Ab B C (harmonic major)
> > > C D E F G# A B C (Ionian augmented)
> > >
> > > Equating the two implies that G# = Ab, which implies 128/125. 128/125
> > > and 81/80 being eliminated means this only works in 12-equal (and also
> > > leads to 648/625 vanishing, as per your ideas). So perhaps if we allow
> > > 81/80 to not vanish, we'd get some kind of augmented temperament, and
> > > combining this with some 7-limit commas that 12-tet supports would
> > > lead to the (or one of the) rank 1 temperaments this supports.
> > >
> > > I really wish I was comfortable enough with the math to work this all out.
> > >
> > > -Mike
> >
>

And, maintaining near-perfect 7-limit Just interpretation of this scale with full transposability and circulation, the comma to temper out is simply 225/224, the difference between (two pure M3+m3) and (septimal tritone+pure fourth), also the difference between (septimal subminor third + pure fourth) and (two pure M3), etc.

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Mike, I should add that an obvious concern with the scale is that without a specific modality, pedalling the C as you do in the x/C chord example you gave, in any tuning with more or less Just fifths, the thing is going to tend to drive, by virtue of historical habits vis a vis fifths as well as spectral tendencies, toward the C# as tonic. So, one approach to a Just ideal would be weight the spectrum toward C (17/16, 19/16, 20/16, 22/16... would do it). But I think the tension of keeping it floating over C is groovier, so I'd got for 7/5 for F#.
>
> Notice that you've got the Viennese trichord on the tonic, LOL. 0,1,6.
>
> -Cameron
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > Personally I'd be concerned with first choosing a Just ideal for the scale, then worrying about the commas, temperament and practical tuning. This scale is very, very nice in 41-edo for example. I got this by starting with the F# as 7/5, and keeping the proportions of s,L, roughly so rather than strictly MOS (roughly s,L, as they'd almost certainly be in a historical Magen Abot mode).
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > Also, keeping with your original train of thought, if we're going to
> > > say that this is a type of diminished temperament, then that implies
> > > 648/625 vanishing. 648/625 = 128/125 * 81/80, so that would imply that
> > > 128/125 * 81/80 ~ 1/1. This means that either one of those intervals
> > > is going to have to be reversed.
> > >
> > > Reversing 128/125 would lead to a situation where 3 major thirds
> > > exceeds the octave rather than falling short of it, which might
> > > naturally lend to equating them with 14/11 or 9/7. Reversing 81/80
> > > would lead to a situation where 4 perfect fifths would be LESS than a
> > > major third (hence 28-et).
> > >
> > > 12-equal is the case where none of these intervals are reversed and
> > > instead both reduced to 0 cents.
> > >
> > > -Mike
> > >
> > > On Fri, Aug 27, 2010 at 4:12 PM, Mike Battaglia <battaglia01@> wrote:
> > > > On Fri, Aug 27, 2010 at 11:13 AM, cityoftheasleep
> > > > <igliashon@> wrote:
> > > >>
> > > >> MOS-wise, it looks like it's just a diminished scale with two of the steps flipped: sLsLLsLs. It's like Paul's "pentachordal" scales, which are not strictly Pajara (they're not MOS) but are derived from it. If you're cool with enharmonics being equivalent, you can play this scale in any EDO divisible by 4...so 16, 20, 24, 28, 32, 36, 40, 44, etc. However, I'm not sure any of these will afford much harmonic advantage. 28-EDO gives basically a pure 5/4, but you also get stuck with the 7-EDO fifth of around 685 cents.
> > > >
> > > > Yeah, that's one of the questions that this has all brought up for me:
> > > > how do you determine whether a scale "comes from a temperament" or
> > > > not? Is the harmonic minor or melodic minor scale not from meantone,
> > > > because they're not MOS?
> > > >
> > > > But as to your suggestion - what if we ended up in a temperament where
> > > > one or more of those minor thirds became a 7/6 instead of a 6/5? Then
> > > > in 12-tet they would "collapse" 4-tet to a diminished chord, because
> > > > 36/35 is eliminated.
> > > >
> > > > Also note that one of the modes is the bebop scale, C D E F G G# A B
> > > > C, which is a major scale with G# added, although it's a lot more
> > > > interesting if you use it to produce chords rather than just as a
> > > > major scale with this one chromatic tendency. But if you look at it
> > > > that way, it's also the composite of these two scales:
> > > >
> > > > C D E F G Ab B C (harmonic major)
> > > > C D E F G# A B C (Ionian augmented)
> > > >
> > > > Equating the two implies that G# = Ab, which implies 128/125. 128/125
> > > > and 81/80 being eliminated means this only works in 12-equal (and also
> > > > leads to 648/625 vanishing, as per your ideas). So perhaps if we allow
> > > > 81/80 to not vanish, we'd get some kind of augmented temperament, and
> > > > combining this with some 7-limit commas that 12-tet supports would
> > > > lead to the (or one of the) rank 1 temperaments this supports.
> > > >
> > > > I really wish I was comfortable enough with the math to work this all out.
> > > >
> > > > -Mike
> > >
> >
>

On Fri, Aug 27, 2010 at 4:39 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Yeah, that's one of the questions that this has all brought up for me:
> > how do you determine whether a scale "comes from a temperament" or
> > not? Is the harmonic minor or melodic minor scale not from meantone,
> > because they're not MOS?
>
> That was a question I was actually faced with when I was classifying the 7-note strictly proper scales of 31et. 31 has so many linear temperaments it supports--each step size from 1 to 15 results in a new temperament if used as a generator. So where, if anywhere, should I stick one of these scales in terms of such a temperament? What I did was to find the span of the scale in terms of each generator. Often one was clearly less than the rest, and so classifying it as belonging to a particular temperament was easy. Sometimes it was closer, but going with the smallest seemed to make the most sense.

That makes sense. I think part of it is also, though, that a
particular scale might have more than one JI rendition, and hence more
than one way to temper it. LLsLLLs scales can be meantone, where the
fifths are tempered narrow and the major thirds 5/4, or they can be
superpyth (or is it dominant?), where the fifths are tempered sharp
and the major thirds 9/7. Or you could equate both of these and get
12-tet.

Is there some way to find some kind of general "temperament class"
that meantone and superpyth are both a part of, since they both
produce 5+2 scales? Or some way to formalize all of that?

> > Also note that one of the modes is the bebop scale, C D E F G G# A B
> > C, which is a major scale with G# added, although it's a lot more
> > interesting if you use it to produce chords rather than just as a
> > major scale with this one chromatic tendency. But if you look at it
> > that way, it's also the composite of these two scales:
> >
> > C D E F G Ab B C (harmonic major)
> > C D E F G# A B C (Ionian augmented)
> >
> > Equating the two implies that G# = Ab, which implies 128/125. 128/125
> > and 81/80 being eliminated means this only works in 12-equal (and also
> > leads to 648/625 vanishing, as per your ideas).
>
> I think analyzing scales makes a lot more sense in 31 than in 12. 12 leaves you scratching your head like this all too often.

That's an idea, making the half steps different sizes. I had taken a
cue from Paul's decatonic scales and hoped to make them the same size,
but perhaps it isn't necessary. Perhaps I should try in 15-equal or
something too, to explore other temperaments where 128/125 vanishes.

> > I really wish I was comfortable enough with the math to work this all out.
>
> I'm not clear on what you want to work out.

I suppose a clearer reformulation of my question here is that I want
to work out some kind of a 7- or higher limit rendition of the scale I
posted. Preferably something that can be constructed with a generator
and a period, or at least as close to having two interval sizes per
octave as possible.

Paul's decatonic scales exist in 12-tet, but sound kind of boring.
Once you put them into 22-tet, all sorts of 7-limit inflections get
involved and the scale really jumps into life. This turns out to be
because the scales are supported by pajara temperament, which is the
12&22 temperament. I was hoping to find some 7-limit or higher rank-1
temperament that supports the scale I posted, a temperament of which
12-equal is an instance.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> That makes sense. I think part of it is also, though, that a
> particular scale might have more than one JI rendition, and hence more
> than one way to temper it. LLsLLLs scales can be meantone, where the
> fifths are tempered narrow and the major thirds 5/4, or they can be
> superpyth (or is it dominant?), where the fifths are tempered sharp
> and the major thirds 9/7. Or you could equate both of these and get
> 12-tet.

You can move from a temperament to a scale, but you can't do the reverse. Scales always span multiple temperaments.

> Is there some way to find some kind of general "temperament class"
> that meantone and superpyth are both a part of, since they both
> produce 5+2 scales? Or some way to formalize all of that?

I would like to know this as well, but signs point to "no". I'm pretty sure there's no comma that 5-EDO, 7-EDO, 12-EDO, 22-EDO, 19-EDO, 31-EDO, 26-EDO, 27-EDO, 33-EDO, etc. all have in common. The trouble is that temperament is explicitly concerned with JI, eg. mapping one prime to something generated by some combination of other primes. The range of a temperament class is defined by error; on one side is JI (0 error), and the other side is open-ended, subject to an arbitrary "maximum" error. In between, there's the question of consistency, which Herman Miller has shown is not fluid but sort of fractal--i.e. not all the generators between 3\7-EDO and 7\12-EDO produce 5-limit-consistent meantone mappings.

So my guess is that to explain why superpyth and meantone scales sound so similar, we have to take a perspective other than temperament.

-Igs

Hi Cameron,

On Sat, Aug 28, 2010 at 7:47 PM, cameron <misterbobro@...m> wrote:
>
> Mike, I should add that an obvious concern with the scale is that without a specific modality, pedalling the C as you do in the x/C chord example you gave, in any tuning with more or less Just fifths, the thing is going to tend to drive, by virtue of historical habits vis a vis fifths as well as spectral tendencies, toward the C# as tonic. So, one approach to a Just ideal would be weight the spectrum toward C (17/16, 19/16, 20/16, 22/16... would do it). But I think the tension of keeping it floating over C is groovier, so I'd got for 7/5 for F#.

Doesn't sound that way to me. I've uploaded a listening example of me
noodling around on it. I'm not in my studio now so I just loaded up
scala and messed with some stuff and edited it - sorry about the
skips, my crappy little comp here is acting up today.

It transposes a chord voicing up magen abot-ically through the scale,
ending on Amaj7#9 -> B9add4 -> Caugmaj7#9, the final chord and the
tonic in this particular case. Then I resolve that to C lydian, and it
doesn't to me feel like it's going to C#.

Now what would be really nice is to render the same thing in some
other ET that supports it and adds higher-limit stuff as well...

-Mike

On Sun, Aug 29, 2010 at 4:14 AM, cameron <misterbobro@...> wrote:
>
> And, maintaining near-perfect 7-limit Just interpretation of this scale with full transposability and circulation, the comma to temper out is simply 225/224, the difference between (two pure M3+m3) and (septimal tritone+pure fourth), also the difference between (septimal subminor third + pure fourth) and (two pure M3), etc.

How did you come up with marvel here, exactly? I'm also noticing that
if the C-F# is intoned 7/5, and if the F#-A-C can both be 6/5's, that
equates 36/25 with 10/7, which tempers out 126/125 - the starling
comma, which I haven't really played around with before. Tempering
both 126/125 and 225/224 means that 81/80 is tempered as well, so
we're back to septimal meantone. I ended up playing around with this
in 31-tet, as per both this and Gene's suggestion, and it sounded
great. So it basically means that it works very well as a subset of
meantone[12]. As for what else it works as, I'm curious, as the
decatonic scales can work as a part of meantone[12] too :)

I just discovered that scala's "fit/mode" function, so fitting it to
the 12-equal version gives 20-equal as having a strictly proper
version. It's 2 3 2 3 3 2 3 2 for those interested. It is apparently
CS too.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> So it basically means that it works very well as a subset of
> meantone[12]. As for what else it works as, I'm curious, as the
> decatonic scales can work as a part of meantone[12] too :)

Which decatonic scales? Paul's Pajara scales? I guess it depends on what you mean by "work as a part of", but with their half-octave period, they explicitly do NOT come up as an exact subset of meantone[12].

> I just discovered that scala's "fit/mode" function, so fitting it to
> the 12-equal version gives 20-equal as having a strictly proper
> version. It's 2 3 2 3 3 2 3 2 for those interested. It is apparently
> CS too.

Which is just one of the 20-EDO diminished scales with half the scale flipped. It could also work as 1 4 1 4 4 1 4 1, and would give better 7-limit harmonies.

-Igs

On Sun, Aug 29, 2010 at 8:32 PM, cityoftheasleep
<igliashon@...> wrote:
>
> > Is there some way to find some kind of general "temperament class"
> > that meantone and superpyth are both a part of, since they both
> > produce 5+2 scales? Or some way to formalize all of that?
>
> I would like to know this as well, but signs point to "no". I'm pretty sure there's no comma that 5-EDO, 7-EDO, 12-EDO, 22-EDO, 19-EDO, 31-EDO, 26-EDO, 27-EDO, 33-EDO, etc. all have in common. The trouble is that temperament is explicitly concerned with JI, eg. mapping one prime to something generated by some combination of other primes. The range of a temperament class is defined by error; on one side is JI (0 error), and the other side is open-ended, subject to an arbitrary "maximum" error. In between, there's the question of consistency, which Herman Miller has shown is not fluid but sort of fractal--i.e. not all the generators between 3\7-EDO and 7\12-EDO produce 5-limit-consistent meantone mappings.
>
> So my guess is that to explain why superpyth and meantone scales sound so similar, we have to take a perspective other than temperament.

I suppose that if we're dealing with rank 2 temperaments, any 5+2
scale will be generated if the fifth is > than 7-tet, and < than
5-tet. Not sure how that works from a regular mapping perspective, but
that is obviously the rule... It starts in meantone, in the middle is
dominant temperament where they overlap, and then there's superpyth at
the end. I know these 3 temperaments are part of the "syntonic
temperament" category - how is this concept formally defined and how
can it be extended to other temperaments as well? Would syntonic
temperament be an actual temperament, or more like a class of
temperaments?

-Mike

On Mon, Aug 30, 2010 at 12:47 AM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > So it basically means that it works very well as a subset of
> > meantone[12]. As for what else it works as, I'm curious, as the
> > decatonic scales can work as a part of meantone[12] too :)
>
> Which decatonic scales? Paul's Pajara scales? I guess it depends on what you mean by "work as a part of", but with their half-octave period, they explicitly do NOT come up as an exact subset of meantone[12].

Yes, Paul's pajara scales. If you're going to treat them as a subset
of meantone, then you won't treat them as though they do have a
half-octave period. You could do that and Unless I'm mistaken, not
every one of Paul's scales is symmetrical about the half-octave either
(the "Standard Pentachordal Major" scale in scala certainly isn't).

But we have the benefit of Paul having laid this stuff out for us in
advance, having figured out that these scales were supported by pajara
and so on. If you were playing around with standard pentachordal major
for the first time in 12-tet, you might notice at first that it's
basically be an unbroken string of 10 fifths, from Bb-C#. So from that
perspective, it certainly seems logical to treat it as meantone[12] -
2 (which is sort of what everyone is noticing with the scale I posted
as well).

But that scale isn't just supported by meantone, it also works as part
of pajara temperament which is also supported by 22. And in 22, its
7-limit tendencies are strengthened, and the whole steps are 164 cents
and start to take on interesting xenharmonic inflections, and so on.
So I'm curious to see what other temperaments might support the scale
I posted, find some other low-numbered ET that gives it interesting
7-limit tendencies or whatever, and find it.

One good candidate I just discovered is augene temperament, which
tempers out 128/125 and 126/125, and hence 39-tet supports this scale
in interesting ways. It adds a little bit of xenharmonic "flair" to it
but in many ways is still not that different from 12. But at least I'm
getting somewhere now...

-Mike

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

> I suppose that if we're dealing with rank 2 temperaments, any 5+2
> scale will be generated if the fifth is > than 7-tet, and < than
> 5-tet. Not sure how that works from a regular mapping perspective, but
> that is obviously the rule... It starts in meantone, in the middle is
> dominant temperament where they overlap, and then there's superpyth at
> the end. I know these 3 temperaments are part of the "syntonic
> temperament" category - how is this concept formally defined and how
> can it be extended to other temperaments as well? Would syntonic
> temperament be an actual temperament, or more like a class of
> temperaments?

"Syntonic Temperament" refers to a temperament which makes 81:80 (the syntonic comma) a unison. I'm not exactly sure how this is worked out; as Sethares uses the term, it refers to anything with a fifth between that of 5-EDO and 7-EDO, but this ignores consistency completely. It's one man's arbitrary delimitation of a temperament class which maps 81:64 and 5:4 to the same note, presuming that a 5/4 can be "approximated" by anything from 480 cents to about 343 cents. As I keep repeating what I've been told, a temperament class is bounded only by arbitrary decisions regarding how much error may be accepted. Any scale can be said to temper any comma, unless you set a maximum allowable error of approximation. But the temperament-smiths here are concerned with minimum errors, not maximum, and to my knowledge no one has worked out maximum errors for any temperament class.

However, consistency is a bigger issue from a "definition of temperament" perspective. I.e. in the range of 720 to about 685 cents, there are a lot of generators which don't produce their best approximation to 5/4 at four iterations. But there are also a lot that do...trouble is, the ones that do are sprinkled sporadically among ones that don't. So you can't give a continuous range of generators for a temperament if you want to care about consistency at all.

-Igs

On Mon, Aug 30, 2010 at 2:22 AM, cityoftheasleep
<igliashon@...> wrote:
>
> "Syntonic Temperament" refers to a temperament which makes 81:80 (the syntonic comma) a unison. I'm not exactly sure how this is worked out; as Sethares uses the term, it refers to anything with a fifth between that of 5-EDO and 7-EDO, but this ignores consistency completely. It's one man's arbitrary delimitation of a temperament class which maps 81:64 and 5:4 to the same note, presuming that a 5/4 can be "approximated" by anything from 480 cents to about 343 cents. As I keep repeating what I've been told, a temperament class is bounded only by arbitrary decisions regarding how much error may be accepted. Any scale can be said to temper any comma, unless you set a maximum allowable error of approximation. But the temperament-smiths here are concerned with minimum errors, not maximum, and to my knowledge no one has worked out maximum errors for any temperament class.

Ah, I see. I was using the definition at
http://en.wikipedia.org/wiki/Syntonic_temperament - perhaps it's
wrong? I haven't done much reading on this subject or know what the
precise terminology is supposed to be.

The article is inconsistent in whether 5 has to be mapped to 4
generators or not, even in cases where the major thirds are clearly
wide enough to be in 9/7 territory. It first says so by saying that
81/80 has to vanish up top, but then also says that Pythagorean tuning
is a type of syntonic temperament, and allows wide fifths all the way
up to 5-equal. It also defines syntonic temperament, at the top of the
page, as sort of just an umbrella term for any rank-2 system where a
"tempered fifth" is the generator. This incorporates narrow fifths,
wide fifths, etc.

It then tries to play the super-wide fifths off as being just a type
of a meantone for inharmonic timbres, but I had always assumed the
"correct" definition was the definition here:

"The syntonic temperament[1] is a system of musical tuning in which
the frequency ratio of each musical interval is a product of powers of
an octave and a tempered perfect fifth, with the width of the tempered
major third being equal to four tempered perfect fifths minus two
octaves and the width of the tempered major second being equal to two
tempered perfect fifths minus one octave (i.e., half the width of the
major third)."

Hence for some generators a "major third" (as in the larger third in
LLsLLLs) could be 9/7, for some it could be 5/4, for some it could be
16/13, etc. But when you pass 5-tet, this isn't the case anymore as
you start to get into father temperament, and when you pass 7-tet it
isn't the case either as you start to get into mavila. I suppose by
that definition the syntonic temperament would be any linear
temperament that produces 5+2 scales.

Then again there's also this:
http://xenharmonic.wikispaces.com/MOSNamingScheme

So if the syntonic temperament is the name for the other thing
(meantone temperaments for inharmonic timbres), then perhaps a good
way to name these "temperament classes" (if they can be rigorously
defined to begin with, that is) would just be to use Graham's names
there. So this would then be the diatonic temperament class, or
something like that. That is, unless this has already been mapped out,
and I'm reinventing the wheel here. If it has, I haven't heard much
discussion about it.

> However, consistency is a bigger issue from a "definition of temperament" perspective. I.e. in the range of 720 to about 685 cents, there are a lot of generators which don't produce their best approximation to 5/4 at four iterations. But there are also a lot that do...trouble is, the ones that do are sprinkled sporadically among ones that don't. So you can't give a continuous range of generators for a temperament if you want to care about consistency at all.

I agree, hence the problems with just defining the syntonic
temperament as an inconsistent meantone. If it's supposed to be a
generalization of meantone for inharmonic timbres, then the above
doesn't end up being a problem. Or if it's supposed to be a
generalization of meantone for LLsLLLs scales, then it also doesn't
end up being a problem. Hopefully it's one of those. :)

-Mike

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

> Yes, Paul's pajara scales. If you're going to treat them as a subset
> of meantone, then you won't treat them as though they do have a
> half-octave period.

Well, if you're going to do that, it begs the question of whether you're actually talking about the same scales.

> You could do that and Unless I'm mistaken, not
> every one of Paul's scales is symmetrical about the half-octave either
> (the "Standard Pentachordal Major" scale in scala certainly isn't).

No, but those that aren't are just variants on those that are, they simply have an L and an s permuted. But those ones cannot be described as rank-2 linear temperaments, because you can't make them with two generators. Pajara temperament uses a period of a half-octave or so and a generator of a fourth or so. In that sense, it's sort of a "half-meantone".

> But we have the benefit of Paul having laid this stuff out for us in
> advance, having figured out that these scales were supported by pajara
> and so on. If you were playing around with standard pentachordal major
> for the first time in 12-tet, you might notice at first that it's
> basically be an unbroken string of 10 fifths, from Bb-C#. So from that
> perspective, it certainly seems logical to treat it as meantone[12] -
> 2 (which is sort of what everyone is noticing with the scale I posted
> as well).

It doesn't work that way in 22-tET, though. In that tuning, it is decidedly NOT a string of 10 fifths. Compare a scale of 10 22-tET fifths to that scale and you'll see it. Do any other meantones have a 10-note MOS that looks like the standard pentachordal major? They actually don't. You'll get three step-sizes in any other meantone. The fact that both sharps and flats are mixed in should make that obvious. So treating the S.P.M. scale as a subset of meantone makes about as much sense as treating the diatonic scale as a subset of Blackwood-10 (i.e. try playing the major scale in 20-EDO).

> So I'm curious to see what other temperaments might support the scale
> I posted, find some other low-numbered ET that gives it interesting
> 7-limit tendencies or whatever, and find it.

If you want it strictly proper, diminished is the only way to go. Though this has more to do with the fact that you have 4L+4s=1200 cents, which means L+s=300 cents, than various commas or what have you. If you don't mind getting 3 step-sizes in the mix, the sky's the limit.

> One good candidate I just discovered is augene temperament, which
> tempers out 128/125 and 126/125, and hence 39-tet supports this scale
> in interesting ways. It adds a little bit of xenharmonic "flair" to it
> but in many ways is still not that different from 12. But at least I'm
> getting somewhere now...

I guess you'd get more helpful suggestions if we knew what you specifically wanted to accomplish and what you specifically wanted to preserve from 12-tET. Any old scale from 12-tET can be interpreted in as many ways as their are temperaments that 12-tET can reasonably said to support. Really, just about any temperament which generates a 12-note MOS can be considered to be compatible with 12-tET, I think? And if you follow Cameron's advice and formulate a JI version of the 12-tET scale, rather than looking at it from an altered-MOS perspective, you can find even more possibilities in other temperaments, since you'd be eschewing the temperamental biases of 12-tET.

-Igs

On Mon, Aug 30, 2010 at 1:15 AM, Mike Battaglia <battaglia01@...> wrote:
>
> One good candidate I just discovered is augene temperament, which
> tempers out 128/125 and 126/125, and hence 39-tet supports this scale
> in interesting ways. It adds a little bit of xenharmonic "flair" to it
> but in many ways is still not that different from 12. But at least I'm
> getting somewhere now...

Three quick things to add to this: the first is that as Igs mentioned,
dimipent is also supported, and 16-tet is a type of dimipent. So is
28-tet. So the interesting situation comes up where this scale is
supported by 12 (a dimipent/meantone temperament) and 16 (a
dimipent/mavila temperament), and the flavor of it doesn't ever
change. That is, the major and minor thirds don't switch in the mavila
version or anything like that.

The second is that perhaps this scale might be viewed better as a
subset (or superset) of another more structurally simple scale
(besides meantone[7]). Right now in 12 it has one approximate
4:5:6:7:9 and a bunch of 4:5:6's and 10:12:15's (some of which might
turn into 14:18:21's and 6:7:9's with a different generator). Perhaps
a 12-note MOS differing from meantone[12] or something. OR, another
possibility, is that it's an altered MOS scale in the same way that
melodic minor is an altered diatonic. I'm not seeing how it could be
immediately altered though...

The third is that I'm amazed at how similar and familiar this scale
sounds in most of these "xenharmonic" edo's - 15, 16, 27, etc. The
diatonic major scale is pretty far from usual out in most of these
(especially 15 and 16), but this scale doesn't have as much variance.
Since I've been so caught up in Rothenberg equivalence classes these
past few days - does anyone have a clue how that would apply here?
This is really a brand new scale that I don't have a very good
built-in "map" for yet (except for the seconds), so how is this
working? Has anyone figured this little piece of musical perception
out? Perhaps the fact that I hear it as lydian aug#2 with an added #1
has something to do with it.

-Mike

Igs wrote:

> "Syntonic Temperament" refers to a temperament which makes 81:80
> (the syntonic comma) a unison. I'm not exactly sure how this
> is worked out; as Sethares uses the term,

It's Milne/Sethares/Plamondon's name for meantone. They had
concerns about conflict with the use of the term in the early
music community. Some have expressed such concerns here also,
and I have posted extensively about the issue so I won't belabor
it now other than to say I think it is: 1. an appropriate and
practical generalization, just as "glass" and "salt" are in
chemistry and 2. the best possible term, because the theory of
intonation was not so foreign to musicians of the meantone era
as it is today, and any effort to reintroduce it to musicians
would do well to remind them of this fact.

They write,
""The syntonic rational continuum extends beyond the narrower
range implied by "meantone," which usually refers to the range
of syntonic tunings providing reasonably pure-sounding tunings
(for sounds with harmonic spectra) and/or which have an
established historical use — a range of approximately 19-TET
to 12-TET.""

The way they do establish the range is: they define a list
of consonances, and when any of their approximations moves
more than half way to a different one, they stop. Too bad
they didn't ask Herman Miller or Manuel Op de Coul. You
know, like, on this list.

They also managed not to cite Graham in either of their
two publications.

-Carl

> It's one man's arbitrary delimitation of a temperament class which maps 81:64 and 5:4 to the same note, presuming that a 5/4 can be "approximated" by anything from 480 cents to about 343 cents.
> a temperament class is bounded only by arbitrary decisions regarding how much error may be accepted.

I don't think this is the case. I think it also has to do with the relationship between the temperament and JI, and then at what points these relationships become invalid.
For example, a fifth from 685 to 720 cents allows a temperament where a the interval defined as 5/4, aka [(3/2)^4 * (2/1)^-2], is always greater than or equal to 6/5 and less than or equal to 4/3.

Above 720 cents the major third becomes larger than the fourth. This fact invalidates both the melodic functions of the temperament (something defined as a third should be smaller than or equal to something defined as a fourth) and the harmonic functions (the harmonic function defined by the relationship between the fourth and third harmonic needs to be larger than or equal to the harmonic function defined by the ratio between the fifth and fourth.)

A less recognizable invalidation is the invalidation of a timbre derived from the temperament. A timbre defined by the temperament will have it's partials defined as locations in that temperament's matrix of notes, and then when you use a generator outside of the temperament's valid tuning range, you will then be putting the partial defined as the fifth harmonic either below the one defined as the fourth or above the one defined as the sixth.

Does that make sense?

John

On Mon, Aug 30, 2010 at 2:43 AM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Yes, Paul's pajara scales. If you're going to treat them as a subset
> > of meantone, then you won't treat them as though they do have a
> > half-octave period.
>
> Well, if you're going to do that, it begs the question of whether you're actually talking about the same scales.

In 12-equal, where all of the relevant commas are eliminated, they are
the same scales. In this case, you could go a level back up and come
up with a generalized meantone version of the scales, or you could go
a level back up and come up with a generalized pajara version of the
scales, or perhaps you could find a generalized rank-2 system besides
those two that supports it well. For whatever reason, the pajara
version turned out to excel in terms of 7-limit harmony, was about as
close to having Myhill's property as you can really get, and ended up
having a reasonably practical ET supporting it - 22.

But I just discovered this scale recently, so I don't yet have the
foresight of knowing what this other direction to generalize it is.
Right now I'm seeing meantone, and after taking Cameron's advice and
making a JI plot, it seemed augene and dimipent might work well too.
Augene has been the best so far with 27-et as a potential ET
candidate, although it's still far from as stellar as pajara and 22
was for Paul.

> No, but those that aren't are just variants on those that are, they simply have an L and an s permuted. But those ones cannot be described as rank-2 linear temperaments, because you can't make them with two generators. Pajara temperament uses a period of a half-octave or so and a generator of a fourth or so. In that sense, it's sort of a "half-meantone".

Why wouldn't it be part of pajara if the relevant commas are still
being tempered out? The 1/5-comma meantone melodic minor scale is
still created from within the meantone framework and eliminates 81/80,
even if it isn't formed from a contiguous chain of fifths...

> It doesn't work that way in 22-tET, though. In that tuning, it is decidedly NOT a string of 10 fifths. Compare a scale of 10 22-tET fifths to that scale and you'll see it.

Yes, I'm aware of that. I'm saying that if you're starting in 12-tet,
with this decatonic scale that you've never thought of before, it
might look just like a 10-note string of meantone fifths at first. To
get to 22-tet, you have to find another way to look at it, which is
what I'm trying to do.

> Do any other meantones have a 10-note MOS that looks like the standard pentachordal major? They actually don't. You'll get three step-sizes in any other meantone. The fact that both sharps and flats are mixed in should make that obvious. So treating the S.P.M. scale as a subset of meantone makes about as much sense as treating the diatonic scale as a subset of Blackwood-10 (i.e. try playing the major scale in 20-EDO).

Indeed. That's what I'm trying to do here - find the equivalent of
pajara for those decatonic scales with the octatonic scale I posted
above. Something that ideally organizes beautifully into an MOS or
almost-MOS, has stellar 7-limit harmonies, a low-numbered ET that
supports it, etc. Augene is the best so far, although I'm wondering if
I can beat the 400 cent major thirds.

It should be mentioned though that at least on my scala
implementation, the SPM isn't MOS by any definition of the word - the
half-octave also comes in 11/8 and 16/11 flavors as well. That is,
every interval has 2 step sizes except for that half-octave one, which
has 3. But if I could find a linear temperament supporting this that
was that close to MOS, I'd be happy. Perhaps it's a subset from some
brilliant 9- or 10- note MOS that I'm not seeing.

> If you want it strictly proper, diminished is the only way to go. Though this has more to do with the fact that you have 4L+4s=1200 cents, which means L+s=300 cents, than various commas or what have you. If you don't mind getting 3 step-sizes in the mix, the sky's the limit.

Proper would be fine here too, but why does it have to be diminished
for strictly proper? Looking at Scala's fit mode stuff you're clearly
right, as all of the SP ET's are multiples of 4, but how did you
deduce that?

> I guess you'd get more helpful suggestions if we knew what you specifically wanted to accomplish and what you specifically wanted to preserve from 12-tET. Any old scale from 12-tET can be interpreted in as many ways as their are temperaments that 12-tET can reasonably said to support. Really, just about any temperament which generates a 12-note MOS can be considered to be compatible with 12-tET, I think? And if you follow Cameron's advice and formulate a JI version of the 12-tET scale, rather than looking at it from an altered-MOS perspective, you can find even more possibilities in other temperaments, since you'd be eschewing the temperamental biases of 12-tET.

Hopefully this post has served to clarify my goals with all of this.
As for Cameron's advice - that's how I came up with the Augene idea
(Starling seemed like a good idea, and getting rid of 128/125 seemed
like a good idea, and hence Augene). Hopefully if I keep at it I'll
stumble across the Best Temperament There Is for this scale, although
it would be nice if there was an algorithm to just get all of the
linear temperaments supporting this scale, find out what the 7-limit
error is for each one, sort them, and go from there. This implies that
there's a way to define this scale outside of any particular
temperament, which the idea of a "temperament class" would be good
for, if it exists.

Gene's 31-et approach was a good start, although I missed Augene from
that, so perhaps there's a way to generalize the concept (maybe 62-et
or something?). Anyone...?

-Mike

On Mon, Aug 30, 2010 at 2:59 AM, Carl Lumma <carl@...> wrote:
>
> Igs wrote:
>
> > "Syntonic Temperament" refers to a temperament which makes 81:80
> > (the syntonic comma) a unison. I'm not exactly sure how this
> > is worked out; as Sethares uses the term,
>
> It's Milne/Sethares/Plamondon's name for meantone. They had
> concerns about conflict with the use of the term in the early
> music community. Some have expressed such concerns here also,
> and I have posted extensively about the issue so I won't belabor
> it now other than to say I think it is: 1. an appropriate and
> practical generalization, just as "glass" and "salt" are in
> chemistry and 2. the best possible term, because the theory of
> intonation was not so foreign to musicians of the meantone era
> as it is today, and any effort to reintroduce it to musicians
> would do well to remind them of this fact.

So it is, then, supposed to be a generalization of meantone for
inharmonic timbres, not a generalization of meantone for harmonic
timbres and 5+2 scales? As in, 5 is always supposed to be mapped to 4
generators, so superpyth is not a type of syntonic temperament, but
dominant temperament is?

> The way they do establish the range is: they define a list
> of consonances, and when any of their approximations moves
> more than half way to a different one, they stop. Too bad
> they didn't ask Herman Miller or Manuel Op de Coul. You
> know, like, on this list.
>
> They also managed not to cite Graham in either of their
> two publications.

Lame. I don't understand though, if they're dealing with inharmonic
timbres, doesn't the whole basis for the map change? If you don't have
3 and 5 anymore, what does 81/80 mean? Do they just map the 3rd
overtone as 3, even though it's not really 3x the frequency, and just
let it all go from there?

-Mike

-Mike

Just realized I forgot to give the link. Here it is:

http://f1.grp.yahoofs.com/v1/8FZ7TNsCo1VSZTFKaGf9cFJ2dyoHTn9MKD6l0SMXFAUf7OOuqMmZN2LVdc_gRYDBu_SROIUCipIx0l88uJPbjsdQZjHJ9mY/MikeBattaglia/magenabotfinal.mp3

This is the 12-tet version, before I discovered augene, but it does
given an example of how this scale might function harmonically. If the
decatonic pajara scales are supposed to be "tonal," this one, in this
incarnation, is definitely "modal." Some 7-limit stuff in here would
be awesome.

-Mike

On Sun, Aug 29, 2010 at 11:51 PM, Mike Battaglia <battaglia01@...> wrote:
> Hi Cameron,
>
> On Sat, Aug 28, 2010 at 7:47 PM, cameron <misterbobro@...> wrote:
>>
>> Mike, I should add that an obvious concern with the scale is that without a specific modality, pedalling the C as you do in the x/C chord example you gave, in any tuning with more or less Just fifths, the thing is going to tend to drive, by virtue of historical habits vis a vis fifths as well as spectral tendencies, toward the C# as tonic. So, one approach to a Just ideal would be weight the spectrum toward C (17/16, 19/16, 20/16, 22/16... would do it). But I think the tension of keeping it floating over C is groovier, so I'd got for 7/5 for F#.
>
>
> Doesn't sound that way to me. I've uploaded a listening example of me
> noodling around on it. I'm not in my studio now so I just loaded up
> scala and messed with some stuff and edited it - sorry about the
> skips, my crappy little comp here is acting up today.
>
> It transposes a chord voicing up magen abot-ically through the scale,
> ending on Amaj7#9 -> B9add4 -> Caugmaj7#9, the final chord and the
> tonic in this particular case. Then I resolve that to C lydian, and it
> doesn't to me feel like it's going to C#.
>
> Now what would be really nice is to render the same thing in some
> other ET that supports it and adds higher-limit stuff as well...
>
> -Mike
>

Mike wrote:

> > It's Milne/Sethares/Plamondon's name for meantone. They had
> > concerns about conflict with the use of the term in the early
> > music community. Some have expressed such concerns here also,
> > and I have posted extensively about the issue so I won't belabor
> > it now other than to say I think it is: 1. an appropriate and
> > practical generalization, just as "glass" and "salt" are in
> > chemistry and 2. the best possible term, because the theory of
> > intonation was not so foreign to musicians of the meantone era
> > as it is today, and any effort to reintroduce it to musicians
> > would do well to remind them of this fact.
>
> So it is, then, supposed to be a generalization of meantone

Which? Syntonic temperament is meantone.

> for inharmonic timbres, not a generalization of meantone for
> harmonic timbres and 5+2 scales? As in, 5 is always supposed
> to be mapped to 4 generators, so superpyth is not a type of
> syntonic temperament, but dominant temperament is?

Huh? They have two ways of matching tunings to a map.
One is the way I mentioned, which they call the "rational"
way. It's an OK method though it's highly dependent on the
concordances they pick, and I don't see where they explain
how they pick them. The other is the scale way, based on
the scale tree as Igs suggested, which they call the "ordinal"
method. They admit neither is perfect.

> if they're dealing with inharmonic
> timbres, doesn't the whole basis for the map change? If you
> don't have 3 and 5 anymore, what does 81/80 mean? Do they
> just map the 3rd overtone as 3, even though it's not really
> 3x the frequency, and just let it all go from there?

Meantone aka syntonic is always 81/80. It makes a lot of
sense if 2:1, 3:1, and 5:1 are consonant in the timbre, and
probably a lot less sense otherwise. Their two publications
hardly mention the issue. Probably because the ear does NOT
perceive inharmonic spectra to be very concordant, no matter
what you do.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Just realized I forgot to give the link.

I think you meant

/tuning/files/MikeBattaglia/magenabotfinal.mp3

-C.

On 30 August 2010 14:59, Carl Lumma <carl@...> wrote:

> They also managed not to cite Graham in either of their
> two publications.

To be fair about this, they didn't really need to. And I did see one
of the papers before publication, and they altered one of the
citations following a quibble I raised.

Graham

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> The way they do establish the range is: they define a list
> of consonances, and when any of their approximations moves
> more than half way to a different one, they stop. Too bad
> they didn't ask Herman Miller or Manuel Op de Coul. You
> know, like, on this list.

On the "Syntonic Temperament" wikipedia page, which cites Sethares almost exclusively, it's defined as running between the fifths of 5-EDO and 7-EDO.

> They also managed not to cite Graham in either of their
> two publications.

Bummer.

-Igs

> As in, 5 is always supposed to be mapped to 4
> generators, so superpyth is not a type of syntonic temperament, but
> dominant temperament is?

In the syntonic temperament 5/4 IS always represented by up four generators, down two octaves.
It is defined as so by the fact that 81/80 is tempered out:

5/4 is the major third
3/2 is the fifth
2/1 is the octave

We define the major third as up four fifths and down two octaves:'

5/4 = [(3/2)^4 * (2/1)^-2]
(major third) = (the fifth)^up-four-times * (the octave)^down two times

Simplifying we eventually arrive at the conclusion that...

5/4 = 81/16 * 1/4
5/4 = 81/64
80/64=81/64
81/80 = 1
The interval 81/80, the syntonic comma, is tempered to unison.

ANY tuning that defines 81/80 to be equal:
1. Defines four 3/2's minus two 2/1's to be equal to 5/4
2. Is a syntonic tuning

The syntonic temperament also defines 4/3>5/4>6/5 because it is at those points that the 5 Large 2 Small diatonic scale exists.
While the 3/2 remains between 685 and 720 cents, this definition remains valid, and so those values are used as the valid tuning range.

John

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
>
> > The way they do establish the range is: they define a list
> > of consonances, and when any of their approximations moves
> > more than half way to a different one, they stop. Too bad
> > they didn't ask Herman Miller or Manuel Op de Coul. You
> > know, like, on this list.
>
> On the "Syntonic Temperament" wikipedia page, which cites
> Sethares almost exclusively, it's defined as running between
> the fifths of 5-EDO and 7-EDO.

That's the "ordinal" definition. The one I describe above
is their "rational" definition. They admit that neither is
perfect.

-Carl

Hi John,

> Simplifying we eventually arrive at the conclusion that...

Who's we? And what is your relationship to Milne/Sethares?
Are you one of Milne's students?

> 2. Is a syntonic tuning

Around here it's called meantone. It's a better term for
various reasons. I suggest you adopt it.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Why wouldn't it be part of pajara if the relevant commas are still
> being tempered out? The 1/5-comma meantone melodic minor scale is
> still created from within the meantone framework and eliminates 81/80,
> even if it isn't formed from a contiguous chain of fifths...

Well, it would still be part of Pajara, but you couldn't get the scale directly by adding generators. What you'd have to do to get the SPM is probably get up to pajara[12], and then take a 10-note subset of that. In 22-EDO, pajara[12] is 2 2 2 2 2 1 2 2 2 2 2 1, so to get a 10-note subset, you just add those 1-step intervals to one of the near-by 2-step intervals; in this way, you could get any of the decatonic scales. But only the "symmetric" scales (in Paul's terms) are MOS, because pajara[10] is 2 2 2 2 3 2 2 2 2 3--because of the half-octave period.

> Yes, I'm aware of that. I'm saying that if you're starting in 12-tet,
> with this decatonic scale that you've never thought of before, it
> might look just like a 10-note string of meantone fifths at first. To
> get to 22-tet, you have to find another way to look at it, which is
> what I'm trying to do.

Look at it in terms of large and small steps. Rank-2 temperaments will always produce MOS scales, because they are defined by a generator and a period. To get a scale from a temperament, you just keep adding generators and normalizing by period. I guess you could stop at a non-MOS point, though. But typically there isn't good reason to do that.

> Proper would be fine here too, but why does it have to be diminished
> for strictly proper? Looking at Scala's fit mode stuff you're clearly
> right, as all of the SP ET's are multiples of 4, but how did you
> deduce that?

Your scale in 12-EDO is 1 2 1 2 2 1 2 1, right? If you switch the first four steps, it's 2 1 2 1 2 1 2 1, which is more generally LsLsLsLs. But that order doesn't even really matter, what matters is ONLY that there are 4 large steps and 4 small steps. As I said above, 4L+4s=1200 cents means L+s=300 cents; you don't need to know any commas to know that. It's basic algebra. So if you want to preserve having two step-sizes, you have to keep L+s=300 cents a constant. You can also say 4L+4s=x-EDO, so L+s=x/4-EDO: that means you will be stuck working in EDOs divisible by 4 unless you're cool with 3 step-sizes...or working with a larger scale. But the only way to get an 8-note scale with 4 large steps and 4 small steps, where the small steps are all the same size and the large steps are all the same size, is to work in an EDO divisible by 4--regardless of what other commas are tempered out.

> Hopefully this post has served to clarify my goals with all of this.
> As for Cameron's advice - that's how I came up with the Augene idea
> (Starling seemed like a good idea, and getting rid of 128/125 seemed
> like a good idea, and hence Augene). Hopefully if I keep at it I'll
> stumble across the Best Temperament There Is for this scale, although
> it would be nice if there was an algorithm to just get all of the
> linear temperaments supporting this scale, find out what the 7-limit
> error is for each one, sort them, and go from there. This implies that
> there's a way to define this scale outside of any particular
> temperament, which the idea of a "temperament class" would be good
> for, if it exists.

The way to define this scale outside of a temperament is the way I outlined above, i.e. as an MOS scale defined by a pattern of large and small steps. This approach works irrespective of commas, mappings, and errors. You simply count the number of large steps, and the number of small steps, then you put it in form nL+ms. If you want to find EDOs that support the scale, you just plug in different integers for L & s; the integers you plug in are the "steps out of the EDO" you're solving for. For instance, your scale is 4L+4s; with L=2 and s=1, we get 12-EDO because 4(2)+4(1)=8+4=12. For L=3 and s=2, we get 4(3)+4(2)=12+8=20-EDO. To get step-sizes in cents, you then just multiply 1200 by 3/20 and 2/20, getting 180 cents for L and 120 cents for s.

But as I keep repeating and you keep ignoring (or maybe you don't believe me?), there's no way to lump a bunch of MOS scales together into one temperament class just because they share the same scale-shape. I went down that road recently, asking very much the same questions you are asking of this list, and that's the answer I got. Temperament classes are defined by two things: mapping and error. Knowing the mapping you want and the specific amount of error is what allows you to find a tuning for the generators of a temperament. If you haven't specified an amount of error, you cannot define the tuning of the generators. With the right (or rather, "wrong") error value, you can make the Mavila scale with a Meantone mapping. Or Gene gave me a great example of using 29 out of 36-EDO as a "meantone" generator to get a "major third" of 44/36, i.e. using an approximate 7/4 as a 3/2 to get an approximate 7/3 as a 5/4. I also gave an example of looking at Mohajira as something like a "16-comma Meantone"; so you see, you can call ANY scale a Meantone if you allow for any amount of error.

To put it bluntly: there is never going to be a 1:1 correspondence between "temperament class" and MOS configuration, the way both are currently defined. Are we clear?

To end on a positive note, let me try to actually help you meet your goal. With a scale of sLsLLsLs, we know some notes are going be constant, because L+s=300 cents no matter what tuning we use, but others will vary:
I: 0 cents
II: s cents
III: 300 cents
IV: 300+s cents
V: 600 cents
VI: 600+L cents
VII: 900 cents
VIII: 900+L cents

You want good 7-limit implications. That's not going to happen in this modal position, because there's no room in any of the variable degrees to get to a 7-limit concordance. Let's switch to the LsLssLsL mode:

I: 0 cents
II: L cents
III: 300 cents
IV: 300+L cents
V: 600 cents
VI: 600+s cents
VII: 900 cents
VIII: 900+s cents

We can make s something like 66.666... cents (round to 66.67) to get a 36-EDO tuning, which gives an L of 233 cents. So we get: 0-233-300-533.33-600-666.67-900-966.67-1200 cents. We could also try making L something like a 7/6 at 266.67 cents, making s 33.33 cents, giving 0-266.67-300-566.67-600-633.33-900-933.33-1200 cents.
Going back to your mode of the scale, the first incarnation gives 0-66.67-300-366.67-600-833.33-900-1133.33-1200 cents. The second incarnation gives 0-33.33-300-333.33-600-866.67-900-1166.67-1200 cents.

All of these suffer in both the 3 and 5 limits compared to the 12-EDO version, because of the fact that the 3rd, 5th, and 7th harmonics are all going to be approximated by deviations from 600, 300, and 900 cents (respectively), and to maintain two step-sizes, the 7th harmonic will have to be as far from 900 cents as the 3rd harmonic is from 600 cents. I.e., the closer you get to a pure 7/4, the further you get from a pure 3/2.

In my next post, however, I'll look at some non-diminished MOS interpretations of your scale.

-Igs

> > Simplifying we eventually arrive at the conclusion that...
>
> Who's we? And what is your relationship to Milne/Sethares?
> Are you one of Milne's students?

Even though in that sentence the function of "we" was meant to be like "one", as in:
> > Simplifying, *one* eventually arrives at the conclusion that...
I'd be happy to fill you in on a little of my background =)

I am not a student of Bill's first of all. I just happen to be a guy who's really curious about the underlying patterns of music, found isomorphic keyboards (or would you "suggest" the term generalized? =P), and began to try to learn what I could about them. This lead me to their fingering invariance not only across keys, but also across all tunings of a given temperament, which peaked my interest in tuning. I have since tried to learn whatever I can about temperament theory, and microtonal theory in general, and have always found my own conclusions aligning themselves with those of Sethares/Milne/Plamondon. I learned a lot from them, but I was also always skeptical of anything I didn't fully understand and have tried not to preach anything until I know the practice.

That's pretty much me in a nutshell.

John Moriarty

Igs wrote:
> Your scale in 12-EDO is 1 2 1 2 2 1 2 1, right? If you switch the first four steps, it's 2 1 2 1 2 1 2 1, which is more generally LsLsLsLs. But that order doesn't even really matter, what matters is ONLY that there are 4 large steps and 4 small steps. As I said above, 4L+4s=1200 cents means L+s=300 cents; you don't need to know any commas to know that. It's basic algebra. So if you want to preserve having two step-sizes, you have to keep L+s=300 cents a constant. You can also say 4L+4s=x-EDO, so L+s=x/4-EDO: that means you will be stuck working in EDOs divisible by 4 unless you're cool with 3 step-sizes...or working with a larger scale. But the only way to get an 8-note scale with 4 large steps and 4 small steps, where the small steps are all the same size and the large steps are all the same size, is to work in an EDO divisible by 4--regardless of what other commas are tempered out.

I see. Perhaps there is some better 9-note or 10-note scale that mine
is a subset of then. I think starling is the way to go with this one -
either starling or something that eliminates 50/49. But I don't think
that diminished is going to lead to any really spectacular 7-limit
harmonies, so perhaps it's better to relax the strict propriety
requirement and just go with regular propriety. The fact that there
are so many "proper" intonations of this scale makes me think that a
strict proper version might come just by adding a few more notes.

> The way to define this scale outside of a temperament is the way I outlined above, i.e. as an MOS scale defined by a pattern of large and small steps. This approach works irrespective of commas, mappings, and errors. You simply count the number of large steps, and the number of small steps, then you put it in form nL+ms.

That's fine, but it might not end up being an MOS at all. It probably
won't be, as Paul's SPM isn't either. It might have 3 interval sizes
for some things, and 2 interval sizes for other things, e.g. harmonic
minor.

> But as I keep repeating and you keep ignoring (or maybe you don't believe me?), there's no way to lump a bunch of MOS scales together into one temperament class just because they share the same scale-shape. I went down that road recently, asking very much the same questions you are asking of this list, and that's the answer I got. Temperament classes are defined by two things: mapping and error. Knowing the mapping you want and the specific amount of error is what allows you to find a tuning for the generators of a temperament.

Right, I think we're miscommunicating because I'm using the wrong
term. I probably should have said something like "Temperament
Superclass," since I didn't realize that "Temperament class" was
already defined. But now that I see that the phrase "temperament
class" is used to refer to things like meantone, augmented, pajara,
etc, I'll use the term "Temperament superclass" from now on.

The idea is that meantone and superpyth are obviously related because
they both produce LLsLLLs scales, although they map the intervals in
them differently. So perhaps it could be rigorously defined that there
is some kind of "diatonic" temperament superclass that these
temperament classes are a part of. The "syntonic temperament" as we
were originally discussing seemed like it was defining things that
way, but then ruined the whole concept by forcing the interpretations
of 4 generators as 5. I think Graham has already worked this out...?

And, coming at it from the other angle, in 12-tet, the decatonic SPM
scale is supported both by pajara and meantone, and perhaps other
things too. From a "pajara angle," it can be viewed as a near-MOS
alteration of the symmetric decatonic scales, or perhaps pajara[12]
minus 2. From a "meantone angle," it can be viewed as a chain of 10
meantone fifths, or perhaps meantone[12] minus 2. Both of these are
alterations of nearby MOS's that different temperaments support, and
they all become the same thing in 12-equal.

One that you've pointed out is that it's very close (only a note or
two off) from a diminished MOS. The only problem is that I'm not sure
diminished is really going to lead to really stellar 7-limit
harmonies, and you could probably also come up with a diminished
interpretation of the 12-tet decatonic scales as well. Looking at it
from an augmented perspective naturally suggests augene, but then the
5-limit harmonies suffer. So I'm wondering what other options there
might be, and if there could be defined some sort of superset of
temperaments that supports these. Perhaps in this instance the concept
is a bit redundand, and the superset of temperaments is just
everything that 12-tet supports, by definition.

> All of these suffer in both the 3 and 5 limits compared to the 12-EDO version, because of the fact that the 3rd, 5th, and 7th harmonics are all going to be approximated by deviations from 600, 300, and 900 cents (respectively), and to maintain two step-sizes, the 7th harmonic will have to be as far from 900 cents as the 3rd harmonic is from 600 cents. I.e., the closer you get to a pure 7/4, the further you get from a pure 3/2.

Thanks for the analysis. That actually turned out to better for the
7-limit than I thought, although as you pointed out the 5-limit
suffered pretty bad. It might just reaffirm that it might be best to
allow more than 2 step sizes in this case.

> In my next post, however, I'll look at some non-diminished MOS interpretations of your scale.

Thanks, I look forward to it!

-Mike

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
> ANY tuning that defines 81/80 to be equal:
> 1. Defines four 3/2's minus two 2/1's to be equal to 5/4
> 2. Is a syntonic tuning

This definition could actually apply to any tuning, unless we account for error and consistency. First we have to define how much you can temper a 3/2 before it stops being a 3/2, and how much we can temper a 5/4 before it stops being 5/4. Otherwise we could call a 7/4 a 3/2, and make our 5/4 something like 7/3, since four 7/4's minus two octaves is something like a 7/3. THEN, to avoid having an inconsistent tuning, we have to stipulate that four 3/2s is a BETTER approximation to 5/4 than any other number of 3/2's in the temperament. If we do this, we can no longer have a continuous "range" of generators in the temperament. Consider that a generator of 690 cents--squarely between 7-EDO and 12-EDO--produces a better 5/4 at a stack of 11, rather than a stack of 4 (690*11=7590 cents, subtract 7200 cents (6 octaves) and you get 390 cents. 690*4=2760 cents, subtract 2400 cents (two octaves) and you get 360 cents. 390 is closer to 5/4 than 360).

> The syntonic temperament also defines 4/3>5/4>6/5 because it is at those points that the 5 Large 2 Small diatonic scale exists.

This actually has nothing to do with tempering the syntonic comma, or those JI ratios. It can be expressed entirely in terms of generators and periods, as: (-1gen+1per)>(4gen-2per)>(-3gen+2per). If we take the period to be an octave and solve for a value of the generator, we see that only generators between 3-of-5-EDO and 4-of-7-EDO fit the bill. Approximation to JI never enters the equation at any point. So the "syntonic temperament" is actually not a temperament but a range of MOS scale generators that will produce a scale of 5L+2s=1 octave.

I don't mean to shoot the messenger, of course. But someone should really clue Bill in to the fact that his (and I guess Milne's et al.) definition is problematic from a temperament perspective.

-Igs

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
I have since tried to learn whatever I can about temperament theory, and microtonal theory in general, and have always found my own conclusions aligning themselves with those of Sethares/Milne/Plamondon.

Which conclusions were those? If you mean the tuning continua paper, that obviously was heavily influenced by tuning-math@yahoo, so it's not another, unrelated, circle of ideas to what goes on here.

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

> The idea is that meantone and superpyth are obviously related because
> they both produce LLsLLLs scales, although they map the intervals in
> them differently. So perhaps it could be rigorously defined that there
> is some kind of "diatonic" temperament superclass that these
> temperament classes are a part of.

They belong to a set of temperaments which all have octave period and fifth generator.

Hi John,

> > Who's we? And what is your relationship to Milne/Sethares?
> > Are you one of Milne's students?
>
> Even though in that sentence the function of "we" was meant to
> be like "one",

Ah. Thanks for answering the question anyway. FYI, it looks
like for some reason your msg posted twice, so I deleted the
extra one from the archives (e-mail users will still get two
copies). -Carl

Hi Mike, I think we're still talking past each other a little bit here.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > The way to define this scale outside of a temperament is the way I outlined above, i.e. as an MOS scale defined by a pattern of large and small steps. This approach works irrespective of commas, mappings, and errors. You simply count the number of large steps, and the number of small steps, then you put it in form nL+ms.
>
> That's fine, but it might not end up being an MOS at all. It probably
> won't be, as Paul's SPM isn't either. It might have 3 interval sizes
> for some things, and 2 interval sizes for other things, e.g. harmonic
> minor.

Any scale with two step-sizes can be arranged as an MOS or not. Any scale with three step-sizes, assuming it's an equal temperament, can be broken down to a larger scale with two step-sizes. So even if a scale is not explicitly MOS, it can be related to an MOS scale, and relating everything back to MOS's is a very easy way to figure out how different tunings will support a given scale. Any tuning that supports a scale of 4L+4s will support any arrangement of those steps, be it ssssLLLL or ssLLsLsL or whatever. But they will all support LsLsLsLs also, which means that if you can find tunings that support the MOS scale, the very same tunings will support the scale's non-MOS permutations. And in fact the only way to "generate" those non-MOS scales is to treat them as permuted MOS scale, or as non-MOS subsets of larger MOS scales. Otherwise you're choosing notes at random.

> > But as I keep repeating and you keep ignoring (or maybe you don't believe me?), there's no way to lump a bunch of MOS scales together into one temperament class just because they share the same scale-shape. I went down that road recently, asking very much the same questions you are asking of this list, and that's the answer I got. Temperament classes are defined by two things: mapping and error. Knowing the mapping you want and the specific amount of error is what allows you to find a tuning for the generators of a temperament.
>
> Right, I think we're miscommunicating because I'm using the wrong
> term. I probably should have said something like "Temperament
> Superclass," since I didn't realize that "Temperament class" was
> already defined.

No, that's not it. Call it whatever you want, define it as broadly as you want, there is no non-arbitrary way to make any category based on tempering have a 1-to-1 correspondence with a category based on scale configuration. Any time you are talking about temperament, be it a specific temperament, a temperament class, or a temperament "super-class", you're stuck with concepts of mappings, commas, errors, and consistencies. To somehow lump all LLsLLLs scales together into a temperament class or super-class, you'd have to do the following things:
1: find some mapping that they all have in common
2: define your error range such it gives generators only between 4/7ths of an octave and 3/5ths of an octave, which will be arbitrary
3: ignore consistency completely

> The idea is that meantone and superpyth are obviously related because
> they both produce LLsLLLs scales, although they map the intervals in
> them differently.

The unifying feature here is the scale shape, not the mapping; temperament is ALL ABOUT MAPPING. If the mapping is different, the temperament is different. OTOH, Moment of Symmetry is ALL ABOUT SCALE SHAPE, and could give a damn about mapping or JI.
> One that you've pointed out is that it's very close (only a note or
> two off) from a diminished MOS. The only problem is that I'm not sure
> diminished is really going to lead to really stellar 7-limit
> harmonies, and you could probably also come up with a diminished
> interpretation of the 12-tet decatonic scales as well. Looking at it
> from an augmented perspective naturally suggests augene, but then the
> 5-limit harmonies suffer. So I'm wondering what other options there
> might be, and if there could be defined some sort of superset of
> temperaments that supports these. Perhaps in this instance the concept
> is a bit redundand, and the superset of temperaments is just
> everything that 12-tet supports, by definition.

And there you have it! Yes, the superset is just everything that 12-tET supports, by definition.

> Thanks for the analysis. That actually turned out to better for the
> 7-limit than I thought, although as you pointed out the 5-limit
> suffered pretty bad. It might just reaffirm that it might be best to
> allow more than 2 step sizes in this case.

Yep. In which case, the sky's the limit! You can find some way to play this scale with 3 or 4 step sizes in probably any EDO above 12. So you could just look at EDOs that have good harmonic characteristics, and shoe-horn your scale into them and see what happens. But you won't get any guiding light from temperament or MOS theory at this point...you're basically looking at raw JI, which can be tempered any which way at all.

-Igs

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> They belong to a set of temperaments which all have octave period and fifth generator.

What about Mavila? That has a fifth generator--albeit a drastically-tempered one--but it produces a very different scale. 4/7ths of an octave and 3/5ths of an octave are not the psychoacoustic boundaries for a recognizable fifth, at any rate. So that definition would lump in scales other than the 5L+2s scales we want.

-Igs

> From: Mike Battaglia

> That makes sense. I think part of it is also, though, that a
> particular scale might have more than one JI rendition, and hence
> more than one way to temper it. LLsLLLs scales can be meantone,
> where the fifths are tempered narrow and the major thirds 5/4, or
> they can be superpyth (or is it dominant?), where the fifths are
> tempered sharp and the major thirds 9/7. Or you could equate both of
> these and get 12-tet.

> Is there some way to find some kind of general "temperament class"
> that meantone and superpyth are both a part of, since they both
> produce 5+2 scales? Or some way to formalize all of that?

Dear Mike (and Caleb),

Thank you for raising a very important question to which I
believe there is a good answer actually defined by Easley
Blackwood, Jr.

All the temperaments you are describing from 7-EDO (685.71c) to
5-EDO (720.00c) are "regular diatonic" tunings. Of course, there
are many other wonderful generators, but this is the range where
a chain of six fifths give us a regular diatonic scale of the
LLsLLLs pattern.

What we'd also like is a term analogous to "meantone," but
describing any tuning in this regular diatonic range where a
"major third" from four fifths up (less two octaves) is formed
from two identically sized "major seconds" or "tones," each in
turn formed from two fifths up (less an octave).

Here I propose the term "isotone," meaning that a major third is
formed from two tones of the "same" size.

To take your example, a fifth at 709 cents produces a major
second at 218 cents, and two of these 218-cent tones gives us a
major third at 436 cents, very close to 9/7 (435.08 cents). This
is an isotone, because we get the 436-cent major third from two
identical 218-cent tones; or, each tone is precisely half the
size of the major third.

While all meantones are isotones, not all isotones are
meantones. Our 709-cent temperament, close to 22-EDO, tempers out
the 64:63, the septimal or Archytan comma (9/7 vs. 81/64), not
the syntonic comma of meantone at 81:80 (5/4 vs. 81/64).

Here are some examples of isotones at different points along the
regular diatonic continuum, with the medieval and later Dorian
mode, LsLLLsL, used to illustrate the common pattern.

Inframeantone, 689c

0 178 333 511 689 867 1022 1200
178 155 178 178 178 155 178
L s L L L s L

In an inframeantone as I define it, the regular larger and
smaller thirds, here 356 and 333 cents, will both be in a range
between 330 and 372 cents which might be termed neutral, or, near
the low and high ends, "semi-neutral" or supraminor/submajor.
Note that the tone (L) and semitone (S) differ in size by only
23 cents, but do fit a regular diatonic pattern! For the 356-cent
major or rather largish neutral third, close for example to 27/22
(355 cents), we might speak of "tempering out" the 33:32 diesis
(53 cents), the difference between four 3/2 fifths up at 81/64
and a 27/22 neutral third.

Meantone, 696c (50-EDO)

0 192 312 504 696 888 1008 1200
192 120 192 192 192 120 192
L s L L L s L

This is a typical meantone not too far from Zarlino's 2/7-comma,
in the "central" range of around 1/3-1/6 comma favored for
keyboards in 16th-17th century Europe. The term "syntonic
temperament," as has been observed, would seem logically
synonymous with meantone, since the 81/80 is tempered out to
produce regular thirds at or close to 5/4 and 6/5.

Pythagorean, 3/2 or 701.955c

1/1 9/8 32/27 4/3 3/2 27/16 16/9 2/1
0 204 294 498 702 906 996 1200
9:8 256:243 9:8 9:8 9:8 256:243 9:8
204 90 204 204 204 90 204

This is the classic Pythagorean diatonic of medieval Europe, for
example, which doesn't temper out the 81:80 but has major and
minor thirds at 408 and 294 cents, a syntonic comma larger and
smaller respectively than 5/4 and 6/5. To call this a "syntonic"
temperament or mapping seems to me very strange!

Superpythagorean undecimal/tredecimal, 704c

0 208 288 496 704 912 992 1200
208 80 208 208 208 80 208
L s L L L s L

This is a superpythagorean temperament of what we might call the
"undecimal/tredecimal" variety -- if we are naming based on the
type of regular thirds produced -- since 416 cents is close to
the "undecimal" 14/11, and 288 cents to the "tredecimal" 13/11,
at 417.5 and 289.1 cents (following Scala). We're tempering out
the 352:351 (13/11 vs 32/27) and 896:891 (14/11 vs. 81/64).

Superpythagorean septimal, 709c (a la Paul Erlich)

0 218 273 491 709 927 982 1200
218 55 218 218 218 55 218
L s L L L s L

This a superpythagorean temperament very close to 22-EDO where,
as you have noted, the major third is close to 9/7 (435 cents),
and the minor third to 7/6 (267 cents) -- here 436 and 273 cents.
The comma being tempered out is 64:63, the septimal or Archytan
comma.

Superpythagorean interseptimal, 714c

0 228 258 486 714 942 972 1200
228 30 228 228 228 30 228
L s L L L s L

This temperament with the fifth at 714 cents might be styled
"interseptimal," because the regular smaller minor or subminor
third at 258 cents is in the "interseptimal" region between 8/7
(231 cents) and 7/6 (267 cents); and the major or supramajor
third likewise at 456 cents in the range between 9/7 (435 cents)
and 21/16 (471 cents). The diatonic semitone at 30 cents is very
small but still recognizable. If we take the 456-cent third as
representing 13/10, then we might speak of tempering out 416:405,
at 46 cents, the difference between 81/64 and 13/10.

For each of these regular diatonic scales, a "major third" from
four fifths up is formed from two whole tones or major seconds of
identical size, the pattern which "isotone" describes in the most
general sense, with "meantone" as one type of isotone:

System 5th tone Maj3

Inframeantone 689c 178c 356c

Meantone 696c 192c 384c

Pythagorean 702c 204c 408c

Superpyth/ 704c 208c 416c undecimal-tredecimal

Superpyth/ 709c 218c 436c
septimal

Superpyth/ 714c 228c 456c interseptimal

Best,

Margo
mschulter@...

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@> wrote:
> I have since tried to learn whatever I can about temperament theory, and microtonal theory in general, and have always found my own conclusions aligning themselves with those of Sethares/Milne/Plamondon.
>
> Which conclusions were those? If you mean the tuning continua paper, that obviously was heavily influenced by tuning-math@yahoo, so it's not another, unrelated, circle of ideas to what goes on here.

I don't mean to say that I agree with them, *as opposed to what the general consensus is here*. My only intent in saying that was as a response to the question as to whether or not I was one of Bill's students. I'm not, but I've gotten a lot of information from him and tend to agree with it and then mirror it in my own discussions, which might explain why it would seem like I were one of his students.
Sorry for the confusion.

John

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:

>My only intent in saying that was as a response to the question
>as to whether or not I was one of Bill's students.

Actually I asked if you were one of Andy's students. :)

-Carl

No one can talk any sense to me for the next little while, because I just tried 46EDO for the first time! I think I'm in love.

I feel like Goldilocks--this tuning is just right.

5ths nicely wide. Sweet thirds. The scale is not too big, not too small. It even has a symmetrical tritone. Familiar scales all sound good in it.

I see that this is a well-known temperament, but for some reason I'd never tried it.

caleb

> ...
> Dear Mike (and Caleb),
>
> ...
> Best,
>
> Margo
> mschulter@...
>
>

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

> What we'd also like is a term analogous to "meantone," but
> describing any tuning in this regular diatonic range where a
> "major third" from four fifths up (less two octaves) is formed
> from two identically sized "major seconds" or "tones," each in
> turn formed from two fifths up (less an octave).
>
> Here I propose the term "isotone," meaning that a major third is
> formed from two tones of the "same" size.

A related idea is to name the MOS pattern, for instance by calling it "Sleepy" or "diatonic". Another way of naming would be to call it "5L2s" or "4/7"; how these two are equivalent is explained here:

http://xenharmonic.wikispaces.com/MOSScales

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
>
> No one can talk any sense to me for the next little while, because I just tried 46EDO for the first time! I think I'm in love.
>
> I feel like Goldilocks--this tuning is just right.
>
> 5ths nicely wide. Sweet thirds. The scale is not too big, not too small. It even has a symmetrical tritone. Familiar scales all sound good in it.
>
> I see that this is a well-known temperament, but for some reason I'd never tried it.

Try it! The only music for it I could find for this article

http://xenharmonic.wikispaces.com/46edo

was by me. It could use someone else, but then, there are so many tuning systems which have not been tried so far as I know.

I like "music for your ears" - especially the 46 edo portions

And chromosounds - a big blues / jazz band treatment is really nice!

Chris

On Mon, Aug 30, 2010 at 5:54 PM, genewardsmith
<genewardsmith@...>wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, caleb morgan
> <calebmrgn@...> wrote:
> >
> >
> > No one can talk any sense to me for the next little while, because I just
> tried 46EDO for the first time! I think I'm in love.
> >
> > I feel like Goldilocks--this tuning is just right.
> >
> > 5ths nicely wide. Sweet thirds. The scale is not too big, not too small.
> It even has a symmetrical tritone. Familiar scales all sound good in it.
> >
> > I see that this is a well-known temperament, but for some reason I'd
> never tried it.
>
> Try it! The only music for it I could find for this article
>
> http://xenharmonic.wikispaces.com/46edo
>
> was by me. It could use someone else, but then, there are so many tuning
> systems which have not been tried so far as I know.
>
>
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I like "music for your ears" - especially the 46 edo portions
>
> And chromosounds - a big blues / jazz band treatment is really nice!

Thanks, Chris!

Dear Caleb,

Congratulations on finding and liking 46-EDO!

By a curious coincidence, ten years ago today, I first used
a 17-note circulating temperament which had its first 12 notes
in an almost identical temperament, with the fifth at
704.377 cents, producing a pure 14/11 major third. As I
later found out, George Secor in 1978 had developed a
17-note well-temperament having eight of its fifths at
this same size, almost identical to 46-EDO.

You mentioned "sweet thirds," and actually there are
four families or varieties of thirds present for you
to explore, here using the Dorian mode for these examples.
Each example shows cents and 46-EDO steps.

Regular or isotone (417/287 cents): L=8, s=3

0 208.7 287.0 495.7 704.3 913.0 991.3 1200.0
0 8 11 19 27 35 38 46
L s L L L s L
8 3 8 8 8 3 8

Submajor/supraminor or Zalzalian (365/339 cents): L=8, M=6, s=5

0 208.7 339.1 495.7 704.3 860.9 991.3 1200.0
0 8 13 19 27 33 38 46
L s M L M s L
8 5 6 8 6 5 8

Septimal or interseptimal (443/261 cents): L=9, M=8, s=2

0 208.7 260.9 495.7 704.3 913.0 965.2 1200.0
0 8 10 19 27 35 37 46
M s L M M s L
8 2 9 8 8 2 9

Pental or 5-limit (391/313 cents): L=8, M=7, s=4

0 208.7 313.0 495.7 704.3 913.0 1017.4 1200.0
0 8 12 19 27 35 39 46
L s M L M s L
8 4 7 8 7 4 8

So there are lots of shadings here, but with the first
version using L=8, s=3, as the only one with only two
step sizes. This is the "isotone" I mentioned, like a
meantone, but with a major third at 14/11 rather than 5/4.

And you might want to try a couple of Near Eastern
modes in 46-EDO also. The intervals of 5 and 6 steps,
or around 130 and 156 cents, will often occur in such
modes. The subtle difference between these two neutral
or middle second steps is something favored by many
traditional performers, although the exact tunings
will, of course, vary.

Rast (Zalzal's medieval tuning)

0 208.7 365.2 495.7 704.3 860.9 991.3 1200.0
0 8 14 19 27 33 38 46
L M s L M s L
8 6 5 8 6 5 8

Rast (most common modern version)

0 208.7 365.2 495.7 704.3 913.0 1069.6 1200.0
0 8 14 19 27 35 41 46
L M s L L M s
8 6 5 8 8 6 5

In short, you have _lots_ of options in 46-EDO, and may
find it interesting to see which you prefer most, and
when.

Best,

Margo
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Dear Caleb,
>
> Congratulations on finding and liking 46-EDO!

It is about as close to perfect as ETs get (along with 12, 22,
31, 41, and 72).

-Carl

On 31 August 2010 04:20, cityoftheasleep <igliashon@...> wrote:
> --- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
>> ANY tuning that defines 81/80 to be equal:
>> 1. Defines four 3/2's minus two 2/1's to be equal to 5/4
>> 2. Is a syntonic tuning
>
> This definition could actually apply to any tuning, unless
> we account for error and consistency.   First we have to
> define how much you can temper a 3/2 before it stops
> being a 3/2, and how much we can temper a 5/4 before
> it stops being 5/4.

It could apply to any tuning, yes, so it defines what I call a
"temperament class" (assuming "be equal" means tempered out") -- and
what Sethares et. al. call a "regular temperament" -- that looks
suspiciously like meantone. The mapping is not a property of a
tuning, so I'd rather not class tunings by mappings. I call a tuning
of a temperament class a "temperament" so this would define a meantone
or syntonic temperament. Unfortunately the term is contentious.

I don't know what "consistency" means here. You keep saying it but I
only know the definition for equal temperaments.

I don't think you need to say where a fifth stops being a 3/2. You
can attach the mapping to any tuning with the appropriate structure,
and call it a meantone. Maybe it's a lousy meantone. The error tells
you how lousy it is and everybody can make their own decision how high
an error means the temperament's so lousy that it makes no sense at
all to apply that mapping.

I noticed on Wikipedia, instead of a fight about whether the meantone
page should be called "meantone" or "syntonic", there are two
different pages with slightly different definitions. That means it
looks like they're both assimilated into the same theory. In fact, I
don't know why you'd need both terms. If you want to talk about
meantones in a given tuning range, you can say "typical meantones" or
something like that. Defining exactly what you mean in context, of
course.

There really isn't a need, in general, to say what error is too high.
If you're talking about meantones, what you say should naturally apply
to the good ones, not the lousy ones.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
I call a tuning
> of a temperament class a "temperament" so this would define a meantone
> or syntonic temperament. Unfortunately the term is contentious.

For instance I call it a tuning, and what Graham calls a "temperament class" a temperament or abstract temperament. I think "temperament class" is a confusing name, and take "2/7 comma meantone" to refer to a kind of meantone, a tuning of meantone temperament, not a member of a class of meantones. So far as I can see, if there's only one meantone then it had better be 1/4 comma, but then what are the others?

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

> By a curious coincidence, ten years ago today, I first used
> a 17-note circulating temperament which had its first 12 notes
> in an almost identical temperament, with the fifth at
> 704.377 cents, producing a pure 14/11 major third. As I
> later found out, George Secor in 1978 had developed a
> 17-note well-temperament having eight of its fifths at
> this same size, almost identical to 46-EDO.

These days we've been calling the corresponding regular temperament "leapday": <<1 21 15 11 8 ...||.

Hi Graham,

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> I don't know what "consistency" means here. You keep saying it but I
> only know the definition for equal temperaments.

I may be misusing it, but I thought it was what Herman was using to say whether a given generator might produce a better approximation to a given prime than the one defined by the mapping. I.e. a meantone with a generator of 690 cents is inconsistent because it gives a better 5/4 at 11 fifths minus 6 octaves than it does at 4 fifths minus 2 octaves. I'll have to go back and re-read that whole section of posts to see if I just read the word wrong or something.

> I don't think you need to say where a fifth stops being a 3/2. You
> can attach the mapping to any tuning with the appropriate structure,
> and call it a meantone. Maybe it's a lousy meantone. The error tells
> you how lousy it is and everybody can make their own decision how high
> an error means the temperament's so lousy that it makes no sense at
> all to apply that mapping.

Right. Well, it may not be necessary, but that doesn't mean people don't want to set those boundaries. Psychoacoustically, it is possible to say when a fifth stops sounding like a fifth, or to at least give a general range where it becomes fuzzy. I know most of you temperament guys are interested in the minimum error temperaments, so it's of little concern to you where one ends and another begins (or if one really "ends" anywhere at all), since the least-error temperament tunings don't exactly straddle the boundary between temperament classes. However, I am not interested in minimal-error temperaments, I'm interested in investigating scalar boundaries. It's undeniable that different temperaments that come from different temperament classes can produce music that sounds similar, and I don't think regular mapping explains this phenomenon adequately--if at all. So I think that it's a misnomer that Sethares et al call the range of generators that runs from 4/7ths of an octave to 3/5ths of an octave the "Syntonic Temperament" continuum. Anything generators in that range have to do with the syntonic comma is incidental, I think.

-Igs

Gene wrote:

> For instance I call it a tuning, and what Graham calls a
> "temperament class" a temperament or abstract temperament.
> I think "temperament class" is a confusing name, and take
> "2/7 comma meantone" to refer to a kind of meantone, a
> tuning of meantone temperament, not a member of a class
> of meantones.

Indeed. -Carl

> Actually I asked if you were one of Andy's students. :)

Oops! Not him either...

On 31 August 2010 10:37, cityoftheasleep <igliashon@...> wrote:

> I may be misusing it, but I thought it was what
> Herman was using to say whether a given generator
> might produce a better approximation to a given
> prime than the one defined by the mapping.  I.e. a
> meantone with a generator of 690 cents is
> inconsistent because it gives a better 5/4 at 11 fifths
> minus 6 octaves than it does at 4 fifths minus
> 2 octaves.  I'll have to go back and re-read that
> whole section of posts to see if I just read the word
> wrong or something.

I thought the objection to that was that an irrational generator will
always give you a better approximation (to any interval that isn't
perfect) if you go far enough. So the definition's incomplete if it's
going to be exact. And I'm not sure it can be made exact in an
entirely satisfactory way.

You can say that Lucy Tuning is plainly a meantone and not a
schismatic, because the meantone approximation is both simpler and
more accurate than the schismatic one, for this tuning. You can also
say that Pythagorean intonation is schismatic and not meantone, but by
what rule? The schismatic approximation is more accurate here, but
the meantone one is still simpler.

> Right. Well, it may not be necessary, but that
> doesn't mean people don't want to set those
> boundaries.  Psychoacoustically, it is possible to
> say when a fifth stops sounding like a fifth, or to
> at least give a general range where it becomes
> fuzzy.  I know most of you temperament guys are
> interested in the minimum error temperaments, so
> it's of little concern to you where one ends and
> another begins (or if one really "ends" anywhere at
> all), since the least-error temperament tunings
> don't exactly straddle the boundary between
> temperament classes. However, I am not interested
> in minimal-error temperaments, I'm interested in
> investigating scalar boundaries.  It's undeniable
> that different temperaments that come from different
> temperament classes can produce music that
> sounds similar, and I don't think regular mapping
> explains this phenomenon adequately--if at all.  So I
> think that it's a misnomer that Sethares et al call
> the range of generators that runs from 4/7ths of an
> octave to 3/5ths of an octave the "Syntonic
> Temperament" continuum.  Anything generators in
> that range have to do with the syntonic comma is
> incidental, I think.

Yes, I understand that people want to set boundaries. What I'll say
is that there are no natural boundaries that everybody will agree on,
and so if you want do draw your own you can use your own rules and
there's no need to discuss it here. If you want to bring in
psychoacoustics, you'll have to research it. From what I've seen it
doesn't address this problem.

We're interested in good temperaments. Why would you be care about
bad ones? If you're going to draw boundaries, you'll want the good
tunings to lie within them. You can also have fuzzy boundaries, and
say that Pythagorean intonation may be heard as a meantone but is more
likely to be schismatic.

Regular mappings would suggest that scales sound similar if they
approximate the same intervals.

Things that temper out the syntonic comma, and sound reasonable, will
all lie within this Syntonic Temperament continuum. Maybe the
boundaries are too wide, but that's not an interesting thing to argue
about.

Diaschismic temperaments are more interesting. I cover them here,
with outdated terminology:

http://x31eq.com/diaschis.htm

There's a graph showing two different approximations to 7:4, with the
natural boundary at 34-equal (itself inconsistent). What I left off
at the time was the simpler Pajara (Erlich decatonic) approximation
because at the time I thought it was peculiar to 22-equal. But the
consensus now is that the whole range to the right of 34-equal also
contains valid Pajara tunings, and the optimal tuning is somewhere in
the middle. So at this point there are two different 7-limit
approximations that work for the same tuning, both of them valid, and
both of them usable in the same piece. I see no reason to draw a line
between them.

Another thing about that graph is that 12-equal is off the left-hand
side, and so appears to be way outside the Pajara range. But in so
far as it works with 7-limit harmony (and some think it does) it's
consistent with Pajara. In fact, because Pajara's 7:4 is so simple,
it doesn't change dramatically with tuning, and so for small fifths
it's once again the best of the three approximations. So we could
segment Diaschismic tunings into:

Pajara | 58&46 | mixed | Pajara

Perhaps the conclusion is that tuning ranges don't imply mappings.
You have to make the mapping implicit, and then some kind of tuning
range (that may overlap with other mappings) follows from it.

Graham

On Mon, Aug 30, 2010 at 4:50 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Any scale with two step-sizes can be arranged as an MOS or not. Any scale with three step-sizes, assuming it's an equal temperament, can be broken down to a larger scale with two step-sizes.

So how do you take 72-equal's "just" major scale and break it down
into a larger scale with two step-sizes? Of course, you could also
cheat and just say it's a subset of 72-equal itself, with 1/72 as the
generator.

> So even if a scale is not explicitly MOS, it can be related to an MOS scale, and relating everything back to MOS's is a very easy way to figure out how different tunings will support a given scale. Any tuning that supports a scale of 4L+4s will support any arrangement of those steps, be it ssssLLLL or ssLLsLsL or whatever. But they will all support LsLsLsLs also, which means that if you can find tunings that support the MOS scale, the very same tunings will support the scale's non-MOS permutations. And in fact the only way to "generate" those non-MOS scales is to treat them as permuted MOS scale, or as non-MOS subsets of larger MOS scales. Otherwise you're choosing notes at random.

Right. I think what this is pointing to is that the sLsLLsLs 12-tet
incarnation of this would probably be better off having 3 step sizes,
two of which are equated in 12-equal, and the comma between them
tempered out.

> No, that's not it. Call it whatever you want, define it as broadly as you want, there is no non-arbitrary way to make any category based on tempering have a 1-to-1 correspondence with a category based on scale configuration.

Why? We've already done it. The category of linear temperaments with
generators between 3/5 and 4/7 and periods of an octave ends up
meeting the criteria.

> To somehow lump all LLsLLLs scales together into a temperament class or super-class, you'd have to do the following things:
> 1: find some mapping that they all have in common
> 2: define your error range such it gives generators only between 4/7ths of an octave and 3/5ths of an octave, which will be arbitrary
> 3: ignore consistency completely

I disagree that a common mapping is necessary. Superpyth and meantone
have two completely different mappings, but they both produce 5+2
scales, even if those aren't specifically tied into the 5-limit
anymore.

> The unifying feature here is the scale shape, not the mapping; temperament is ALL ABOUT MAPPING. If the mapping is different, the temperament is different. OTOH, Moment of Symmetry is ALL ABOUT SCALE SHAPE, and could give a damn about mapping or JI.

I still don't get what you're so resistant to this idea for. Nothing
I'm saying changes anything about regular mapping, it's just that you
can show how temperaments are related through the MOS's they produce.
The fact that superpyth aeolian and meantone aeolian have as much of a
similarity as they do suggests that this would be a perceptually
useful grouping to make.

I think, after doing some "light reading" (lol, that's a joke), I
think this is really the same basic thing that Graham has proposed by
giving the MOS scales names to begin with. (And what names they are.)

So perhaps a database of what MOS's are supported by what linear
temperaments would be helpful. I need to work on that.

> And there you have it! Yes, the superset is just everything that 12-tET supports, by definition.

Yeah, I realize that now. I guess that approach to organizing
temperaments isn't the way to go - I'd rather go with the above
approach instead.

> Yep. In which case, the sky's the limit! You can find some way to play this scale with 3 or 4 step sizes in probably any EDO above 12. So you could just look at EDOs that have good harmonic characteristics, and shoe-horn your scale into them and see what happens. But you won't get any guiding light from temperament or MOS theory at this point...you're basically looking at raw JI, which can be tempered any which way at all.

Indeed. What I really wish I had were some tools to make all of this
easier. Maybe tonalsoft's the way to go... What have you been using?

-Mike

Graham wrote:

> Igs wrote:
>
> > I may be misusing it, but I thought it was what
> > Herman was using to say whether a given generator
> > might produce a better approximation to a given
> > prime than the one defined by the mapping.  I.e. a
> > meantone with a generator of 690 cents is
> > inconsistent because it gives a better 5/4 at 11 fifths
> > minus 6 octaves than it does at 4 fifths minus
> > 2 octaves.  I'll have to go back and re-read that
> > whole section of posts to see if I just read the word
> > wrong or something.
>
> I thought the objection to that was that an irrational
> generator will always give you a better approximation (to any
> interval that isn't perfect) if you go far enough. So the
> definition's incomplete if it's going to be exact. And I'm not
> sure it can be made exact in an entirely satisfactory way.

If I may interject, you read it right, Igs. You forgot
above the part about reporting the length of the chain
required to invalidate the map. But Graham may have been
thinking of that anyway, since it's not entirely satisfactory,
since it produces the nested boundaries we discussed.

However it occurs to me that instead of using error, why
not try badness? Graham all but suggests it:

> You can also say that Pythagorean intonation is schismatic
> and not meantone, but by what rule? The schismatic
> approximation is more accurate here, but the meantone one
> is still simpler.

As I am packing my car currently for morning departure,
I don't have time to compute examples, but it seems that
badness might make the boundaries in Herman's charts more
uh, convex. After all, we used badness to name the
temperaments in the first place, so there ought to be a
way to use it to recover tuning ranges for them.

Peace!

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> So how do you take 72-equal's "just" major scale and break it down
> into a larger scale with two step-sizes? Of course, you could also
> cheat and just say it's a subset of 72-equal itself, with 1/72 as the
> generator.

There are a few ways. Simply put, break each of the three step-sizes down until you only have 2 step sizes. The steps in 72's Just Major are 12, 11, and 7; all can be reduced to a series of 2's and 3's. So one configuration is 14L+15s, if you make 12 into 3+2+3+2+2; another is 20L+6s if you make 12 into 3+3+3+3. They're pretty big and you could make smaller subsets of them if you wanted, but there you have it. You could tune the former scale in 43-EDO and the latter variation in 46-EDO. But I guess this goes to show you that sometimes, when dealing with scales with more than 2 step-sizes (what's the word for only having two step-sizes again? I can never keep all those terms straight) and especially in higher EDOs, there can be more than one "parent" scale.

> Right. I think what this is pointing to is that the sLsLLsLs 12-tet
> incarnation of this would probably be better off having 3 step sizes,
> two of which are equated in 12-equal, and the comma between them
> tempered out.

Whatever you desire, Mike. It's your scale, you can take it anywhere!

> Why? We've already done it. The category of linear temperaments with
> generators between 3/5 and 4/7 and periods of an octave ends up
> meeting the criteria.

Defined that way, that's not a category of linear temperaments. At least, not in the way the term "linear temperament" is used here.

> > To somehow lump all LLsLLLs scales together into a temperament class or super-class, you'd have to do the following things:
> > 1: find some mapping that they all have in common
> > 2: define your error range such it gives generators only between 4/7ths of an octave and 3/5ths of an octave, which will be arbitrary
> > 3: ignore consistency completely

> I disagree that a common mapping is necessary. Superpyth and meantone
> have two completely different mappings, but they both produce 5+2
> scales, even if those aren't specifically tied into the 5-limit
> anymore.

Without a mapping, we're no longer talking about temperament, or anything related to temperament. You keep trying to use the term "temperament" in a way that is inconsistent with how it's defined here.

> > The unifying feature here is the scale shape, not the mapping; temperament is ALL ABOUT MAPPING. If the mapping is different, the temperament is different. OTOH, Moment of Symmetry is ALL ABOUT SCALE SHAPE, and could give a damn about mapping or JI.
>
> I still don't get what you're so resistant to this idea for. Nothing
> I'm saying changes anything about regular mapping, it's just that you
> can show how temperaments are related through the MOS's they produce.
> The fact that superpyth aeolian and meantone aeolian have as much of a
> similarity as they do suggests that this would be a perceptually
> useful grouping to make.

I'm not resistant to anything, except imprecise or misused language. You keep wanting to call an MOS family a temperament super-class, it seems, and I don't understand why. Temperament classes have no natural boundaries, but MOS families do. Anything can be a meantone, anything can be a superpyth, but not everything can be 5L+2s. It happens to be true that there are several temperaments whose optimal generators all fall in the same range that produces MOS scales of 5L+2s, but those temperaments also have non-optimal generators that fall within every other MOS family. And not nearly every generator in the 5L+2s MOS family will be an optimal or near-optimal generator for a temperament-class.

But sure, you could certainly look at a list of optimal and near-optimal generators for different temperament classes and index them according to MOS families they belong to.
It'd be a bit tedious, though. And MOS families are actually somewhat difficult to pin down unless you set a maximum number of notes. It is interesting to note, for instance, that 19-EDO and 22-EDO share the same 7-note LLsLLLs scale, but have opposite 12-note scales (19's is LsLsLLsLsLsL, or 7L+5s, whereas 22's is LsLssLsLsLss, or 5L+7s)--are they REALLY in the same MOS family? Likewise, 22 and 27 have the same 12-note MOS, but a different 22-note MOS. So if we group only according to a 7-note MOS, we may be blurring important distinctions. So now we have the burden of arbitrarily defining a limit to MOS size, or else indexing each tuning according to multiple levels of MOS.

> I think, after doing some "light reading" (lol, that's a joke), I
> think this is really the same basic thing that Graham has proposed by
> giving the MOS scales names to begin with. (And what names they are.)

Actually, those names were a result of me asking the list if there was a nice precise "scientific-sounding" way to name different MOS families several years back. There wasn't, so Graham furnished the next best thing. After years of consideration and trying (and failing) to find better alternatives, I've come to prefer the nL+ms approach. I do occasionally nurture a pipe-dream of concocting a true taxonomy of MOS scales, with 3- to 5-note scales being the "Kingdom", 6- to 9-note scales being the "Phylum", etc. But the exactly-equal scales throw me off, since they are between branches.

> So perhaps a database of what MOS's are supported by what linear
> temperaments would be helpful. I need to work on that.

"What MOS'S are supported by what optimal (or near-optimal) *tunings* of what linear temperaments" is how you should have wrote that. Remember, a temperament is just a mapping; you need to define error to get a tuning, and without having the tuning of a temperament, there's no way to categorize it in terms of MOS family.

> Yeah, I realize that now. I guess that approach to organizing
> temperaments isn't the way to go - I'd rather go with the above
> approach instead.

I wonder what the name for that approach is...sort of brute-forcing a JI scale to the nearest approximation each step has in a given tuning, regardless of whether the tuning is consistent to the same limit as the JI scale.

> Indeed. What I really wish I had were some tools to make all of this
> easier. Maybe tonalsoft's the way to go... What have you been using?

Uh...TextEdit and Calculator? When I had a PC, I would occasionally use Scala, but not for anything analytical or what have you, just to tune up different scales to play on my computer keyboard (before I had any other retunable synths and MIDI controllers). If I can't work it out with a pencil, paper, and calculator, I'm not interested in it. And my math education ended at Calculus 1. Does that explain me a little better?

-Igs

On Tue, Aug 31, 2010 at 3:16 AM, cityoftheasleep
<igliashon@...> wrote:
>
> (what's the word for only having two step-sizes again? I can never keep all those terms straight)

Isn't that MOS?

> > Why? We've already done it. The category of linear temperaments with
> > generators between 3/5 and 4/7 and periods of an octave ends up
> > meeting the criteria.
>
> Defined that way, that's not a category of linear temperaments. At least, not in the way the term "linear temperament" is used here.

Why not? If it has a single generator and a single period, it's a
linear temperament. And I just placed them within a category, with my
category superpowers. Besides, after Margo just fleshed all of that
out... what's the debate about at this point? :)

> > I disagree that a common mapping is necessary. Superpyth and meantone
> > have two completely different mappings, but they both produce 5+2
> > scales, even if those aren't specifically tied into the 5-limit
> > anymore.
>
> Without a mapping, we're no longer talking about temperament, or anything related to temperament. You keep trying to use the term "temperament" in a way that is inconsistent with how it's defined here.

I never said that we weren't going to have a mapping. I said that not
every temperament that produces 5+2 scales has to have a common
mapping. Superpyth and meantone are two examples of temperaments with
different mappings that still produce 5+2 scales. You seem to be
agreeing with me, so I don't know why you're disagreeing with me.

> > I still don't get what you're so resistant to this idea for. Nothing
> > I'm saying changes anything about regular mapping, it's just that you
> > can show how temperaments are related through the MOS's they produce.
> > The fact that superpyth aeolian and meantone aeolian have as much of a
> > similarity as they do suggests that this would be a perceptually
> > useful grouping to make.
>
> I'm not resistant to anything, except imprecise or misused language. You keep wanting to call an MOS family a temperament super-class, it seems, and I don't understand why.

If the term for what I am describing already exists and is called "MOS
family," and meantone and superpyth can be said to belong to the
diatonic MOS family, then I'll use that term instead. I used the
phrase "temperament class" to indicate that the concept I was getting
across was a way of categorizing temperaments into groups or "classes"
with perceptual similarities. But then it turns out that Graham had
already used the phrase "temperament class" to describe something like
meantone or porcupine, so I started using the term "temperament
superclass" to indicate that it was a group of "temperament classes"
with perceptual similarities. Now you're saying that it's called an
MOS family, so if it is then I'll start using that term. I don't
really care what it's called.

The reason I picked the term "temperament superclass" is that
superpyth and meantone are obviously very much perceptually related in
a way that meantone and porcupine are not, which lends itself to a
taxonomy whereby there is some larger concept of which meantone and
superpyth are subcategories.

> Temperament classes have no natural boundaries, but MOS families do. Anything can be a meantone, anything can be a superpyth, but not everything can be 5L+2s. It happens to be true that there are several temperaments whose optimal generators all fall in the same range that produces MOS scales of 5L+2s, but those temperaments also have non-optimal generators that fall within every other MOS family. And not nearly every generator in the 5L+2s MOS family will be an optimal or near-optimal generator for a temperament-class.

Right, I understand very well that you could say that a meantone
generator is 1199 cents and the period is 1200 cents. I'm not dodging
that point because I don't understand it, I'm dodging it because I
think we all have a decent "common sense" definition of what
constitutes a meantone and what doesn't, and that rigorously defining
it isn't necessary at this point to flesh out how temperaments fall
into larger perceptual categories.

> But sure, you could certainly look at a list of optimal and near-optimal generators for different temperament classes and index them according to MOS families they belong to.

aaaaaaa HAAAA! You admit it! :)

> It'd be a bit tedious, though. And MOS families are actually somewhat difficult to pin down unless you set a maximum number of notes. It is interesting to note, for instance, that 19-EDO and 22-EDO share the same 7-note LLsLLLs scale, but have opposite 12-note scales (19's is LsLsLLsLsLsL, or 7L+5s, whereas 22's is LsLssLsLsLss, or 5L+7s)--are they REALLY in the same MOS family?

And likewise, mavila and meantone share the same 5 note ssLsL scale,
but have opposite 7-note MOS's. So mavila and meantone and superpyth
would both be in the 2+3 MOS family, but mavila wouldn't be in the 5+2
MOS family, and the remaining two would diverge for the 12-note MOS
family.

> Likewise, 22 and 27 have the same 12-note MOS, but a different 22-note MOS. So if we group only according to a 7-note MOS, we may be blurring important distinctions. So now we have the burden of arbitrarily defining a limit to MOS size, or else indexing each tuning according to multiple levels of MOS.

That is a good point. What do you think would be best?

An idea I just had - perhaps that's one common sense way to delimit a
temperament's range. So we could say that meantone is only meantone if
the generator produces both 5+2 and 7+5 scales, (or equal variants
thereof). This means that a lower bound for meantone is 7-tet,
exclusive, and an upper bound for meantone is 12-tet, inclusive.

Yes... I like this idea a lot :) Seems like it might lead to something
interesting. I also wonder, just as you can specify a linear
temperament by two ET's, what can you specify by two MOS's? Some
higher-limit generalization of MOS?

> Actually, those names were a result of me asking the list if there was a nice precise "scientific-sounding" way to name different MOS families several years back. There wasn't, so Graham furnished the next best thing. After years of consideration and trying (and failing) to find better alternatives, I've come to prefer the nL+ms approach. I do occasionally nurture a pipe-dream of concocting a true taxonomy of MOS scales, with 3- to 5-note scales being the "Kingdom", 6- to 9-note scales being the "Phylum", etc. But the exactly-equal scales throw me off, since they are between branches.

What really gets me is how really fundamental to musical perception
the scale taxonomy is. I would support this initiative and would like
to do some work on it myself. After spending all of this time with
regular mapping, my head was turned completely upside down when Paul
pointed out that superpyth minor scales and meantone minor scales
sounded perceptually "the same." So after thinking that JI was some
kind of underlying reality that regular mapping was "warping," I don't
even think that anymore.

That is, the question really is - do JI intervals and triads have
their own inherent "identity?" Is it that tempering "connects"
different pieces of the JI lattice and smear identities together? Or
is there something else fundamentally going on? Paul's example
certainly seems to suggest the latter. Thus your proposed MOS taxonomy
might be a more useful way of categorizing this particular element of
musical perception, with regular mapping then taking the role of
finding the generator that produces the most concordant chords - which
is a completely different level of perception, and may be more
reducible to a simple notion of "concordant" and "discordant" with
different JI intonations just implying subtly different flavors.

Perhaps a larger discussion of "the big picture" is really necessary
to produce your taxonomy.

> > So perhaps a database of what MOS's are supported by what linear
> > temperaments would be helpful. I need to work on that.
>
> "What MOS'S are supported by what optimal (or near-optimal) *tunings* of what linear temperaments" is how you should have wrote that. Remember, a temperament is just a mapping; you need to define error to get a tuning, and without having the tuning of a temperament, there's no way to categorize it in terms of MOS family.

Well, if we tie meantone down to having generators that can produce
both the 5+2 and 7+5 families, there goes that problem :) And I think
that's pretty close to how we all intuitively decide to stop calling
something meantone anyway.

So if we place those limits on what are allowable generators for a
linear temperament, then it implies a natural taxonomy by itself.
Meantone and superpyth would be related in that they're both part of
the "diatonic" (or isotonic if we use Margo's terminology) family, but
meantone would be a part of the "chromatic" family, whereas superpyth
would be a part of the inverted chromatic family. And there you go,
taxonomy :)

> Uh...TextEdit and Calculator? When I had a PC, I would occasionally use Scala, but not for anything analytical or what have you, just to tune up different scales to play on my computer keyboard (before I had any other retunable synths and MIDI controllers). If I can't work it out with a pencil, paper, and calculator, I'm not interested in it. And my math education ended at Calculus 1. Does that explain me a little better?
>
> -Igs

Haha. Well, you have a very good conceptual head about all of this,
and your explanations have been much appreciated.

-Mike

On Mon, Aug 30, 2010 at 5:10 PM, Margo Schulter <mschulter@...> wrote:
>
> What we'd also like is a term analogous to "meantone," but
> describing any tuning in this regular diatonic range where a
> "major third" from four fifths up (less two octaves) is formed
> from two identically sized "major seconds" or "tones," each in
> turn formed from two fifths up (less an octave).
>
> Here I propose the term "isotone," meaning that a major third is
> formed from two tones of the "same" size.

Sounds good to me. Either that or "diatonic" works. But is a
superpythagorean LLsLLLs still considered "diatonic" around these
parts?

> This temperament with the fifth at 714 cents might be styled
> "interseptimal," because the regular smaller minor or subminor
> third at 258 cents is in the "interseptimal" region between 8/7
> (231 cents) and 7/6 (267 cents); and the major or supramajor
> third likewise at 456 cents in the range between 9/7 (435 cents)
> and 21/16 (471 cents). The diatonic semitone at 30 cents is very
> small but still recognizable.

And how remarkable that last statement is. What implications it has
for psychoacoustics I wish I fully understood.

Thanks for working all of this out, Margo, this is exactly what I had in mind.

-Mike

I have not been happy with how the MOS family thing has come about. If you see where these scales fall on the scale tree this confusion disappears for me as one knows that a particular scale template is found on all underneath but not vice versa.
Perhaps MOS 'branch' i think is a better idea than 'family'.
So maybe temperment branch wouldn't bother me not that class really does.
I n example
2/5-3/7 branch which also would also include these generalized keyboards

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 31/08/10 6:41 PM, Mike Battaglia wrote:

This is why the term family is confusing.
if you look at page 8 on the scale tree http://anaphoria.com/sctree.PDF
both the 8)19 scale and the 9)22 scales are both under the 3)7 in the same direction but they branch into different direction once one gets to the 5)12 which is why the L and s trade places

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 31/08/10 6:41 PM, Mike Battaglia wrote:
>
>
>
> aaaaaaa HAAAA! You admit it! :)
>
> > It'd be a bit tedious, though. And MOS families are actually > somewhat difficult to pin down unless you set a maximum number > of notes. It is interesting to note, for instance, that 19-EDO > and 22-EDO share the same 7-note LLsLLLs scale, but have > opposite 12-note scales (19's is LsLsLLsLsLsL, or 7L+5s, > whereas 22's is LsLssLsLsLss, or 5L+7s)--are they REALLY in > the same MOS family?
>

On 31 August 2010 20:54, Kraig Grady <kraiggrady@...> wrote:
>
> I have not been happy with how the MOS family thing has come
> about. If you see where these scales fall on the scale tree this
> confusion disappears for me as one knows that a particular scale
> template is found on all underneath but not vice versa.

It's all about the scale tree. How many names you assign depends on
how far down you go.

>  Perhaps MOS 'branch' i think is a better idea than 'family'.
>  So maybe temperment branch wouldn't bother me not that class
> really does.

You can have a family tree, can't you? Tree structures are often
described in terms of parent/child relationships. And they're usually
drawn with the roots to one side, or in the sky, so the "tree"
metaphor doesn't hold very well.

Graham

Gene Ward Smith wrote:

>> Dear Caleb,
>
>> Congratulations on finding and liking 46-EDO!

> These days we've been calling the corresponding regular
> temperament "leapday": <1 21 15 11 8... ||.

Dear Gene,

This reminds me of a question I've had for some time:
is there some special meaning to the name "leapday"?

Knowing some of the conventions of the regular
mapping school, I found that "leapyear" suggested
to me something like 2/29 (the notion for leapyear
day, February 29, in an American style with the
month first), and I wondered whether or how it
could have something to do with 2/29 octave, the
regular diatonic semitone or limma of 29-EDO?

Of course, a name need not have the logic of
something like "beatles" 19/64 or "orwell"
19/84, as witness "peppermint" for
a rank 3 temperament (2/1, 704.096, 58.680),
where the "pepper" part is for Keenan Pepper
as a rediscoverer of the rank 2 temperament
(2/1, 704.096) on which it's based, but why
the "mint"?

And I suspect that the story of "leapday"
may throw some light on either the temperament
or the occasion on which it was named, maybe
both.

Best,

Margo

It was a mere suggestion. There is absolutely nothing to be gained by the name.
Yes i understand the tree . that is what my other examples uses if you notice.
I thought you objected to 'class', so you are now objecting what someone might call the alternative?

the thing is already called the scale tree and it separates into branches more than families. why don't you insist it be called roots it the top bottom thing bothers you so much? One oftens refers to branches in a family tree.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 31/08/10 11:37 PM, Graham Breed wrote:
>
> On 31 August 2010 20:54, Kraig Grady <kraiggrady@... > <mailto:kraiggrady%40anaphoria.com>> wrote:
> >
> > I have not been happy with how the MOS family thing has come
> > about. If you see where these scales fall on the scale tree this
> > confusion disappears for me as one knows that a particular scale
> > template is found on all underneath but not vice versa.
>
> It's all about the scale tree. How many names you assign > depends on
> how far down you go.
>
> > Perhaps MOS 'branch' i think is a better idea than 'family'.
> > So maybe temperment branch wouldn't bother me not that class
> > really does.
>
> You can have a family tree, can't you? Tree structures are often
> described in terms of parent/child relationships. And they're > usually
> drawn with the roots to one side, or in the sky, so the "tree"
> metaphor doesn't hold very well.
>
> Graham
>
>

On Tue, Aug 31, 2010 at 4:41 AM, Mike Battaglia <battaglia01@...> wrote:
>
> An idea I just had - perhaps that's one common sense way to delimit a
> temperament's range. So we could say that meantone is only meantone if
> the generator produces both 5+2 and 7+5 scales, (or equal variants
> thereof). This means that a lower bound for meantone is 7-tet,
> exclusive, and an upper bound for meantone is 12-tet, inclusive.
//snip
> Well, if we tie meantone down to having generators that can produce
> both the 5+2 and 7+5 families, there goes that problem :) And I think
> that's pretty close to how we all intuitively decide to stop calling
> something meantone anyway.

To quickly add to this, I just realized that all temperaments that
produce both 5+2 and 7+5 scales would have to produce 5+2 scales that
are Rothenberg equivalence classes of one another. That is, any 5+2
scale that has a generator that also produces a 7+5 chromatic scale -
so a generator between 7-tet and 12-tet exclusive - will have the same
rank-order matrix for any other LLsLLLs scale in the same range.

In another freak coincidence that I'm sure is not a coincidence, this
is also the range in which the LLsLLLs is strictly proper. For 12-tet,
it's just proper. I wonder if the same applies to chromatic scales -
for 7+5 scales to be equivalence classes of one another, that their
19-note MOS's would have to be the same.

The thought also just occured to me that a generator producing 5+2
scales that produces 5+7 (let's call this an anti-chromatic scale for
now) will likely have some interval reversed as opposed to one
producing 5+2 and 7+5 scales (let's call this a chromatic scale). So
perhaps this suggests a way to tie in Rothenberg's stuff with regular
temperament as well.

I know that the notion that equivalence classes are a fundamental
concept for musical perception is popular here, so perhaps that will
lend some momentum to this. Or even better, maybe it means that it's
already been done, haha.

Whoo... the stars are aligning tonight :)

-Mike

PS: I have to ask, in the larger context of music theory, when we
propose a taxonomy of MOS scales - what exactly are we doing here?
When we start talking about MOS families as being outside of the realm
of mappings or harmonic intervals, then we're clearly stepping outside
of regular temperament entirely. Is it that we formulating a
microtonal approach to music set theory that abstractly uses linear
temperaments as its basis instead of equal temperaments? Seems the
case to me, and if so, it might prove infinitely more perceptually
useful than existing music set theory as I understand it.

And with my idea to bound the generator range of a temperament class
to a combination of MOS families (e.g. meantone being tied to
generators that produce 5+2 and 7+5 scales), are we getting into
mathematical territory where set theory connects with regular
temperament? That would be very neat if so.

I know that Rothenberg's works are sometimes presented as being part
of "diatonic set theory," so perhaps that's what this is all getting
into.

Graham Breed wrote:

> You can say that Lucy Tuning is plainly a meantone and not a
> schismatic, because the meantone approximation is both simpler and
> more accurate than the schismatic one, for this tuning. You can also
> say that Pythagorean intonation is schismatic and not meantone, but by
> what rule? The schismatic approximation is more accurate here, but
> the meantone one is still simpler.

Dear Graham,

Thank you for touching on what I consider an important question:
how broadly should the term "schismatic" be used? I would say
that it might best apply, somewhat broadly in my view, from
12-EDO to 29-EDO, with these two tunings seen as limits rather
than themselves squarely within the category.

Strictly speaking, to say that tunings in the immediate
vicinity of Pythagorean, and that just tuning itself, are
"schismatic" (or "schismic") would seem to me to refer
to two relationships: the 12-vector, as I'll call it,
here at or close to the Pythagorean comma, approximates
either the syntonic comma or the septimal or Archytan
comma. Thus, for example, +4 fifths is around 81/64;
-8 fifths is around 5/4; and +16 fifths is around 9/7.
The latter two approximations show the two "schismatic"
relationships.

At the lower boundary, 12-EDO, the regular (+4 fifths)
and schismic (-8 fifths) major thirds are an identical
400 cents, either of which could be seen as an approximation
of either 81/64 or 5/4. Thus 12-EDO is on the boundary
between meantone and a region which could be called schismic
or schismatic, although with the qualification that it
doesn't have any close septimal approximations.

The center of the schismic region would be around Pythagorean,
including also things like 135-EDO and 41-EDO. Here +4 fifths
give a good approximation or 81/64 (or exactly yield it in
Pythagorean); while -8 fifths yield an approximate 5/4.

When we go as far as 29-EDO, however, the equation has
changed enough that I would say we are at the upper limit
of schismic or schismatic and the lower limit of a new
region.

First, -8 fifths gives us around 372 cents, which some might
regard as a rather inaccurate approximation of 5/4 (386 cents),
but which could also be regarded as a reasonable approximation
of 26/21, for example (370 cents). Ozan Yarman and I have
independently concluded that 29-EDO is around the point where
submajor/supraminor thirds start to be reasonably approximated
by the same pathway, -8/+9 fifths (diminished fourth for submajor
and augmented second for supraminor), as yields 5-limit
approximations in the schismic range (e.g. Pythagorean).

Secondly, the 12-vector at some 41 cents is much larger
than 81:80 or 64:63, and in fact a bit larger than the
128:125 or 1/4-comma meantone! And we no longer get the
septimal approximations of Pythagorean, 135-EDO, or 41-EDO,
but what I term "interseptimal" approximations very close
to 13:10 and 15:13. For example, in Pythagorean, +14 fifths
(9:8 plus 12-vector) gives a slightly narrow 8:7; here
it gives a near-just 15:13.

Thus I might say that 29-EDO is the beginning of a
"Zalzalian-diesic" region. This means that thirds of
-8/+9 fifths are in a "Zalzalian" (neutral/submajor/supraminor)
range of about 330-372 cents, rather than close to 5-limit
in the schismic range or 12-EDO to 29-EDO (400-372 cents).
And the "diesic" part of the name tells us that the 12-vector
has grown from a comma to a diesis -- in fact, one comparable
in size to that of a classic meantone in the range of negative
tunings.

Another landmark at 29-EDO is that -13 fifths, for example,
becomes as good an approximation of 9/7 as +16 fifths: another
sign telling us that we're entering a new realm.

The Zalzalian-diesic region would go from 29-EDO to 17-EDO,
the point where the diminished fourth and augmented second
are the same size, and also where +4 fifths becomes as good
an approximation of 9/7, for example, as -13 fifths.

Note that my criteria for drawing lines are in part
subjective -- "Isn't -8 fifths close enough to 26/21
that it could at least as well be described as a
submajor third as an inaccurate 5/4 major third?"
And in part they're objective: "Here is where +15 fifths
is as good a 7/4 as -14 fourths."

Defining and naming regions can certainly reflect one's
priorities! I'd suspect how one views 29-EDO might depend
in part on how often one seeks and uses a 26/21 third
by comparison to a 5/4 third, for example.

Best,

Margo

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

> This reminds me of a question I've had for some time:
> is there some special meaning to the name "leapday"?

Herman named it. Then reason he gave is that the smallest leapday MOS containing a full 13-limit hexad has 29 notes to the octave.

> Knowing some of the conventions of the regular
> mapping school, I found that "leapyear" suggested
> to me something like 2/29 (the notion for leapyear
> day, February 29, in an American style with the
> month first), and I wondered whether or how it
> could have something to do with 2/29 octave, the
> regular diatonic semitone or limma of 29-EDO?

That's what I thought, but Herman otherwise.

On Tue, Aug 31, 2010 at 5:09 AM, Mike Battaglia <battaglia01@...> wrote:
> In another freak coincidence that I'm sure is not a coincidence, this
> is also the range in which the LLsLLLs is strictly proper. For 12-tet,
> it's just proper. I wonder if the same applies to chromatic scales -
> for 7+5 scales to be equivalence classes of one another, that their
> 19-note MOS's would have to be the same.

Sorry for the rapid-fire messages, but one last very important thing:
I just worked this out, and it turns out that if we do that, and we
limit the meantone generator to only that which produces
LLsLLsLLsLsLLsLLsLs (12+7), the bounds become 19-tet and 12-tet,
exclusive. I'm not commenting either way on whether we should DEFINE
meantone as being between 19 and 12, but I definitely think we can all
agree that meantones with fifths flatter than 19-tet (like 26-tet,
which is awesome) have a very different sound than meantones with
fifths sharper than 19-tet (like 12-tet, which is awesome). So by
dividing meantone into further categories based on MOS families, we
can distinguish further "shades" of meantone which do correlate with
very clear perceptual differences. These MOS groupings are also
correlated with how well the scale approximates 5-limit harmonies, so
I'm curious if there's some overarching mathematical structure that
relates everything here, and if it applies to temperaments besides
meantone or if meantone is a special mathematical case.

This is also the range in which the chromatic MOS is strictly proper
(in 19-tet it's just proper). I assume that if we only look at
meantones where the 19-MOS is 12+7 and strictly proper, we'll get an
even narrower range (with perceptual differences as well), and so on.

Note the pattern for meantone MOS's too (I know Graham has written
about this somewhere):
2+3
5+2
7+5
12+7
19+12
31+19
...

I assume if we come up with the only generator that matches the full
infinite series of these, we'll get some sort of "omega-proper
meantone" in which every MOS produced is strictly proper. I'm also
curious to see how this matches up to TOP.

-Mike

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

> Thank you for touching on what I consider an important question:
> how broadly should the term "schismatic" be used? I would say
> that it might best apply, somewhat broadly in my view, from
> 12-EDO to 29-EDO, with these two tunings seen as limits rather
> than themselves squarely within the category.

In terms of the L-s notation we have been discussing, that's the range of the 12L17s 29-note MOS, and that strikes me as a reasonable range to call schismatic. It would include generators 12/29 < g < 5/12.

> The Zalzalian-diesic region would go from 29-EDO to 17-EDO,
> the point where the diminished fourth and augmented second
> are the same size, and also where +4 fifths becomes as good
> an approximation of 9/7, for example, as -13 fifths.

And that's the range of the 17s12L 29 note MOS, where now 17/29 < g < 10/17.

> Defining and naming regions can certainly reflect one's
> priorities!

I think the above is an example of a general phenomenon, which is that the ranges defined by L-s pairs give good boundries in terms
of mappings, since they are bounded at places where relationships swap around.

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

> I'm curious if there's some overarching mathematical structure that
> relates everything here, and if it applies to temperaments besides
> meantone or if meantone is a special mathematical case.

Meantone is not a special case.

> This is also the range in which the chromatic MOS is strictly proper
> (in 19-tet it's just proper). I assume that if we only look at
> meantones where the 19-MOS is 12+7 and strictly proper, we'll get an
> even narrower range (with perceptual differences as well), and so on.

The children of 12L7s are 19L12s, the proper child, and 12L19s, the improper child. You always get two children, one proper and one improper.

> Note the pattern for meantone MOS's too (I know Graham has written
> about this somewhere):
> 2+3
> 5+2
> 7+5
> 12+7
> 19+12
> 31+19
> ...
>
> I assume if we come up with the only generator that matches the full
> infinite series of these, we'll get some sort of "omega-proper
> meantone" in which every MOS produced is strictly proper. I'm also
> curious to see how this matches up to TOP.

Congratulations; you've just discovered Golden Meantone. It has the property that the ratio L/s of the large step size to the small step size (measured in cents) for any MOS is always the golden ratio, and since that is less than 2, it's always proper. Given any L-s pair you can do something similar.

MikeB>"but I definitely think we can all agree that meantones with fifths
flatter than 19-tet (like 26-tet,
which is awesome) have a very different sound than meantones with fifths sharper
than 19-tet (like 12-tet, which is awesome)"

Flatter than 19TET sounds about right to me as a barrier. That is about
when we become more than 7 cents away from a perfect fifth and it bites your ear
a bit.

Also, excuse my stress on psychoacoustics, but wouldn't it make sense if
meantone "types" where categorized by
A) The highest error of any dyad from the perfect version of that dyad
B) The types of dyads the intervals form

So, for example, 12TET might be categorized
A1) An error of about 14 cents as that's about how far the third and sixth
are from "goal" dyadic ratios they estimate.
A2) Since the fifth is very important in meantone...for meantone scales give
the error of the fifth IE 1.95 cents
B1) A list of goal ratio types of
15/14
9/8
5/4
4/3
7/5
3/2
8/5
27/16 or 5/3
16/9
17/9
2/1
(ideally listing the error of each dyad representation in cents from its
ideal as well)
B2) The general limit the scale tries to capture (IE five in this case), the
two worst-case limit dyads it captures (27 from 27/16 and 17 from 17/9),

Also correct me if I missed something but I still wonder why accuracy in fitting
a certain MOS schemes. You said:

MikeB>"These MOS groupings are also correlated with how well the scale
approximates 5-limit harmonies"

Could you give a few examples of how you think this applies here?

Hi Mike, I think I've found the root of our miscommunications!

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Aug 31, 2010 at 3:16 AM, cityoftheasleep
> <igliashon@...> wrote:
> >
> > (what's the word for only having two step-sizes again? I can never keep all those terms straight)
>
> Isn't that MOS?

No, MOS is a special case. To be MOS, the scale has to be directly generated by a series of generators that wrap around at a period, and the moment of symmetry occurs any time the generator sequence makes a scale of only two step-sizes. But for instance the melodic minor has two step-sizes but is not an MOS, because it is not made by a single chain of generators in a given period. To get the melodic minor, you have to take the diatonic scale and swap a step around. So a scale can have two step-sizes but not be MOS.

> > Defined that way, that's not a category of linear temperaments. At least, not in the way the term "linear temperament" is used here.
>
> Why not? If it has a single generator and a single period, it's a
> linear temperament.

Actually, no, it's not. To be a temperament, it has to have a mapping and an amount of error. Otherwise it's just a plain ol' MOS. If you don't believe me, ask Gene or Graham about it.

> And I just placed them within a category, with my
> category superpowers. Besides, after Margo just fleshed all of that
> out... what's the debate about at this point? :)

Margo named some temperaments that coincide with some MOS scales in what she calls the "isotonic" range, which I call 3\5-4\7. What IS the debate about at this point? Whether you should be allowed to call two temperaments with different mappings the same temperament?

> If the term for what I am describing already exists and is called "MOS
> family," and meantone and superpyth can be said to belong to the
> diatonic MOS family, then I'll use that term instead. I used the
> phrase "temperament class" to indicate that the concept I was getting
> across was a way of categorizing temperaments into groups or "classes"
> with perceptual similarities. But then it turns out that Graham had
> already used the phrase "temperament class" to describe something like
> meantone or porcupine, so I started using the term "temperament
> superclass" to indicate that it was a group of "temperament classes"
> with perceptual similarities. Now you're saying that it's called an
> MOS family, so if it is then I'll start using that term. I don't
> really care what it's called.

It's just that "temperament" implies a specific approach, i.e. one involving mappings. The fact that multiple temperaments can fit in one MOS family has nothing to do explicitly with what comma(s) they temper out (i.e. what mapping(s) they have).
Let me try an analogy: you're at a dog show, and you notice that the representatives of some breeds have black coats, while others have white coats or brown or spotted. You propose to the American Kennel Club that they should classify breeds into "super-breeds" where all the dogs with black coats belong to one super-breed, all the dogs with brown coats to another, etc. You clearly are missing the point that "breed" has nothing to do with "color", i.e. that one breed can be many colors and that "coat color" is not a determining factor in what breed a dog is in the first place. Calling all dogs with the same coat-color a "super-breed" misses the point about what a breed is in the first place.

Calling it a "temperament super-class" would make sense if you were lumping together all the temperaments that had something about their mappings in common, i.e. that they all temper out 81/80 (but may temper out different additional commas). In fact, this already seems to be routinely done, if you look at Gene's Wiki articles on various temperaments and their "children".

> The reason I picked the term "temperament superclass" is that
> superpyth and meantone are obviously very much perceptually related in
> a way that meantone and porcupine are not, which lends itself to a
> taxonomy whereby there is some larger concept of which meantone and
> superpyth are subcategories.

Yes, but the relationship has nothing to do with their properties as temperaments, but rather, their properties as MOS scales. It is the case that a specific tuning for a rank-2 temperament can also be used to define an MOS scale, and that's probably why you're confused about the difference between the two. It's just that if you ONLY have the tuning for a temperament, and not its mapping, there's no way to say what temperament it actually is.

> Right, I understand very well that you could say that a meantone
> generator is 1199 cents and the period is 1200 cents. I'm not dodging
> that point because I don't understand it, I'm dodging it because I
> think we all have a decent "common sense" definition of what
> constitutes a meantone and what doesn't, and that rigorously defining
> it isn't necessary at this point to flesh out how temperaments fall
> into larger perceptual categories.

Is our common-sense definition enough? Is 686 cents a meantone generator? Is 690 cents? If they are, then why isn't 703 cents? If they aren't, what are they? When you get away from TOP meantone (or some other optimized meantone), it becomes very difficult to say what is and isn't a meantone, even by common-sense definitions.

> > But sure, you could certainly look at a list of optimal and near-optimal generators for different temperament classes and index them according to MOS families they belong to.
>
> aaaaaaa HAAAA! You admit it! :)

Note the words "optimal" and "near-optimal". You need to have specific tunings for each temperament before any such indexing is possible.

> > It'd be a bit tedious, though. And MOS families are actually somewhat difficult to pin down unless you set a maximum number of notes. It is interesting to note, for instance, that 19-EDO and 22-EDO share the same 7-note LLsLLLs scale, but have opposite 12-note scales (19's is LsLsLLsLsLsL, or 7L+5s, whereas 22's is LsLssLsLsLss, or 5L+7s)--are they REALLY in the same MOS family?
>
> > Likewise, 22 and 27 have the same 12-note MOS, but a different 22-note MOS. So if we group only according to a 7-note MOS, we may be blurring important distinctions. So now we have the burden of arbitrarily defining a limit to MOS size, or else indexing each tuning according to multiple levels of MOS.
>
> That is a good point. What do you think would be best?

I have no solid ideas as of yet.

> An idea I just had - perhaps that's one common sense way to delimit a
> temperament's range. So we could say that meantone is only meantone if
> the generator produces both 5+2 and 7+5 scales, (or equal variants
> thereof). This means that a lower bound for meantone is 7-tet,
> exclusive, and an upper bound for meantone is 12-tet, inclusive.

Why inclusive of 12-tET? 12-tET doesn't support either 5+7 or 7+5. If you're going to exclude 7, you should exclude 12, or else you should include both. But it doesn't make sense to say that 7-EDO is "neither" 5+2 nor 2+5 but that 12-tET is "both" 7+5 and 5+7.

I went down this road a while back, though. I'll post a spreadsheet graphic I made where I listed the MOS families defined by branches on the Stern-Brocot tree, looking for patterns that would correlate to temperament classes. I too thought that we could define the range of a temperament class by MOS families of sufficiently-large size, and I too got very excited by this idea, thinking I'd found the missing link between linear temperaments and MOS families. But after some probing of the list and some long meditation, I came to the conclusion that I had not.

Why did I conclude this? Because there are too many ways to invalidate the perceptual similarities on which the grouping is supposed to be based. Consider the problem that from 3\5-4\7, when you pass a certain point heading toward either end of the spectrum, the scale STOPS sounding like 5L+2s and starts sounding like 5-EDO or 7-EDO, so the perceptual grouping also stops making sense. I have not succeeded in finding that point.

Also, there is the fact that whatever limit we impose on temperament classes is going to be arbitrary from a temperament perspective. We don't have a way to validate our distinctions, IOW. And as Herman suggested, and I keep trying to get through to you, temperament tunings vary in consistency across a given pitch-range. 690 cents is an inconsistent tuning for Meantone, equally as inconsistent as 709 cents is, in fact. From a perspective of both error and consistency, neither should be considered a meantone, yet 690 cents is squarely between 12-EDO and 7-EDO.

> Yes... I like this idea a lot :) Seems like it might lead to something
> interesting. I also wonder, just as you can specify a linear
> temperament by two ET's, what can you specify by two MOS's? Some
> higher-limit generalization of MOS?

No, giving two MOS's is pointless because the higher one implies the lower one.

> What really gets me is how really fundamental to musical perception
> the scale taxonomy is. I would support this initiative and would like
> to do some work on it myself. After spending all of this time with
> regular mapping, my head was turned completely upside down when Paul
> pointed out that superpyth minor scales and meantone minor scales
> sounded perceptually "the same." So after thinking that JI was some
> kind of underlying reality that regular mapping was "warping," I don't
> even think that anymore.
>
> That is, the question really is - do JI intervals and triads have
> their own inherent "identity?" Is it that tempering "connects"
> different pieces of the JI lattice and smear identities together? Or
> is there something else fundamentally going on? Paul's example
> certainly seems to suggest the latter. Thus your proposed MOS taxonomy
> might be a more useful way of categorizing this particular element of
> musical perception, with regular mapping then taking the role of
> finding the generator that produces the most concordant chords - which
> is a completely different level of perception, and may be more
> reducible to a simple notion of "concordant" and "discordant" with
> different JI intonations just implying subtly different flavors.

If such a taxonomy proves to be possible, I have no doubt that it may prove useful. But I have my doubts about its possibility. I haven't given up on it, though, and it's nice to see someone else interested in the project (finally)!

> Well, if we tie meantone down to having generators that can produce
> both the 5+2 and 7+5 families, there goes that problem :) And I think
> that's pretty close to how we all intuitively decide to stop calling
> something meantone anyway.

Not quite. See my example of a 690-cent generator above.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I assume if we come up with the only generator that matches the full
> infinite series of these, we'll get some sort of "omega-proper
> meantone" in which every MOS produced is strictly proper. I'm also
> curious to see how this matches up to TOP.

That'd be the "golden generator". Man, you REALLY need to look at Wilson's diagrams of the scale tree. I recommend:
http://www.anaphoria.com/sctree.PDF

Meditate on this, and possibly this:
http://www.anaphoria.com/line.PDF

And you shall find what you are seeking, if it's there to be found.

-Igs

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

> The children of 12L7s are 19L12s, the proper child, and 12L19s, the improper child. You always get two children, one proper and one improper.

Three children: you forgot the "equal" child.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > The children of 12L7s are 19L12s, the proper child, and 12L19s, the improper child. You always get two children, one proper and one improper.
>
> Three children: you forgot the "equal" child.

Nope, I did not forget it. It's not a MOS.

Don't know "marvel", anybody care to describe it to me? If it tempers out 81/80 though, it's not what I'm after. I think the scale you present not only doesn't need to temper out 81/80, but doing so will weaken it and the non-meantone possiblities of modulation from and to it. If I get time tomorrow I'll post a demonstration.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Aug 29, 2010 at 4:14 AM, cameron <misterbobro@...> wrote:
> >
> > And, maintaining near-perfect 7-limit Just interpretation of this scale with full transposability and circulation, the comma to temper out is simply 225/224, the difference between (two pure M3+m3) and (septimal tritone+pure fourth), also the difference between (septimal subminor third + pure fourth) and (two pure M3), etc.
>
> How did you come up with marvel here, exactly? I'm also noticing that
> if the C-F# is intoned 7/5, and if the F#-A-C can both be 6/5's, that
> equates 36/25 with 10/7, which tempers out 126/125 - the starling
> comma, which I haven't really played around with before. Tempering
> both 126/125 and 225/224 means that 81/80 is tempered as well, so
> we're back to septimal meantone. I ended up playing around with this
> in 31-tet, as per both this and Gene's suggestion, and it sounded
> great. So it basically means that it works very well as a subset of
> meantone[12]. As for what else it works as, I'm curious, as the
> decatonic scales can work as a part of meantone[12] too :)
>
> I just discovered that scala's "fit/mode" function, so fitting it to
> the 12-equal version gives 20-equal as having a strictly proper
> version. It's 2 3 2 3 3 2 3 2 for those interested. It is apparently
> CS too.
>
> -Mike
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Don't know "marvel", anybody care to describe it to me? If it tempers out 81/80 though, it's not what I'm after. I think the scale you present not only doesn't need to temper out 81/80, but doing so will weaken it and the non-meantone possiblities of modulation from and to it. If I get time tomorrow I'll post a demonstration.

Marvel does not temper out 81/80, and is quite a bit more accurate than any temperament which does. It is a planar, or rank three, temperament, tempering out 225/224. Hence 16/15 and 15/14 are identifed into a secor, two of which give 8/7, which relates marvel to miracle, but it's also a touch more accurate than miracle. In the 11-limit you can also temper out 385/384, and since this makes little difference to the tuning, which wants slightly flat fifths and major thirds, little or no error for 7, and slightly flat 11s, you can also consider that marvel (or unidecimal marvel, if you like.)

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

> Marvel does not temper out 81/80, and is quite a bit more accurate than any temperament which does. It is a planar, or rank three, temperament, tempering out 225/224.

I should have added that as a rank three temperament, it has three generators, which you can take as 2, a slightly flat 3, and a slightly flat 5. For minimal 7-limit error, you can use 3 and 5 flat by (225/224)^(1/4), and for minimal 9-limit error, 3 flat by (225/224)^(1/6) and 5 flat by (225/224)^(1/3).

> http://www.anaphoria.com/sctree.PDF

Even though it might make me look like a total n00b, may I ask for a quick explanation of the scale tree, or a good reference about it? Just a really quick overview of what the each number is representing and how one gets from one level to another would be really helpful.

Thanks!

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
>
> > http://www.anaphoria.com/sctree.PDF
>
> Even though it might make me look like a total n00b, may I ask for a quick explanation of the scale tree, or a good reference about it? Just a really quick overview of what the each number is representing and how one gets from one level to another would be really helpful.

See if this helps:

http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree

About the "consistency" of meantone/syntonic tunings:

It has been argued that tunings that have a better approximation of 5/4 beyond the fourth stack of the generator are inconsistent if one still chooses to use the fourth stack.

But don't you HAVE to use the fourth stack in meantone/syntonic, or else that tuning will no longer be tempering the interval 81/80?

For example, if you choose to use the 11th stack of the fifth as you said is a more accurate in some tunings, instead of setting equal

5/4 = [(3/2)^4 * (2/1)^-2]

and arriving at

81/64=80/64
81/80=1

you instead set equal

5/4 = [(3/2)^11 * (2/1)^-6]

and end up with

5/4 = 177147/131072

A tuning that used any number of generators other than four to approximate 5/4 would no longer be meantone/syntonic, would it?

John

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
> you instead set equal
>
> 5/4 = [(3/2)^11 * (2/1)^-6]
>
> and end up with
>
> 5/4 = 177147/131072

Sorry, didn't finish it:
and temper the interval
177147/163840=1

>
> A tuning that used any number of generators other than four to approximate 5/4 would no longer be meantone/syntonic, would it?
>
> John
>

Think about it. Each instrument has its first overtone at 2/1.

So 3:4:5 has these overtones as 6:8:10...forming dyads with the root including
6/3 = 2/1, 8/3, 10/3.
Meanwhile 3:5:7 has 6:10:14 giving dyads of 2/1, 10/3, 14/3.

Now compare 4:5:6's overtones of 8:10:12...where dyads to the root include 2/1,
5/2, 3/1.

Would the fact the dyads of 4:5:6 are much lower limit have anything to do with
its having an apparent lower discordance?

Hi John

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
>
> About the "consistency" of meantone/syntonic tunings:
>
> It has been argued that tunings that have a better approximation of 5/4 beyond the fourth
> stack of the generator are inconsistent if one still chooses to use the fourth stack.
>
> But don't you HAVE to use the fourth stack in meantone/syntonic, or else that tuning will
> no longer be tempering the interval 81/80?
[snip]
> A tuning that used any number of generators other than four to approximate 5/4 would no > longer be meantone/syntonic, would it?

Yes, and that's the point. With these tunings, either you're inconsistent or you're using a different temperament. So the idea is that the range of the syntonic temperament is not continuous, but rather made up of "patches" where the temperament is consistent separated by patches where it's inconsistent but another temperament *is* consistent. See Herman Miller's post from a few days ago where he linked to some graphs of consistency for different generators for various temperament classes. They look like fractals.

-Igs

On Tue, Aug 31, 2010 at 11:51 AM, genewardsmith
<genewardsmith@...t> wrote:
>
> Congratulations; you've just discovered Golden Meantone. It has the property that the ratio L/s of the large step size to the small step size (measured in cents) for any MOS is always the golden ratio, and since that is less than 2, it's always proper. Given any L-s pair you can do something similar.

*snif* my first rediscovery... Does this always tend to lead to a
near-optimal generator? It's obviously within the same ballpark as TOP
and such.

-Mike

From a scales perspective, each node on the scale tree is a fraction of a period, usually taken to be an octave. The tree itself is a recurrent sequence: you start with the nodes 0/1, 1/1, and 1/0. The next nodes are 0+1/1+1 and 1+1/1+0, i.e. 1/2 and 2/1, which can be read "1 out of 2-EDO" and "2 out of 1-EDO" if we're dealing with scales of an octave period. Then you make the next branches as 1+1/2+1, 1+0/2+1, 2+1/1+1, and 2+1/1+0, or 2/3, 1/3, 3/2, and 3/1. At this point it should be clear that the nodes of a given level are defined as the classical mediants between the nodes of the previous level.

The important thing about the scale tree is that it is a visual representation of all possible MOS families. Between any node and its parent node is an MOS family. See my folder in the "Files" section for a document called "MOS spreadsheet".

You can also use the tree as a chart of JI intervals, but that's another story.

-Igs

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
>
> > http://www.anaphoria.com/sctree.PDF
>
> Even though it might make me look like a total n00b, may I ask for a quick explanation of the scale tree, or a good reference about it? Just a really quick overview of what the each number is representing and how one gets from one level to another would be really helpful.
>
> Thanks!
>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> Nope, I did not forget it. It's not a MOS.
>
I believe Kraig would disagree...didn't he just post a little while ago that equal scales are a kind of MOS?

-Igs

On Tue, Aug 31, 2010 at 12:19 PM, Michael <djtrancendance@...> wrote:
>
>    Also, excuse my stress on psychoacoustics, but wouldn't it make sense if meantone "types" where categorized by
> A)  The highest error of any dyad from the perfect version of that dyad
> B) The types of dyads the intervals form

Sure. That's how these temperaments are already categorized, I think.
I'm just taking an alternate perspective on things, just for fun. I
think Erv Wilson has already fleshed this out, so I'm going to have to
go back to the Stern-Brocot tree.

> So, for example, 12TET might be categorized
>    A1) An error of about 14 cents as that's about how far the third and sixth are from "goal" dyadic ratios they estimate.
>    A2) Since the fifth is very important in meantone...for meantone scales give the error of the fifth IE 1.95 cents
>    B1) A list of goal ratio types of
>     15/14
>     9/8
>     5/4
>     4/3
>     7/5
>     3/2
>     8/5
>     27/16 or 5/3
>     16/9
>     17/9
>     2/1
>      (ideally listing the error of each dyad representation in cents from its ideal as well)

I think this is something similar to woolhouse-optimal, where the
dyads are 5/4, 6/5, and 3/2. But if your goal was to do that above,
you could probably work it out as an algebraic expression and get the
least-squares error or something like that.

> Also correct me if I missed something but I still wonder why accuracy in fitting a certain MOS schemes.  You said:
>
> MikeB>"These MOS groupings are also correlated with how well the scale approximates 5-limit harmonies"
>
>     Could you give a few examples of how you think this applies here?

I don't know why it relates, I'm just looking for some overarching
structure. The example is that with fifths east of 19-tet, you have
the proper 19-note MOS. With fifths west of 19-tet, you have the
improper 19-note MOS. And I'm pointing out that we can probably all
agree that 19-tet does serve as a pretty useful "barrier," and that
meantones with fifths less than that (what Margo just called
"inframeantones") have a characteristic sound that is different than
fifths greater than that (what she called just regular "meantones").

So there are two properties that correlate with this change in sound:
1) That inframeantones have an improper 19-note MOS, which means that
E# will be lower than Fb
1a) And meantones have a proper 19-note MOS, which means that E# will
be higher than Fb. Hence there is an interval that is being reversed
here.
2) That inframeantones tend to more poorly approximate 5-limit
harmonies compared to regular meantones.

Whether #2 is just a coincidence or comes naturally from #1 (or vice
versa) I'm not sure - just something I'm pointing out.

-Mike

Gene wrote:

>> This reminds me of a question I've had for some time: is there
>> some special meaning to the name "leapday"?

> Herman named it. Then reason he gave is that the smallest leapday
> MOS containing a full 13-limit hexad has 29 notes to the octave.

This gives me an idea, by analogy to sets of a temperament like
Miracle with names like Blackjack (21) and Canasta (31).

If we're thinking of the same kinds of hexads and heptads, which
I learned from George Secor, then Herman's leapday 29-MOS might
actually support a 13-limit heptad: 4:5:6:7:9:11:13. Of course,
if we stick to primes and leave out 9 (which makes a more
isoharmonic structure with all differences of 1 or 2), then we'd
indeed have a 2-3-5-7-11-13 hexad.

And we could also recognize a 17-MOS, maybe "St. Patrick's Day"
(March 17), supporting the hexad 4:6:7:9:11:13.

So we'd have "St. Patrick's Day" (17) for 2-3-7-11-13, and
"leapday" (29) for 2-3-5-7-11-13.

Typically I use a set of 24, which mostly functions as
St. Patrick's Day with some extra notes to provide the more
remote intervals of the 17-MOS in more locations, with nine of
the 2-3-7-9-11-13 hexads available.

Anyway, having a name for the 17-MOS to complement Herman's
leapday might make the meaning of his name clearer, as well point
out the "neomedieval" 2-3-7-9-11-13 and "full 13" 2-3-5-7-9-11-13
perspectives on this fine regular temperament.

Best,

Margo

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> From a scales perspective, each node on the scale tree is a fraction of a period, usually taken to be an octave.

You really need to take the period to be whatever it is, which often enough is an aliquot fraction of an octave.

> The important thing about the scale tree is that it is a visual representation of all possible MOS families.

For a given period. But an important decision must first be made: does the fraction represent the middle of the family, in the sense of being the mediant of its endpoints, or is it the right hand endpoint? I prefer the latter, which gives a 1-1 relationship between L-s pairs in the given period and the nodes of the tree, where each fraction q represents the range of generators from q to the right-hand parent p of q, so that for instance 4/7 represents 4/7 < g < 3/5. That way, 4/7 corresponds exactly to 5L2s. Of course, if the period is 1/2 octave, then we will probably want it to correspond to 10L4s instead.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > Nope, I did not forget it. It's not a MOS.
> >
> I believe Kraig would disagree...didn't he just post a little while ago that equal scales are a kind of MOS?

I don't know, but formerly IIRC he said the opposite. Anyway, since it doesn't have Myhill's property I think trying to include it will only confuse things.

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

> I think this is something similar to woolhouse-optimal, where the
> dyads are 5/4, 6/5, and 3/2. But if your goal was to do that above,
> you could probably work it out as an algebraic expression and get the
> least-squares error or something like that.

Woolhouse is the least squares tuning, and 1/4-comma meantone is the minimax tuning.

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

> For a given period. But an important decision must first be made: does the fraction represent the middle of the family, in the sense of being the mediant of its endpoints, or is it the right hand endpoint? I prefer the latter

Except that I meant to say left hand endpoint.

To group, including Graham, G.W., George S., Margo S:

Thanks very much for the files and info and help.

I'm a little overwhelmed with all the things to be tried and tested, but I appreciate the help very much.

Checking out the Secor 29&41 right now...

Caleb

! secor29htt.scl
!
George Secor's 29-tone 13-limit high-tolerance temperament (5/4 & 7/4
exact)
29
!
58.08980
97.04984
140.19633
179.15637
207.15739
265.24719
296.73557
347.35372
5/4
414.31478
472.40458
496.42131
554.51111
593.47114
633.37025
679.56197
703.57869
761.66849
800.62853
843.77502
882.73506
910.73608
7/4
992.84261
1050.93241
1089.89245
1117.89347
1175.98327
2/1

! secor41htt.scl
!
George Secor's 13-limit high-tolerance temperament superset (5/4 &
7/4 exact)
41
!
30.08878
58.08980
97.04984
116.17960
140.19633
179.15637
207.15739
237.24617
265.24719
296.73557
323.33699
347.35372
5/4
414.31478
444.40355
472.40458
496.42131
526.51008
554.51111
593.47114
612.60091
636.61764
679.56197
703.57869
733.66747
761.66849
800.62853
819.75830
843.77502
882.73506
910.73608
940.82486
7/4
992.84261
1026.91568
1050.93241
1089.89245
1117.89347
1147.98225
1175.98327

The original MOS paper shows EDO as an MOS and uses them to show the subsets that are also MOS.
We also have http://anaphoria.com/MOSedo.PDF

--- In tuning@yahoogroups.com </tuning/post?postID=X302I3Kc11ylYp7OPxvK0DxR_WxLA1IbmMy-zaM6MzeY49bintjjCxSp9SSzdEJ3dIHsS-GGEq0JIB4l>, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com </tuning/post?postID=X302I3Kc11ylYp7OPxvK0DxR_WxLA1IbmMy-zaM6MzeY49bintjjCxSp9SSzdEJ3dIHsS-GGEq0JIB4l>, "genewardsmith" <genewardsmith@> wrote:
> > Nope, I did not forget it. It's not a MOS.
> >
> I believe Kraig would disagree...didn't he just post a little while ago that
equal scales are a kind of MOS?

I don't know, but formerly IIRC he said the opposite. Anyway, since it doesn't
have Myhill's property I think trying to include it will only confuse things.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

Also http://anaphoria.com/meantone-mavila.PDF

--- In tuning@yahoogroups.com </tuning/post?postID=4rVYEQrmaJsqP1VSQF7fOku4I-d1Cmu86YvhzpgysB9HTRjPVbZELLmhVT7yesEcIoK5cQTFjMx46aLnNEE>, Mike Battaglia <battaglia01@...> wrote:
> I assume if we come up with the only generator that matches the full
> infinite series of these, we'll get some sort of "omega-proper
> meantone" in which every MOS produced is strictly proper. I'm also
> curious to see how this matches up to TOP.

That'd be the "golden generator". Man, you REALLY need to look at Wilson's
diagrams of the scale tree. I recommend:
http://www.anaphoria.com/sctree.PDF

Meditate on this, and possibly this:
http://www.anaphoria.com/line.PDF

And you shall find what you are seeking, if it's there to be found.

-Igs
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
>
> The original MOS paper shows EDO as an MOS and uses them to show
> the subsets that are also MOS.
> We also have http://anaphoria.com/MOSedo.PDF

Carey and Clampitt in their 1989 paper "Aspects of Well-Formed Scales" calls those "degenerate well-formed scales". I suggest calling them degenerate MOS if you need to include them for some reason. In general, mathematical definitions should be ones which make stating and proving theorems easiest, and this is really a math topic, so I think it boils down to which definition makes things easier. If you try to define MOS including the degenerate MOS, it seems to me it makes things harder. It also gives overlapping ranges on the scale tree.

If the 700 cent 5 tone scale is an MOS and a 7 tone one also. I see no problem why the 12 would not be also.
I would hate to reduce your own work to working with ETs to propose that you start with a bunch of Degenerate MOS scales in order to find other proper ones.

No. 5 might actually cover all MOS. the others defining how it occurs

Definition of Moment of Symmetry
A moment of Symmetry is a scale that consists of:
1. A generator (of any size, for example a 3/2 or a fifth in 12 equal temperament) which is repeatedly superimposed but reduced within the
2. Interval of Equivalence (of any size for example most commonly the octave)
3. where each scale degree or units will be represented by no more than two sizes and two sizes only (Large and small=L s)
4. and where in turn every number of units within the scale will also occur in no more than two different sizes.
5. Every interval will be one step size and one step size only.

A constant structure (commonly of a higher limit) is a scale where every occurrence of a ratio will always be the same number of steps or units.]

--- In tuning@yahoogroups.com </tuning/post?postID=CJwLLaOCjU6Xj_C6Ws05KSapclpP21q_EpXJG56b8VjWQa52KzYWA-oKJC3ZUDFB2nwH0ltT04YnD3EImA>, Kraig Grady <kraiggrady@...> wrote:
>
>
> The original MOS paper shows EDO as an MOS and uses them to show
> the subsets that are also MOS.
> We also have http://anaphoria.com/MOSedo.PDF

Carey and Clampitt in their 1989 paper "Aspects of Well-Formed Scales" calls
those "degenerate well-formed scales". I suggest calling them degenerate MOS if
you need to include them for some reason. In general, mathematical definitions
should be ones which make stating and proving theorems easiest, and this is
really a math topic, so I think it boils down to which definition makes things
easier. If you try to define MOS including the degenerate MOS, it seems to me it
makes things harder. It also gives overlapping ranges on the scale tree.
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
>
>
> If the 700 cent 5 tone scale is an MOS and a 7 tone one also. I
> see no problem why the 12 would not be also.

Because it's annoying and silly to constantly be saying "X is true for all MOS but equal ones" or however you would prefer to put it.
All MOS have Myhill's property, except when they don't. All MOS have two specific intervals for every general interval, except when they don't. All MOS are found by stopping after a point where only two step sizes exist, except when they are not found in that way. All MOS belong to a specific place on the scale tree, except when they don't. All MOS have both a generator and a period, except when they don't.

I don't buy it. It seems to be it's just not a good definition.

The other problem of omitting the MOS at the EDO is that MOS is conceived within the concept of a continuum of possible generators which is illustrated graphically in numerous charts.
To omit these points seem mathematically more troublesome than including them.
It is like saying you can can have an MOS at any size and every size intervals except when it creates an EDO.

Actually this is incorrect as the tritone in the 7 tone scale of 700 cents in example.

5. Every interval will be one step size and one step size only.

-

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

cityoftheasleep wrote:
> Hi Graham,
> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >> I don't know what "consistency" means here. You keep saying it but
>> I only know the definition for equal temperaments.
> > I may be misusing it, but I thought it was what Herman was using to
> say whether a given generator might produce a better approximation to
> a given prime than the one defined by the mapping. I.e. a meantone
> with a generator of 690 cents is inconsistent because it gives a
> better 5/4 at 11 fifths minus 6 octaves than it does at 4 fifths
> minus 2 octaves. I'll have to go back and re-read that whole section
> of posts to see if I just read the word wrong or something.

When the topic originally came up, I believe I was using "consistency" as a shorthand for "consistency range" or "consistency limit", something along those lines. So the consistency limit of a meantone tuning with a 690 cent generator would be 11 (meaning that any scale with 11 or fewer notes in a contiguous chain of generators is consistent, but if you add a 12th note, the 11-fifths interval would make it inconsistent). On the other hand, any scale with a 680-cent generator is inconsistent with meantone, since 4 octaves up minus 3 fifths down is a better fifth than the "meantone" version.

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> The other problem of omitting the MOS at the EDO is that MOS
> is conceived within the concept of a continuum of possible
> generators which is illustrated graphically in numerous charts.

A generator by itself doesn't lead to an edo; you need to have a generator which is a rational portion of the period and even then you don't necessarily get an edo.

genewardsmith wrote:
> > --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>
>> No one can talk any sense to me for the next little while, because I just tried 46EDO for the first time! I think I'm in love.
>>
>> I feel like Goldilocks--this tuning is just right.
>>
>> 5ths nicely wide. Sweet thirds. The scale is not too big, not too small. It even has a symmetrical tritone. Familiar scales all sound good in it.
>>
>> I see that this is a well-known temperament, but for some reason I'd never tried it.
> > Try it! The only music for it I could find for this article
> > http://xenharmonic.wikispaces.com/46edo
> > was by me. It could use someone else, but then, there are so many tuning systems which have not been tried so far as I know.

Interesting, I see it's a starling temperament (tempers out 126/125) as well as tempering out the diaschisma 2048/2025. With a 9/46 generator you get rodan, and 13/46 is a variety of amity (which I've called "hitchcock" as it has an MOS of 39 steps). You also get shrutar with a 2/23 generator and a half-octave period. And it contains 23-ET which is a good mavila tuning.

I've generally overlooked these larger ET's, but this one sounds like it has some nice properties.

genewardsmith wrote:
> > --- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
> >> This reminds me of a question I've had for some time:
>> is there some special meaning to the name "leapday"?
> > Herman named it. Then reason he gave is that the smallest leapday MOS containing a full 13-limit hexad has 29 notes to the octave.
> >> Knowing some of the conventions of the regular
>> mapping school, I found that "leapyear" suggested
>> to me something like 2/29 (the notion for leapyear
>> day, February 29, in an American style with the
>> month first), and I wondered whether or how it
>> could have something to do with 2/29 octave, the
>> regular diatonic semitone or limma of 29-EDO?
> > That's what I thought, but Herman otherwise.

That might have made more sense, I guess. On the other hand, the temperament with a 2/29 generator already had a name (nautilus). I believe it got that name because its horogram reminded someone (Paul?) of a nautilus shell.

I guess we ought to have a list somewhere of name origins for temperaments, but it might be tricky to dig some of them up from the archives. I've already forgotten why "sentinel" was called that, and that was only a couple of months ago.

On Tue, Aug 31, 2010 at 1:04 PM, cityoftheasleep
<igliashon@...> wrote:
>
> > And I just placed them within a category, with my
> > category superpowers. Besides, after Margo just fleshed all of that
> > out... what's the debate about at this point? :)
>
> Margo named some temperaments that coincide with some MOS scales in what she calls the "isotonic" range, which I call 3\5-4\7. What IS the debate about at this point? Whether you should be allowed to call two temperaments with different mappings the same temperament?

I'm not sure what we're debating. I never said the last point, just
that it might be useful to group temperaments according to what MOS's
they produce, assuming you use a generator that's near-optimal. I
think we're in agreement at this point.

> > The reason I picked the term "temperament superclass" is that
> > superpyth and meantone are obviously very much perceptually related in
> > a way that meantone and porcupine are not, which lends itself to a
> > taxonomy whereby there is some larger concept of which meantone and
> > superpyth are subcategories.
>
> Yes, but the relationship has nothing to do with their properties as temperaments, but rather, their properties as MOS scales. It is the case that a specific tuning for a rank-2 temperament can also be used to define an MOS scale, and that's probably why you're confused about the difference between the two. It's just that if you ONLY have the tuning for a temperament, and not its mapping, there's no way to say what temperament it actually is.

As per your definition above, superpyth and meantone do not have any
specific properties as MOS scales, since they aren't tied down to a
specific generator. Then again, by that same logic, you can't really
say much about MOS scales either. The pajara MOS scales have a period
of half an octave, but who's to say that that period isn't just a
really detuned octave? And the LLsLLLs scales that we're used to -
who's to say that the period shouldn't be a third of an octave, and
the octave period isn't just a really sharp third-of-an-octave? Who's
to say that 13-equal temperament isn't actually 12-equal with a really
flat octave? If we aren't going to recognize "common-sense"
delineations for the purposes of my example, then why recognize them
anywhere?

> > Right, I understand very well that you could say that a meantone
> > generator is 1199 cents and the period is 1200 cents. I'm not dodging
> > that point because I don't understand it, I'm dodging it because I
> > think we all have a decent "common sense" definition of what
> > constitutes a meantone and what doesn't, and that rigorously defining
> > it isn't necessary at this point to flesh out how temperaments fall
> > into larger perceptual categories.
>
> Is our common-sense definition enough? Is 686 cents a meantone generator? Is 690 cents? If they are, then why isn't 703 cents? If they aren't, what are they? When you get away from TOP meantone (or some other optimized meantone), it becomes very difficult to say what is and isn't a meantone, even by common-sense definitions.

The idea to limit meantone to only the generators that produce 7+5
scales seemed like a fairly rigorous way to express an intuitive
common sense definition of it.

> > > But sure, you could certainly look at a list of optimal and near-optimal generators for different temperament classes and index them according to MOS families they belong to.
> >
> > aaaaaaa HAAAA! You admit it! :)
>
> Note the words "optimal" and "near-optimal". You need to have specific tunings for each temperament before any such indexing is possible.

You could start with the TOP tuning. Or start with the TOP tuning, and
then move to the nearest golden mean generator.

> > An idea I just had - perhaps that's one common sense way to delimit a
> > temperament's range. So we could say that meantone is only meantone if
> > the generator produces both 5+2 and 7+5 scales, (or equal variants
> > thereof). This means that a lower bound for meantone is 7-tet,
> > exclusive, and an upper bound for meantone is 12-tet, inclusive.
>
> Why inclusive of 12-tET? 12-tET doesn't support either 5+7 or 7+5. If you're going to exclude 7, you should exclude 12, or else you should include both. But it doesn't make sense to say that 7-EDO is "neither" 5+2 nor 2+5 but that 12-tET is "both" 7+5 and 5+7.

Because 12-tet produces a 5+2 scale and a 12-note scale which can be
viewed as a special case of 7+5. 7-tet produces a 7-note scale which
can be viewed as a special case of 5+2, but no 12-note scale at all.

> I went down this road a while back, though. I'll post a spreadsheet graphic I made where I listed the MOS families defined by branches on the Stern-Brocot tree, looking for patterns that would correlate to temperament classes. I too thought that we could define the range of a temperament class by MOS families of sufficiently-large size, and I too got very excited by this idea, thinking I'd found the missing link between linear temperaments and MOS families. But after some probing of the list and some long meditation, I came to the conclusion that I had not.
>
> Why did I conclude this? Because there are too many ways to invalidate the perceptual similarities on which the grouping is supposed to be based. Consider the problem that from 3\5-4\7, when you pass a certain point heading toward either end of the spectrum, the scale STOPS sounding like 5L+2s and starts sounding like 5-EDO or 7-EDO, so the perceptual grouping also stops making sense. I have not succeeded in finding that point.

That point will likely (as Rothenberg puts it) be listener-dependent
and have to do with some listener-dependent tolerance factor. Harmonic
Entropy is the same way, and the solution was to include a free "s"
parameter to roughly estimate listener sensitivity. I don't think it
kills the whole project though.

> And as Herman suggested, and I keep trying to get through to you

In all honesty, Igs, I'm getting tired of the patronizing. I am not an
idiot, and my stance on things should be perfectly clear by now. The
reason that I disagree with you that it's impossible to come up with a
structure that relates temperaments and MOS families isn't that I
don't understand the concept that temperament error is unbounded. It's
that I'm proposing a way to bound it, and you just keep telling me
that temperament error is unbounded, so it's impossible. So then I
respond with a way to bound it, and your response is usually that
temperament error is unbounded, so it's impossible. Then I respond
with a way to bound it...

> temperament tunings vary in consistency across a given pitch-range. 690 cents is an inconsistent tuning for Meantone

How so? How is it "just inconsistent?" If I take any meantone
generator that doesn't form a closed chain, a better approximation to
5 will come up somewhere down the line than 4 generators, unless the
generator is exactly that of 1/4-comma meantone. And a better
approximation to 3 will come up somewhere down the line unless the
generator is exactly 3/2.

> equally as inconsistent as 709 cents is, in fact. From a perspective of both error and consistency, neither should be considered a meantone, yet 690 cents is squarely between 12-EDO and 7-EDO.

I am suggesting an alternate way to bound the meantone generator range
besides the concept of consistency, because I have objections to it.
The biggest objection I have is that any meantone, except for a few
objections, is only consistent out to a finite number of generators.
If we're going to limit things to going out a certain number of
generators, then is that not a similar concept to limiting things to
certain MOS's, which is equally as arbitrary?

> > Yes... I like this idea a lot :) Seems like it might lead to something
> > interesting. I also wonder, just as you can specify a linear
> > temperament by two ET's, what can you specify by two MOS's? Some
> > higher-limit generalization of MOS?
>
> No, giving two MOS's is pointless because the higher one implies the lower one.

Ah, my fault.

-Mike

On Tue, Aug 31, 2010 at 1:13 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > I assume if we come up with the only generator that matches the full
> > infinite series of these, we'll get some sort of "omega-proper
> > meantone" in which every MOS produced is strictly proper. I'm also
> > curious to see how this matches up to TOP.
>
> That'd be the "golden generator". Man, you REALLY need to look at Wilson's diagrams of the scale tree. I recommend:
> http://www.anaphoria.com/sctree.PDF
>
> Meditate on this, and possibly this:
> http://www.anaphoria.com/line.PDF
>
> And you shall find what you are seeking, if it's there to be found.
>
> -Igs

Thanks for the reference. I've seen this stuff before, but never
understood it - it was one of the first tuning-related documents I
saw, back before I understood any of the concepts at all. I'll check
it out again and hopefully make more sense of it this time.

Just another reason why I want to get a wikiversity course up...

-Mike

On Tue, Aug 31, 2010 at 5:59 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I think this is something similar to woolhouse-optimal, where the
> > dyads are 5/4, 6/5, and 3/2. But if your goal was to do that above,
> > you could probably work it out as an algebraic expression and get the
> > least-squares error or something like that.
>
> Woolhouse is the least squares tuning, and 1/4-comma meantone is the minimax tuning.

Isn't woolhouse the tuning with least squares error for 5/4, 6/5, and
3/2? That's what I thought I read on the tonalsoft encyclopedia. If
you wanted to calculate the least squares error for all of the
intervals Michael mentioned, would it still be a form of woolhouse?

-Mike

On Tue, Aug 31, 2010 at 9:53 PM, Herman Miller <hmiller@...> wrote:
>
> When the topic originally came up, I believe I was using "consistency"
> as a shorthand for "consistency range" or "consistency limit", something
> along those lines. So the consistency limit of a meantone tuning with a
> 690 cent generator would be 11 (meaning that any scale with 11 or fewer
> notes in a contiguous chain of generators is consistent, but if you add
> a 12th note, the 11-fifths interval would make it inconsistent). On the
> other hand, any scale with a 680-cent generator is inconsistent with
> meantone, since 4 octaves up minus 3 fifths down is a better fifth than
> the "meantone" version.

You mean 4 octaves up minus 3 fifths down is a better 5/4 than the
"meantone" version, right?

But even so, wouldn't that just make a 680 cent "meantone" consistent
out to something like 4 generators? 4 generators gives you C-G-D-A,
and the C-A will be closer to 8/5 than 5/3, but there's no "better"
8/5 with which to compare. So 5 has been introduced indirectly, but
not directly yet.

Although I suppose you could define it whichever way you want -
perhaps a tuning could be said to be "absolutely inconsistent" if it
ends up being inconsistent right when one of the basis primes gets
first introduced.

-Mike

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

> > Woolhouse is the least squares tuning, and 1/4-comma meantone is the minimax tuning.
>
> Isn't woolhouse the tuning with least squares error for 5/4, 6/5, and
> 3/2? That's what I thought I read on the tonalsoft encyclopedia. If
> you wanted to calculate the least squares error for all of the
> intervals Michael mentioned, would it still be a form of woolhouse?

Which intervals are those?

I understand that for the most part EDO are rarer.
the chart that I.Jones referred to earlier is a good example of how MOS was conceived
http://anaphoria.com/line.PDF

But in a way it seems on some philosophical level you might tend to view the continuum as one of an infinite number of EDO which is just as valid as what might be my own JI or a recurrent sequence.

If you think it should have a special name i don't necessary object to that, but "degenerate" if one saw the continuum as EDO would be a not the best place to start.

--- In tuning@yahoogroups.com </tuning/post?postID=_whwcGCVyrzFlzftIqfkd8dmWpVTJUr_izbZqXOp3lyx6zFyJ4d8qOkyD18vu_3UzVXtFL3zCVAx0Im0_A>, Kraig Grady <kraiggrady@...> wrote:
>
> The other problem of omitting the MOS at the EDO is that MOS
> is conceived within the concept of a continuum of possible
> generators which is illustrated graphically in numerous charts.

A generator by itself doesn't lead to an edo; you need to have a generator which
is a rational portion of the period and even then you don't necessarily get an
edo.
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I'm not sure what we're debating. I never said the last point, just
> that it might be useful to group temperaments according to what MOS's
> they produce, assuming you use a generator that's near-optimal. I
> think we're in agreement at this point.

Indeed, at least on this level.

> > Yes, but the relationship has nothing to do with their properties as temperaments, but rather, their properties as MOS scales. It is the case that a specific tuning for a rank-2 temperament can also be used to define an MOS scale, and that's probably why you're confused about the difference between the two. It's just that if you ONLY have the tuning for a temperament, and not its mapping, there's no way to say what temperament it actually is.
>
> As per your definition above, superpyth and meantone do not have any
> specific properties as MOS scales, since they aren't tied down to a
> specific generator.

They are if optimized.

> Then again, by that same logic, you can't really
> say much about MOS scales either. The pajara MOS scales have a period
> of half an octave, but who's to say that that period isn't just a
> really detuned octave?

Because the period is not defined abstractly but by a cents-value. It makes no sense to refer to 600 as a "really small 1200". Periods have unique identities that aren't of necessity defined in relation to other identities. IOW we can speak of flat octaves because the interval of an octave has some perceptual flex, but only an interval of 1200 cents can be an interval of 1200 cents.

But yes, the period has to be specified before you can group MOS's, which is why (when I'm being rigorous), I give MOS's in the form "nL+ms=period" or "nL+ms=x-EDO".

> And the LLsLLLs scales that we're used to -
> who's to say that the period shouldn't be a third of an octave, and
> the octave period isn't just a really sharp third-of-an-octave?

Because cents-values, unlike mappings, are non-arbitrary.

> The idea to limit meantone to only the generators that produce 7+5
> scales seemed like a fairly rigorous way to express an intuitive
> common sense definition of it.

Useful? Yes. Rigorous? Not hardly.

> You could start with the TOP tuning. Or start with the TOP tuning, and
> then move to the nearest golden mean generator.

But where would you stop?

> Because 12-tet produces a 5+2 scale and a 12-note scale which can be
> viewed as a special case of 7+5.

Why? Why can it be viewed this way?

> That point will likely (as Rothenberg puts it) be listener-dependent
> and have to do with some listener-dependent tolerance factor. Harmonic
> Entropy is the same way, and the solution was to include a free "s"
> parameter to roughly estimate listener sensitivity. I don't think it
> kills the whole project though.

It doesn't? How do you propose to get around it? The same way it's done in H.E.?

> In all honesty, Igs, I'm getting tired of the patronizing. I am not an
> idiot, and my stance on things should be perfectly clear by now. The
> reason that I disagree with you that it's impossible to come up with a
> structure that relates temperaments and MOS families isn't that I
> don't understand the concept that temperament error is unbounded. It's
> that I'm proposing a way to bound it, and you just keep telling me
> that temperament error is unbounded, so it's impossible. So then I
> respond with a way to bound it, and your response is usually that
> temperament error is unbounded, so it's impossible. Then I respond
> with a way to bound it...

But your ways of bounding it are arbitrary and based on concepts unrelated to temperament. In your last post, you clearly demonstrated that you didn't know what a temperament actually *is*, at least as the term is used here. I certainly don't think you're an idiot, and I don't blame you for my failure to communicate properly. It's just that I'm trying as best I can and I don't know why I keep failing to articulate this. The point I'm trying to make is not that you can't impose boundaries on temperaments, just that there is no non-arbitrary way of doing so. Any attempt at defining tuning ranges for temperaments beyond the "optimum tuning" is problematic. Maybe this isn't an issue to you, and that as long as a generalization about a temperament is true often enough to be useful, the cases where it fails to be true are irrelevant. I'm holding out for a higher standard of rigor, if I can find one, and I think we all should do the same. If we fail to find a more rigorous and complete taxonomic method, we can always come back and settle for any of the various incomplete possibilities.

> How so? How is it "just inconsistent?" If I take any meantone
> generator that doesn't form a closed chain, a better approximation to
> 5 will come up somewhere down the line than 4 generators, unless the
> generator is exactly that of 1/4-comma meantone.

Not true. Are you familiar with strange attractors? It could be the case that a generator only gets so close before it starts getting further away. Maybe Gene can shed more light on it (maybe prove me wrong, even!).

> > equally as inconsistent as 709 cents is, in fact. From a perspective of both error and consistency, neither should be considered a meantone, yet 690 cents is squarely between 12-EDO and 7-EDO.
>
> I am suggesting an alternate way to bound the meantone generator range
> besides the concept of consistency, because I have objections to it.

And yet consistency is itself an objection to your proposal. A mexican stand-off.

> The biggest objection I have is that any meantone, except for a few
> objections, is only consistent out to a finite number of generators.
> If we're going to limit things to going out a certain number of
> generators, then is that not a similar concept to limiting things to
> certain MOS's, which is equally as arbitrary?

Right. It's all arbitrary. And it's all problematic. Should we really settle for boundaries that we know are inconsistent or problematic in some way? Is that really any more useful than not bothering to draw boundaries at all and just "winging it"? BTW, I'm not being rhetorical with this question.

> > No, giving two MOS's is pointless because the higher one implies the lower one.
>
> Ah, my fault.
>
BUT you can specify an MOS by giving a sequence of EDOs, just like with temperaments.

-Igs

On Wed, Sep 1, 2010 at 12:01 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > Woolhouse is the least squares tuning, and 1/4-comma meantone is the minimax tuning.
> >
> > Isn't woolhouse the tuning with least squares error for 5/4, 6/5, and
> > 3/2? That's what I thought I read on the tonalsoft encyclopedia. If
> > you wanted to calculate the least squares error for all of the
> > intervals Michael mentioned, would it still be a form of woolhouse?
>
> Which intervals are those?

Here's the list:

> So, for example, 12TET might be categorized
> A1) An error of about 14 cents as that's about how far the third and sixth are from "goal" dyadic ratios they estimate.
> A2) Since the fifth is very important in meantone...for meantone scales give the error of the fifth IE 1.95 cents
> B1) A list of goal ratio types of
> 15/14
> 9/8
> 5/4
> 4/3
> 7/5
> 3/2
> 8/5
> 27/16 or 5/3
> 16/9
> 17/9
> 2/1
> (ideally listing the error of each dyad representation in cents from its ideal as well)

Not sure why 17/9 is in there instead of 15/8.

-Mike

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

If I take any meantone
> > generator that doesn't form a closed chain, a better approximation to
> > 5 will come up somewhere down the line than 4 generators, unless the
> > generator is exactly that of 1/4-comma meantone.
>
> Not true. Are you familiar with strange attractors? It could be the case that a generator only gets so close before it starts getting further away. Maybe Gene can shed more light on it (maybe prove me wrong, even!).

Sorry, but he's right. Successively better approximations will appear, a fact you can relate to continued fractions or the Stern-Brocot tree.

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

> Not sure why 17/9 is in there instead of 15/8.

It all seems pretty arbitrary.

[I would also add as a guide for generalized keyboards for scales falling underneath.]

Finnamore's intro here on the horograms which are generated by the scale tree is quite good and fun written.
http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm

From a scales perspective, each node on the scale tree is a fraction of a
period, usually taken to be an octave. The tree itself is a recurrent sequence:
you start with the nodes 0/1, 1/1, and 1/0. The next nodes are 0+1/1+1 and
1+1/1+0, i.e. 1/2 and 2/1, which can be read "1 out of 2-EDO" and "2 out of
1-EDO" if we're dealing with scales of an octave period. Then you make the next
branches as 1+1/2+1, 1+0/2+1, 2+1/1+1, and 2+1/1+0, or 2/3, 1/3, 3/2, and 3/1.
At this point it should be clear that the nodes of a given level are defined as
the classical mediants between the nodes of the previous level.

The important thing about the scale tree is that it is a visual representation
of all possible MOS families. Between any node and its parent node is an MOS
family. See my folder in the "Files" section for a document called "MOS
spreadsheet".

You can also use the tree as a chart of JI intervals, but that's another story.

-Igs

--- In tuning@yahoogroups.com </tuning/post?postID=agjyAqA3UCEfPqxPHTt00pkJ12vxNejODx8S5X3A6AMm6kChct8yM9BEHswhWaN5eWBRAHmjkxZ7MotwuRvBGj8>, "John Moriarty" <JlMoriart@...> wrote:
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

About defining edges to temperaments:
I think another form of temperament may provide the answers to where one should or shouldn't draw lines for the edges of a given temperament.

The Syntonic Temperament (perhaps not meantone, but at least syntonic =P) is defined not just by the tempering of JI ratios to unison, but also the tempering of partials of a harmonic timbre.

For example:
The fundamental and its n octaves are defined as being (0,n) in the syntonic temperament: f, 2f, 4f, 8f, etc are defined as the octaves of (0,0)
The third partial and all its n octaves are defined as (1,1+n): 3f, 6f, 9f, etc are defined as the octaves of (1,1).
The fifth partial and all its n octaves above it are defined as (4,n): 5f, 10f, 15f, 20f, etc are defined as the octaves of (4,0).
These harmonic partial temperaments define the "5-limit timbre" for the syntonic temperament, and they can be derived from the pre-defined comma sequence.

Could it be that, given a timbre defined by a prime limit temperament, one could then define the values of a generator over which that timbre will be valid? One would have to define what a valid timbre is, but I have a suggestion.

Let us define a valid timbre as one whose ratios between two successive partials are always greater than the ratio between the following pair of partials. I admit I arrived at this definition partially intuitively and partially because of where it will take us, but does it not make sense that the distance between the third and fourth partials be greater than the distance between the fourth and fifth partials, and that the distance between the fourth and fifth partials be greater than the distance between the fifth and sixth? I could be talking gibberish, but if I'm not then the constraint for a valid 5 limit timbre would be that 4/3>5/4>6/5. This statement is true from a generator of 685 cents to 720 cents, or from 5-edo to 7-edo. That is the same definition that keeps the diatonic scale (5L, 2s) valid.

Then a seven-limit timbre, which would introduce the temperament of the seventh partial and all its n octaves to (10,-3+n) could be defined as being valid where 6/5>7/6>8/7 is true. This would mean that the augmented second must remain smaller than the minor third, and that the diminished third must remain smaller than the augmented second. This is the case from a generator of 700 cents to 695, or from 12-edo to 19-edo, which is the location containing the (7L, 12s) scale is valid.

It appears that the MOS scales are *very tightly linked* to temperamental theory, when one accounts for the temperament of a timbre defined by comma sequence of a temperament. As one tempers away more comma's, one descends deeper into scale families (right?), and how each prime limit is tempered defines the commas which define over which generators the MOS scales will validate the timbre.
If I were to follow the pattern, I'd say that defining the 11-limit timbre would cut the remaining section of the syntonic temperament into an even smaller piece over which another more complex MOS would remain valid, but I apparently don't have to, because I just noticed on these diagrams from the TransformSynth:
http://img704.imageshack.us/f/continuac.png/
They already have the areas of generator sizes I've been describing labeled (with the orange, red, and blue bars) according to the limit they support!

I think we can define things this way:
5-limit meantone (tempering 81/80) is valid from 685 to 720, along with a corresponding MOS scale (diatonic: 5L, 2s).
7-limit meantone (tempering 59049/57344) is valid from 695 to 700 along with an MOS (7L,12s).
11-limit meantone (tempering 16777216/17537553) is valid from 695 to 697 along with a corresponding MOS scale.
Etc, etc.

If different comma's were tempered out to give us 7 and 11 limit, we would end up with different subsets of the entire 5 limit Syntonic Temperament.

John Moriarty

Ps. Now think about making the Syntonic temperament lower limit, like three limit. This would require that 2/1>3/2>4/3 be true for a tempered timbre to be valid. This would be true from 600 cents to 1200 cents, wouldn't it?

At the risk of eliciting howls of derision, I'm admitting to experimenting with 46EDO with every fourth note removed, starting on 26.08 cents, for a total of 34 pitches.

I expect, though, that I will find I need those omitted pitches.

If this is "Hitchcock", then maybe this subset is "Psycho", or more ridiculous, "Amityville Horror", or "Shutter Island".

Seriously, if anyone is interested and can spot the most obvious problem with this subset beyond the fact that it doesn't allow equal playing in all keys, lemme know.

caleb

! 46et.scl "Shutter Island"
46et ev-4th gone starting 26.08
34
! 0
52.17391 1
78.261 2
104.34 3 b2
156.52 4
182.609 5
!
208.696 6 2
260.8696 7
286.9565 8 b3 narrow
313.0435 9 b3 wide
365.2174 10
391.3043 11 3
!
417.3913 12
469.5652 13
495.6522 14 4
521.7391 15
573.913 16
600 17 #4/b5 middle-!tritone=
!
626.087 18
678.2609 19
704.3478 20 5 !fifth=
730.4348 21
782.6087 22
808.6957 23 b6
!
834.7826 24
886.9565 25
913.0435 26 6
939.1304 27
991.3043 28
1017.391 29 b7
!
1043.478 30
1095.652 31 mj7
1121.739 32
1147.826 33
1200 34

On Aug 31, 2010, at 10:20 PM, Herman Miller wrote:

> genewardsmith wrote:
> >
> > --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
> >>
> >> No one can talk any sense to me for the next little while, because I just tried 46EDO for the first time! I think I'm in love.
> >>
> >> I feel like Goldilocks--this tuning is just right.
> >>
> >> 5ths nicely wide. Sweet thirds. The scale is not too big, not too small. It even has a symmetrical tritone. Familiar scales all sound good in it.
> >>
> >> I see that this is a well-known temperament, but for some reason I'd never tried it.
> >
> > Try it! The only music for it I could find for this article
> >
> > http://xenharmonic.wikispaces.com/46edo
> >
> > was by me. It could use someone else, but then, there are so many tuning systems which have not been tried so far as I know.
>
> Interesting, I see it's a starling temperament (tempers out 126/125) as
> well as tempering out the diaschisma 2048/2025. With a 9/46 generator
> you get rodan, and 13/46 is a variety of amity (which I've called
> "hitchcock" as it has an MOS of 39 steps). You also get shrutar with a
> 2/23 generator and a half-octave period. And it contains 23-ET which is
> a good mavila tuning.
>
> I've generally overlooked these larger ET's, but this one sounds like it
> has some nice properties.

It had its 7/4 removed, rendering it more docile, but less interesting.

Oh well, back to all 46.

On Sep 1, 2010, at 8:53 AM, caleb morgan wrote:

>
>
>
>
> Seriously, if anyone is interested and can spot the most obvious problem with this subset beyond the fact that it doesn't allow equal playing in all keys, lemme know.
>
> caleb
>
> ! 46et.scl "Shutter Island"
> 46et ev-4th gone starting 26.08
> 34
> ! 0
> 52.17391 1
> 78.261 2
> 104.34 3 b2
> 156.52 4
> 182.609 5
> !
> 208.696 6 2
> 260.8696 7
> 286.9565 8 b3 narrow
> 313.0435 9 b3 wide
> 365.2174 10
> 391.3043 11 3
> !
> 417.3913 12
> 469.5652 13
> 495.6522 14 4
> 521.7391 15
> 573.913 16
> 600 17 #4/b5 middle-!tritone=
> !
> 626.087 18
> 678.2609 19
> 704.3478 20 5 !fifth=
> 730.4348 21
> 782.6087 22
> 808.6957 23 b6
> !
> 834.7826 24
> 886.9565 25
> 913.0435 26 6
> 939.1304 27
> 991.3043 28
> 1017.391 29 b7
> !
> 1043.478 30
> 1095.652 31 mj7
> 1121.739 32
> 1147.826 33
> 1200 34
>
>
> On Aug 31, 2010, at 10:20 PM, Herman Miller wrote:
>
>> genewardsmith wrote:
>> >
>> > --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>> >>
>> >> No one can talk any sense to me for the next little while, because I just tried 46EDO for the first time! I think I'm in love.
>> >>
>> >> I feel like Goldilocks--this tuning is just right.
>> >>
>> >> 5ths nicely wide. Sweet thirds. The scale is not too big, not too small. It even has a symmetrical tritone. Familiar scales all sound good in it.
>> >>
>> >> I see that this is a well-known temperament, but for some reason I'd never tried it.
>> >
>> > Try it! The only music for it I could find for this article
>> >
>> > http://xenharmonic.wikispaces.com/46edo
>> >
>> > was by me. It could use someone else, but then, there are so many tuning systems which have not been tried so far as I know.
>>
>> Interesting, I see it's a starling temperament (tempers out 126/125) as
>> well as tempering out the diaschisma 2048/2025. With a 9/46 generator
>> you get rodan, and 13/46 is a variety of amity (which I've called
>> "hitchcock" as it has an MOS of 39 steps). You also get shrutar with a
>> 2/23 generator and a half-octave period. And it contains 23-ET which is
>> a good mavila tuning.
>>
>> I've generally overlooked these larger ET's, but this one sounds like it
>> has some nice properties.
>
>
>

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
>
> About defining edges to temperaments:
> I think another form of temperament may provide the answers to where one should or shouldn't draw lines for the edges of a given temperament.
>
> The Syntonic Temperament (perhaps not meantone, but at least syntonic =P) is defined not just by the tempering of JI ratios to unison, but also the tempering of partials of a harmonic timbre.
>

I really don't see why we have to bring the tempering of partials into this. I have plenty of issues with Sethares' ideas about matching timbres to scales, but those notwithstanding, I don't think it is necessary to build into a temperament a set of timbres for which the temperament is valid, since it will be a circular definition anyway (you'll have to know how the temperament is tuned to know how to tune the partials, and how to tune the partials to know how to tune the temperament!)

> Could it be that, given a timbre defined by a prime limit temperament, one could then
> define the values of a generator over which that timbre will be valid? One would have to > define what a valid timbre is, but I have a suggestion.

All timbres are valid for all temperaments. There's really no sense in trying to tell people what instruments or sounds they can and can't use in making music with a given temperament.

> Let us define a valid timbre as one whose ratios between two successive partials are
> always greater than the ratio between the following pair of partials.

So, uh, inharmonic timbres will never be valid?

> I could be talking gibberish, but if I'm not then the constraint for a valid 5 limit timbre
> would be that 4/3>5/4>6/5. This statement is true from a generator of 685 cents to
> 720 cents, or from 5-edo to 7-edo. That is the same definition that keeps the diatonic
> scale (5L, 2s) valid.

This is another place the argument gets circular. I say that once you get past a certain generator size, the intervals produced by 4gen-2per and -3gen+2per no longer are no longer close enough to 5/4 and 6/5 (respectively) to be psychoacoustically identified as those ratios, so it no longer makes sense to call them that. Then YOU say that we have to call them that, because this is the syntonic temperament and our generator is still a recognizable fifth, ergo 81/64 (four fifths minus two octaves) equals 5/4. Then I say that maybe the syntonic temperament cannot be validly extended as far as you want it to, assuming musical intervals are considered to have psychoacoustic identities based on JI that are finite in size. Then YOU say that psychoacoustics doesn't matter here, because we're talking about temperament and of course interval identities are going to get warped and conflated with one another.

The problem is that we're both insisting on contradictory ways to define intervals. I think it makes the most sense to defined "5/4" according to some distance, harmonic-entropy-wise, from the cents-value of the 5/4 frequency ratio. By definition, this places it between 4/3 and 6/5. The way you're defining it for the syntonic temperament implies not just that a 5/4 can be ANYTHING so long as it is between the 4/3 and 6/5, but that in a near-by temperament like Father or Mavila, a 5/4 might actually be HIGHER in pitch than 4/3 (Father) or LOWER in pitch than 6/5 (Mavila). If you're going to suggest this, then I really don't know what the point is of naming these intervals after JI frequency ratios.

A much BETTER way to define these inequalities that determine the bounds of the diatonic range, one that doesn't lead to psychoacoustic absurdity, is to use generator-period coordinates. Instead of 4/3>5/4>6/5, you can use (-1gen+1per)>(4gen-2per)>(-3gen+2per). In Father temperament, you get (4gen-2per)>(-1gen+1per)>(-3gen+2per), and in Mavila, you get (-1gen+1per)>(-3gen+2per)>(4gen-2per). With these definitions, we don't even have to look at psychoacoustics, and to boot it's also the case that these inequalities can be SOLVED for either generator or period (or both, if you're crazy like that), so the inequality can actually be read as the formula that defines the family.

If you want to narrow down the family further, you can add other inequalities to the picture. For instance, between 7-EDO and 12-EDO, an augmented 2nd (9gen-5per) is lower than a minor third, but from 12-EDO to 5-EDO, it is higher. So you could specify what a lot of people call the "meantone range" by the inequality (-1gen+1per)>(4gen-2per)>(-3gen+2per)>(9gen-5per), and the "schismatic range" as (-1gen+1per)>(4gen-2per)>(9gen-5per)>(-3gen+2per).

We could also bring the diminished 3rd into it if we wanted, since you like defining 7-limit diatonics by relationship between minor 3rd, diminished 3rd, and augmented 2nd. The diminished third is (-10gen+6per), so now we want the inequality to (-1gen+1per)>(4gen-2per)>(-3gen+2per)>(9gen-5per)>(-10gen+6per). At 12-EDO, (-3gen+2per)=(9gen-5per), but both are >(-10gen+6per). At 19-EDO, (9gen-5per)=(-10gen+6per), but both are <(-3gen+2per); so 19-EDO is the upper limit and 12-EDO the lower. 31-EDO is right smack in the middle, being the classical mediant, and the noble mediant--the "Golden Meantone"--is between 19-EDO and 31-EDO. If you look at the Scale Tree, it's plain as day.

If I could find a quick and easy way of generating these inequalities for each level of the Scale Tree, I'd be in hog-heaven.

> Then a seven-limit timbre, which would introduce the temperament of the seventh
> partial and all its n octaves to (10,-3+n) could be defined as being valid where
> 6/5>7/6>8/7 is true.

Look at 22-EDO in terms of your inequalities. In 22-EDO, 5/4>7/6>6/5>9/8>8/7...but that 5/4 is psychoacoustically 9/7 at 436.36 cents, the 7/6 is psychoacoustically 5/4 at 381.82 cents, the 6/5 is psychoacoustically 7/6 at 272.73 cents, and the 8/7 is psychacoustically about 16/15 at 109.09 cents. Only the 9/8, at 218.18 cents, can reasonably be identified with an actual 9/8 frequency ratio. 22-EDO is in the range of the syntonic temperament, but defining its intervals in the way you propose makes a huge mess of their psychoacoustic identities.

> It appears that the MOS scales are *very tightly linked* to temperamental theory, when
> one accounts for the temperament of a timbre defined by comma sequence of a
> temperament. As one tempers away more comma's, one descends deeper into scale
> families (right?), and how each prime limit is tempered defines the commas which define
> over which generators the MOS scales will validate the timbre.

Not just the commas, but also the error. You can temper as many commas as you like, but if you allow for large error, the "range" will still remain wide.

> If I were to follow the pattern, I'd say that defining the 11-limit timbre would cut the
> remaining section of the syntonic temperament into an even smaller piece over which
> another more complex MOS would remain valid [snip]

It depends on how you define the 11-limit! There are many, many ways to to define 11 in terms of 3, 5, and 7, if you're actually concerned about the 11th harmonic in your temperament sounding like the actual 11th harmonic.

> If different comma's were tempered out to give us 7 and 11 limit, we would end up with > different subsets of the entire 5 limit Syntonic Temperament.

Yes, exactly!

> Ps. Now think about making the Syntonic temperament lower limit, like three limit. This > would require that 2/1>3/2>4/3 be true for a tempered timbre to be valid. This would > be true from 600 cents to 1200 cents, wouldn't it?

It would make no sense to call this Syntonic, because the syntonic comma is by definition 5-limit.

From a gen/per coordinate perspective, however, this does make sense: you'd have (0gen+1per)>(1gen+0per)>(-1gen+1per).

-Igs

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:

> These harmonic partial temperaments define the "5-limit timbre" for the syntonic temperament, and they can be derived from the pre-defined comma sequence.

What timbre? I see no definition of a timbre in the above.

> Let us define a valid timbre as one whose ratios between two successive partials are always greater than the ratio between the following pair of partials.

For timbres derived from periodic waveforms, this isn't an issue. What has this to do with meantone?

> It appears that the MOS scales are *very tightly linked* to temperamental theory, when one accounts for the temperament of a timbre defined by comma sequence of a temperament.

First you need to define "timbre defined by comma sequence" which you haven't done.

> I think we can define things this way:
> 5-limit meantone (tempering 81/80) is valid from 685 to 720, along with a corresponding MOS scale (diatonic: 5L, 2s).
> 7-limit meantone (tempering 59049/57344) is valid from 695 to 700 along with an MOS (7L,12s).

You can also define it as tempering 126/125 or 225/224. [81/80, 59049/57344] is what I've termed the normal comma list.

> 11-limit meantone (tempering 16777216/17537553) is valid from 695 to 697 along with a corresponding MOS scale.
> Etc, etc.

If you invert that and append it to the previous list, you get the normal comma list for what's sometimes called "meanpop" temperament. The reason for the name is that there is another, competing version of 11-limit meantone, whose normal list is obtained by appending 387420489/369098752 to the 7-limit list.

Meanpop could be described as what you get by combining meantone with 11-limit marvel. You can get it by adding 385/384 to the commas in place of using the normal list, but the normal list is useful for sorting out the relationships between competing versions of a temperament such as we have here.

Why? you ask. Because I could, I answer.

Li'l Miss Scale Oven is a tremendous improvement over a hand-calculator, and I'm only using one little part--the knead and fold feature.

This should make it possible, also, to investigate some of the tunings that Graham Breed mentions.

But today I just had fun, and took a subset of 94EDO pitches to make a 48-note, nearly epimorphic scale.

To my ear now, a little beating sounds better than the buzzy, slightly teeth-clenching, slightly finger-in-socket effect of perfect JI.

No, I don't really know what I'm doing. I just looked up 96EDO in the Xenharmonic Wiki:

http://xenharmonic.wikispaces.com/edo

I give you...The "Epidural".

! caleb epidural from94edo.scl
from94edo, caleb's epidural with 48 tones
48
!0
51.064
76.596
102.128 b2
114.894
140.426
153.191
178.723
!
204.255 2
229.787
268.085
293.617
319.149
344.681
357.447
382.979 3
!
408.511
421.277
446.808
472.34
497.872 4/3
536.17
561.702
587.234
!
600.0 symmetric tritone
612.766
651.064
676.596
702.128 5th
753.191
778.723
804.255
!
817.021
842.553
855.319
880.851
906.383
931.915
970.213
995.745
!
1021.277
1046.808
1059.574
1085.106 mj 7th
1110.638
1123.404
1148.936
1174.468
!
1200.

>
>

First of all, thanks everyone for the help with the scale tree.

> All timbres are valid for all temperaments. There's really no sense in trying to tell people what instruments or sounds they can and can't use in making music with a given temperament.

Maybe valid wasn't a great word. I'll reword a few paragraphs:

Instead of:
> > Could it be that, given a timbre defined by a prime limit temperament, one could then
> > define the values of a generator over which that timbre will be valid? One would have to > define what a valid timbre is, but I have a suggestion.
and
> > Let us define a valid timbre as one whose ratios between two successive partials are
> > always greater than the ratio between the following pair of partials.

I should say instead:
Could it be that, given a variable timbre defined by a prime limit temperament, one could then define the values of a generator over which that timbre will still function harmonically? One would have to define what functioning harmonically is, but I have a suggestion.

Let us define a harmonically functioning timbre as one whose ratios between two successive partials are always greater than the ratio between the following pair of partials.

> So, uh, inharmonic timbres will never be valid?

Valid was a bad choice, so I'll say this. Given timbre mapping, there is a goal: minimize sensory dissonance while retaining a timbre that has the recognizable function of a fully harmonic timbre.
Though certainly not useless, a timbre of a snare drum playing a 4/5/6 major triad wouldn't do much for me, you know?

> I say that once you get past a certain generator size, the intervals produced by 4gen-2per and -3gen+2per no longer are no longer close enough to 5/4 and 6/5 (respectively) to be psychoacoustically identified as those ratios, so it no longer makes sense to call them that. Then YOU say that we have to call them that, because this is the syntonic temperament and our generator is still a recognizable fifth, ergo 81/64 (four fifths minus two octaves) equals 5/4. Then I say that maybe the syntonic temperament cannot be validly extended as far as you want it to, assuming musical intervals are considered to have psychoacoustic identities based on JI that are finite in size. Then YOU say that psychoacoustics doesn't matter here, because we're talking about temperament and of course interval identities are going to get warped and conflated with one another.
> The problem is that we're both insisting on contradictory ways to define intervals. I think it makes the most sense to defined "5/4" according to some distance, harmonic-entropy-wise, from the cents-value of the 5/4 frequency ratio. By definition, this places it between 4/3 and 6/5. The way you're defining it for the syntonic temperament implies not just that a 5/4 can be ANYTHING so long as it is between the 4/3 and 6/5, but that in a near-by temperament like Father or Mavila, a 5/4 might actually be HIGHER in pitch than 4/3 (Father) or LOWER in pitch than 6/5 (Mavila). If you're going to suggest this, then I really don't know what the point is of naming these intervals after JI frequency ratios.

It isn't that psychoacoustics doesn't matter anymore. It's that, given a tempered timbre who's fifth partial and it's octaves are mapped to coincide with that very "inaccurate" 5/4, that inaccurate 5/4 which is actually closer in ratio to 9/7 will still sound, harmonically, like a 5/4 psychoacoustically (because of the way the mapped timbres line up). The definition of just ratios is interesting, because you could define the interval 5/4 as you do, as an exact ratio, or you could define 5/4 as I do, as the ratio between the fifth and fourth partials, whatever they may be, and link it to the same perceptual/musical connotation that the exact 5/4 ratio gives when used with a fully harmonic timbre.

I suggest that it still makes sense to call an interval what it is defined as by the temperament even if it is closer to a different JI ratio because it will still sound perceptually like its defined function due to the tempered timbre.

> This is another place the argument gets circular.

I know that on the surface this seems like circular logic. I define the timbre by the temperament, and then use the timbre to allow the temperament to function as I define it. I don't think it's circular, however. I think it is powerful, and consistent.

A friend of mine once countered the explanation for evolution by natural selection by defining it as circular logic:
"So only those that can survive will reproduce, and so then only those that reproduced have the kids that can survive? It's circular logic!"

But I think one can pretty easily see that the theory of evolution does not use a false or circular logic. It just follows a logical progression where some propositions have conclusions that support other conclusions.

>The way you're defining it for the syntonic temperament implies not just that a 5/4 can be ANYTHING so long as it is between the 4/3 and 6/5, but that in a near-by temperament like Father or Mavila, a 5/4 might actually be HIGHER in pitch than 4/3 (Father) or LOWER in pitch than 6/5 (Mavila). If you're going to suggest this, then I really don't know what the point is of naming these intervals after JI frequency ratios.

They way I'm defining it, it may work out that the interval supposed to function as a 5/4 will end up being larger than the exact ratio 4/3 in some other temperament (I don't know, I haven't checked), but I do know that the interval defined as 5/4 will never be larger than the interval defined as 4/3 in that temperament (because the functional range of a 5 limit temperament is where that temperament can define a harmonically functional timbre. That forces 4/3>5/4>6/5 to be true). And given the timbres defined by that temperament, I suggest that the interval defined as 5/4 will retain its function across that temperament, regardless of it's actual relation to the exact ratio 5/4, as long as the temperament is in a range where the defined timbre functions harmonically. (That is, again, a location in the temperament where 4/3>5/4>6/5)

> > It appears that the MOS scales are *very tightly linked* to temperamental theory, when
> > one accounts for the temperament of a timbre defined by comma sequence of a
> > temperament. As one tempers away more comma's, one descends deeper into scale
> > families (right?), and how each prime limit is tempered defines the commas which define
> > over which generators the MOS scales will validate the timbre.
>
> Not just the commas, but also the error. You can temper as many commas as you like, but if you allow for large error, the "range" will still remain wide.

Before, we had no objective way to define where exactly the error was too great, because deviation away from the just ratio of the generator was subjective, and the definition of MOS scales had no direct link to temperament. But now we can define the error as too great once a timbre defined by the temperament ceases to function harmonically.

---
*This means that one CAN determine the range of generators, given the tempered intervals*
---

John Moriarty

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:

> Before, we had no objective way to define where exactly the error was too great, because deviation away from the just ratio of the generator was subjective, and the definition of MOS scales had no direct link to temperament. But now we can define the error as too great once a timbre defined by the temperament ceases to function harmonically.

I can and have produced timbres with partials determined by a tuning, but how do I produce one defined by a temperament? And why is "ceases to function harmonically" any better than "too far out of tune" which we already have?

> > These harmonic partial temperaments define the "5-limit timbre" for the syntonic temperament, and they can be derived from the pre-defined comma sequence.
>
> What timbre? I see no definition of a timbre in the above.

The parameters below define a variable timbre, one that is only given exact value when a generator and period are specified in the syntonic/meantone temperament:

The fundamental and its n octaves are defined as being (0,n) in the syntonic
temperament: f, 2f, 4f, 8f, etc are defined as the octaves of (0,0)
The third partial and all its n octaves are defined as (1,1+n): 3f, 6f, 9f, etc
are defined as the octaves of (1,1).
The fifth partial and all its n octaves above it are defined as (4,n): 5f, 10f,
15f, 20f, etc are defined as the octaves of (4,0).

Listen here for a demonstration I just uploaded:
http://f1.grp.yahoofs.com/v1/4Mx-TIJPGBZOelStpBpfKhovtS3uriwp-M9tVzPlt5cOBoAW_gjOUnA_Sg28Q87zY1r66eWcM-BMyXeGdno36p3xLObP/John%20Moriarty/TransFormSynth%20Demonstration.mp3

> > Let us define a valid timbre as one whose ratios between two successive partials are always greater than the ratio between the following pair of partials.
>
> What has this to do with meantone?

Like I stated above, the timbre's partials are mapped to coordinates in the syntonic/meantone temperament.

>
> > It appears that the MOS scales are *very tightly linked* to temperamental theory, when one accounts for the temperament of a timbre defined by comma sequence of a temperament.
>
> First you need to define "timbre defined by comma sequence" which you haven't done.
>
> > I think we can define things this way:
> > 5-limit meantone (tempering 81/80) is valid from 685 to 720, along with a corresponding MOS scale (diatonic: 5L, 2s).
> > 7-limit meantone (tempering 59049/57344) is valid from 695 to 700 along with an MOS (7L,12s).
>
> You can also define it as tempering 126/125 or 225/224. [81/80, 59049/57344] is what I've termed the normal comma list.

I do recognize, later in that post actually, that different commas could be tempered for each limit. And given those different commas, different sections of the original 5-limit temperament are able to support a functionally harmonic timbre.

John

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@> wrote:
>
> > Before, we had no objective way to define where exactly the error was too great, because deviation away from the just ratio of the generator was subjective, and the definition of MOS scales had no direct link to temperament. But now we can define the error as too great once a timbre defined by the temperament ceases to function harmonically.
>
> I can and have produced timbres with partials determined by a tuning, but how do I produce one defined by a temperament?

One defines certain locations in the temperament's coordinates as the partials' pitches, like I posted in the other post where I just responded to your response to my first post (=P).
For example, because the interval 81/80 is tempered in meantone/syntonic, the fifth partial is mapped to "up four fifths", or (4,0). Wherever the defined just ratio is in the temperament (in terms of coordinates), adjust the octave to the appropriate location in the harmonic series and that becomes the definition of that partials location in the temperament.

> And why is "ceases to function harmonically" any better than "too far out of tune" which we already have?

It is better because the "ceases to function harmonically" is objective, and the "is too far out of tune" is not. You can't define for me exactly where "too far out of tune" is because it is a personal preference that varies from you, to me, to everyone else. A timbre's ceasing to function harmonically can be explicitly defined as when the ratio between any successive pair of partials becomes greater than the ratio between the preceding pair of partials. In 5-limit, harmonically functioning timbres function under this guideline:
4/3>5/4>6/5
That is, the ratio from the sixth to the fifth partial is never greater than the ratio between the fifth and the fourth, and the ratio between the fifth and fourth partial is never greater than the ratio between the fourth and third.
Also, in syntonic/meantone, this can be defined as the range where the major third is not larger than the perfect fourth, nor is it smaller than the major third.
In 7-Limit, harmonically functioning timbres function under this guideline:
6/5>7/6>8/7
Etc. etc.

John M

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
> Valid was a bad choice, so I'll say this. Given timbre mapping, there is a goal: minimize
> sensory dissonance while retaining a timbre that has the recognizable function of a fully
> harmonic timbre.

> Though certainly not useless, a timbre of a snare drum playing a 4/5/6 major triad
> wouldn't do much for me, you know?

Not all inharmonic timbres are "noise". Gongs, glockenspiels, gamelans, marimbas, vibraphones, and even pianos (to some extent) are not fully harmonic, i.e. do not meet your criteria. And at any rate, your approach actually rules out all temperaments, so it's self-defeating. See below a ways.

> I know that on the surface this seems like circular logic. I define the timbre by the
> temperament, and then use the timbre to allow the temperament to function as I define
> it. I don't think it's circular, however. I think it is powerful, and consistent.

I'm afraid it IS circular logic. You said above that you define 5/4 as the ratio between the 4th and 5th partials, but you can't know what that ratio is until you first define the timbre. To know the timbre, in your case, you need the temperament. But to get the temperament, you have to know what 3/2 and 5/4 are. But to know what those frequency ratios are, you need to look at the partials of the timbre. IT'S A CIRCLE.

> They way I'm defining it, it may work out that the interval supposed to function as a 5/4 > will end up being larger than the exact ratio 4/3 in some other temperament (I don't
> know, I haven't checked), but I do know that the interval defined as 5/4 will never be
> larger than the interval defined as 4/3 in that temperament (because the functional
> range of a 5 limit temperament is where that temperament can define a harmonically
> functional timbre. That forces 4/3>5/4>6/5 to be true). And given the timbres defined
> by that temperament, I suggest that the interval defined as 5/4 will retain its function
> across that temperament, regardless of it's actual relation to the exact ratio 5/4, as long > as the temperament is in a range where the defined timbre functions harmonically. (That > is, again, a location in the temperament where 4/3>5/4>6/5)

In "Father" temperament, 16/15 is tempered out, which means, among other things, 5/4=4/3 and 3/2=8/5. Or, in your terms, the distance between the 3rd and 4th partials equals the distance between the 4th and 5th partials. Gene doesn't like calling it a temperament because it has such high error, but it is a valid temperament nonetheless, and it would explicitly violate your conditions.

Similarly, if you temper out 25/24, you get 6/5 and 5/4 mapped to the same note, meaning the distance between the 4th and 5th partials should be different from the 5th and 6th partials. Temper out 49/48, same thing happens with partials 6, 7, and 8. And, for the clincher: in Meantone, the difference between 9/8 and 10/9 is tempered out, causing the same thing to happen with partials 8, 9, and 10. So really, it's impossible to make a "harmonically functional" timbre out of a temperament, so far as I can tell, because at some point the ratio between two consecutive harmonics will NOT shrink. And the more commas you temper, the more partials will end up in the same ratios.

> Before, we had no objective way to define where exactly the error was too great, because > deviation away from the just ratio of the generator was subjective, and the definition of > MOS scales had no direct link to temperament. But now we can define the error as too
> great once a timbre defined by the temperament ceases to function harmonically.

I believe my above example handily disproves this. Sorry, John.

-Igs

On Wed, Sep 1, 2010 at 1:59 AM, cityoftheasleep <igliashon@...> wrote:
> > As per your definition above, superpyth and meantone do not have any
> > specific properties as MOS scales, since they aren't tied down to a
> > specific generator.
>
> They are if optimized.

Hmm, I could have sworn I suggested that as a starting point before... :)

> > Then again, by that same logic, you can't really
> > say much about MOS scales either. The pajara MOS scales have a period
> > of half an octave, but who's to say that that period isn't just a
> > really detuned octave?
>
> Because the period is not defined abstractly but by a cents-value. It makes no sense to refer to 600 as a "really small 1200". Periods have unique identities that aren't of necessity defined in relation to other identities. IOW we can speak of flat octaves because the interval of an octave has some perceptual flex, but only an interval of 1200 cents can be an interval of 1200 cents.

I would assume that you would treat an LLsLLLs scale with a period of
1199 cents as having the same name as an LLsLLLs scale with a period
of 1200 cents. Likewise, you would probably put an LLsLLLs scale with
a period of 600 cents in a different category. At least I'd hope.

> But yes, the period has to be specified before you can group MOS's, which is why (when I'm being rigorous), I give MOS's in the form "nL+ms=period" or "nL+ms=x-EDO".

Fine. but if you're going to assume x-EDO, that assumes a pure octave.
The octaves can be detuned and 2 can be mapped any way you'd like,
with an unlimited amount of error. The point is that you could say
that 6-edo is just 12-edo with really stretched octaves. You could map
things that way too.

> > And the LLsLLLs scales that we're used to -
> > who's to say that the period shouldn't be a third of an octave, and
> > the octave period isn't just a really sharp third-of-an-octave?
>
> Because cents-values, unlike mappings, are non-arbitrary.

So you would say that LLsLLLs with a period of 1200 is a different
scale than LLsLLLs with a period of 1199? Why does the size of one
generator matter, but the other one not?

> > The idea to limit meantone to only the generators that produce 7+5
> > scales seemed like a fairly rigorous way to express an intuitive
> > common sense definition of it.
>
> Useful? Yes. Rigorous? Not hardly.

It's no less rigorous than saying that it has to have a consistency of
more than 12 or whatever.

> > You could start with the TOP tuning. Or start with the TOP tuning, and
> > then move to the nearest golden mean generator.
>
> But where would you stop?

You don't have to stop. This is just another way to organize the
generator space. In one of my messages, I proposed that anything
producing 5+2 scales be called an isotone, that anything producing 7+5
scales be called a meantone (and anything producing 5+12 scales be
called a superpyth). Then I pointed out that we can further subdivide
it and that anything producing 12+7 scales be called a meantone (or a
"regular meantone" if you prefer) and anything producing 7+12 scales
be called an inframeantone, etc. I basically just stole these names
from Margo and we can keep subdividing it as much as we want, and stop
when it seems perceptually useless.

> > Because 12-tet produces a 5+2 scale and a 12-note scale which can be
> > viewed as a special case of 7+5.
>
> Why? Why can it be viewed this way?

Because it can. Because I'm creating a mathematical structure where it
makes sense to view 12-et as a special case of 7+5. Because in one of
your last emails, you SAID that equal temperaments are a special type
of MOS. Because Kraig has also said this. Because existence is such
that one can take this perspective. What is wrong with this? At this
point it seems like you're just playing devil's advocate.

The point is that if we take as an axiom that ET's are a special case
of MOS's, then 12-tet produces 5+2 and 7+5 scales. 7-tet only produces
5+2 scales. There is a chromatic scale in 12-tet, and none in 7-tet.
Whether you want to say that 12-tet is both a member of the 7+5 and
5+7 MOS's, or neither, is really arbitrary and I don't care. It seems
to make sense the first way, but whatever works works. But that would
mean that ET's aren't a type of MOS.

> > That point will likely (as Rothenberg puts it) be listener-dependent
> > and have to do with some listener-dependent tolerance factor. Harmonic
> > Entropy is the same way, and the solution was to include a free "s"
> > parameter to roughly estimate listener sensitivity. I don't think it
> > kills the whole project though.
>
> It doesn't? How do you propose to get around it? The same way it's done in H.E.?

You could formulate, I suppose, a mathematical structure in which
notes aren't represented with exact pitches, but represent a range of
pitches that goes out by some listener tolerance epsilon in either
direction. I'm not sure exactly how the math would work, and it
probably wouldn't have to do with discrete set theory anymore. Feel
free to try and work it out, it's a little bit over my head. But I
think if you read Rothenberg's stuff, you'll find he's basically
worked a lot of it out already, and just dodged around the issue by
noting that there'd be this listener-dependent tolerance factor. I
found that to be acceptable.

> But your ways of bounding it are arbitrary and based on concepts unrelated to temperament.

What ways of bounding it aren't arbitrary? The human auditory system
is arbitrary! Some concept of convention is going to have to come into
it at some point, right? And even if we're going to go with
consistency, why should that matter any more than what MOS scales are
produced? It seems like the scales being produced are the more
relevant category anyway, being as the point where the meantone
generator really flips around your perception of things is when it
gets flatter than 7-tet.

> In your last post, you clearly demonstrated that you didn't know what a temperament actually *is*, at least as the term is used here.

Then I don't know how to demonstrate it any clearer. I understand that
you can give meantone and porcupine and pajara all generators of 324
cents and periods of 12931023 cents, and just map things out so that x
amount of generators is viewed as being this prime and so on. What I'm
saying is that at this point, it's a little bit silly for you to start
throwing at me this notion that my ideas lack mathematical rigor,
because there is no math and no concrete ideas yet. They haven't been
fully formed. I just posted them to get some discussion on how I COULD
make them rigorous, since I think that they would be perceptually
useful.

In fact, what I really was hoping was to find that this idea I was
posing had already been worked out, and see how so. And it seems like
a lot of it is territory that has been stumbled into before anyway. At
this stage, admitting that "common sense" distinctions do exist and
fleshing them out is probably key, and then working out retroactively
what the rigorous pattern in the common sense was can come later.

> Any attempt at defining tuning ranges for temperaments beyond the "optimum tuning" is problematic. Maybe this isn't an issue to you, and that as long as a generalization about a temperament is true often enough to be useful, the cases where it fails to be true are irrelevant. I'm holding out for a higher standard of rigor, if I can find one, and I think we all should do the same. If we fail to find a more rigorous and complete taxonomic method, we can always come back and settle for any of the various incomplete possibilities.

Fair enough, but I still remain hopeful that there's some sort of
brilliant million dollar idea that will tie it all together. We can,
of course, still call meantone with a generator of 2380 cents and a
period of 1234908102398102983 cents a meantone, but it seems prudent
to simultaneously explore this other space, and perhaps see what
connections this could lead to later on.

One idea I had was to make it work the other way - for any MOS, see
which mappings would give the lowest badness. Hence 2+3 would give a
few mappings with low badness (mavila, meantone, superpyth), and 5+2
would give a few more (meantone, superpyth), and then 5+7 would best
be mapped to superpyth and 7+5 best be mapped to meantone, and so on.
This would have to be done within a certain limit. It's an interesting
idea I'm going to explore further, I think.

> > The biggest objection I have is that any meantone, except for a few
> > objections, is only consistent out to a finite number of generators.
> > If we're going to limit things to going out a certain number of
> > generators, then is that not a similar concept to limiting things to
> > certain MOS's, which is equally as arbitrary?
>
> Right. It's all arbitrary. And it's all problematic. Should we really settle for boundaries that we know are inconsistent or problematic in some way? Is that really any more useful than not bothering to draw boundaries at all and just "winging it"? BTW, I'm not being rhetorical with this question.

No, we shouldn't. But the problem is that music is, unfortunately,
partially subjective, partially due to cognitive differences in
listeners (as per Rothenberg's ideas about subjective maps), and
partially due to how the auditory system is tweaked. We can use math
to describe things, but not proscribe them. And in this case, we find
that there is clearly some kind of cognitive or perceptual order to
uncover when you plow into making some kind of scale taxonomy, and
that the idea is sound on some level. Most importantly, it seems to
reflect an organization of a subtle perceptual quality that I think
deserves some looking into. Perhaps when we delve further into the
taxonomy, we'll start to see patterns and then the rigor and
overarching order will become clear. Perhaps not, and it'll be a red
herring, similar to how the "JI is the foundation for all of music"
idea was a red herring. Perhaps it's already been figured out.

-Mike

Mike Battaglia wrote:
> On Tue, Aug 31, 2010 at 9:53 PM, Herman Miller <hmiller@...> wrote:
>> When the topic originally came up, I believe I was using "consistency"
>> as a shorthand for "consistency range" or "consistency limit", something
>> along those lines. So the consistency limit of a meantone tuning with a
>> 690 cent generator would be 11 (meaning that any scale with 11 or fewer
>> notes in a contiguous chain of generators is consistent, but if you add
>> a 12th note, the 11-fifths interval would make it inconsistent). On the
>> other hand, any scale with a 680-cent generator is inconsistent with
>> meantone, since 4 octaves up minus 3 fifths down is a better fifth than
>> the "meantone" version.
> > You mean 4 octaves up minus 3 fifths down is a better 5/4 than the
> "meantone" version, right?

Right, I was thinking 5/1 and it came out "fifth" for some reason. With 1200-cent octaves it's the same as looking for the best 5/4, but I was looking at tunings with tempered octaves as well.

> But even so, wouldn't that just make a 680 cent "meantone" consistent
> out to something like 4 generators? 4 generators gives you C-G-D-A,
> and the C-A will be closer to 8/5 than 5/3, but there's no "better"
> 8/5 with which to compare. So 5 has been introduced indirectly, but
> not directly yet.

You could look at it that way. It could be useful to check all the intervals for consistency and not just the primes, but if you've got tempered octaves, you'd have to check inversions and octave transpositions as well.

> Although I suppose you could define it whichever way you want -
> perhaps a tuning could be said to be "absolutely inconsistent" if it
> ends up being inconsistent right when one of the basis primes gets
> first introduced.

Possibly so, if that turns out to be a useful thing to measure. I'm starting to think checking all the intervals within an odd-limit might be a better idea (there's nothing special musically about the primes). You could just check all intervals, but many temperaments have complex, dissonant intervals within the first few generators, and it might be undesirable to check those for consistency.

caleb morgan wrote:
> At the risk of eliciting howls of derision, I'm admitting to
> experimenting with 46EDO with every fourth note removed, starting on
> 26.08 cents, for a total of 34 pitches.
> > I expect, though, that I will find I need those omitted pitches.
> > If this is "Hitchcock", then maybe this subset is "Psycho", or more
> ridiculous, "Amityville Horror", or "Shutter Island".

Actually it looks like it's more related to diaschismic, with a generator of 4/46 and a half-octave period. Basically in the first 1/2-octave you've got up to 11 generators down and 5 up. In the second 1/2-octave you've got 5 generators down and 11 up. So it could be a little tedious figuring out which intervals are available in different keys. But the general idea is a good one.

Generator mapping: [<2, 3, 5, 7], <0, 1, -2, -8]>

This is related to pajara (Paul Erlich's temperament with the decatonic scales), but has a different mapping of the 7/1. You can bring in 11-limit intervals as well.

[<2, 3, 5, 7, 9], <0, 1, -2, -8, -12]>

Okay Mike, let me stop picking nits. I think it would be a worthwhile endeavor to index the TOP generators for the multitudes of temperaments out there according to their MOS configurations. That way, it would be easy to find the harmonically-optimal tuning for all our favorite MOS scales, or find ways to transform between temperaments using common MOS scales as a bridge.

That said,

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I would assume that you would treat an LLsLLLs scale with a period of
> 1199 cents as having the same name as an LLsLLLs scale with a period
> of 1200 cents. Likewise, you would probably put an LLsLLLs scale with
> a period of 600 cents in a different category. At least I'd hope.

I would not treat the scale with the period of 1199 cents as the same as one with a period of 1200 cents, any more than I would treat Caleb's 45-Equal Divisions of 5/2 the same as I would 34-EDO, which it very closely resembles. 1200 cents is 1200 cents, accept no substitutes!

> > But yes, the period has to be specified before you can group MOS's, which is why (when I'm being rigorous), I give MOS's in the form "nL+ms=period" or "nL+ms=x-EDO".
>
> Fine. but if you're going to assume x-EDO, that assumes a pure octave.

Excuse me, I meant x-EDPeriod.

> The octaves can be detuned and 2 can be mapped any way you'd like,
> with an unlimited amount of error. The point is that you could say
> that 6-edo is just 12-edo with really stretched octaves. You could map
> things that way too.

Are we talking about temperaments or MOS here? There's no mapping involved in MOS scales. And this is why I have zero tolerance for deviating the period of MOS scales. But it is interesting to note how MOS's with the same shape but different periods occasionally may have similar properties.

> So you would say that LLsLLLs with a period of 1200 is a different
> scale than LLsLLLs with a period of 1199? Why does the size of one
> generator matter, but the other one not?

Because the classification scheme is based on MOS's with a common period and different generators. If we make it two-dimensional and let both generators vary, we are going to complicate the taxonomy by orders of magnitude.

> It's no less rigorous than saying that it has to have a consistency of
> more than 12 or whatever.

Right. I agree. So far, I don't think there has been a truly rigorous definition for Meantone that's been proposed.

> You don't have to stop. This is just another way to organize the
> generator space. In one of my messages, I proposed that anything
> producing 5+2 scales be called an isotone, that anything producing 7+5
> scales be called a meantone (and anything producing 5+12 scales be
> called a superpyth). Then I pointed out that we can further subdivide
> it and that anything producing 12+7 scales be called a meantone (or a
> "regular meantone" if you prefer) and anything producing 7+12 scales
> be called an inframeantone, etc. I basically just stole these names
> from Margo and we can keep subdividing it as much as we want, and stop
> when it seems perceptually useless.

Okay, I think I finally get what you're trying to do. You're using "Meantone" not to refer specifically to the linear temperament called "Meantone" but to a category of MOS scales that bears some resemblance to the temperament. Or you're using the term "temperament" in the colloquial sense, rather than the more esoteric mathematical sense that gets used here. Man, stupid WORDS....

> > > Because 12-tet produces a 5+2 scale and a 12-note scale which can be
> > > viewed as a special case of 7+5.
> >
> > Why? Why can it be viewed this way?
>
> Because it can. Because I'm creating a mathematical structure where it
> makes sense to view 12-et as a special case of 7+5. Because in one of
> your last emails, you SAID that equal temperaments are a special type
> of MOS.

Yes, a special type of MOS where L=s. But if you agree that 12 is also a special case of 5+7, then I'm on board. When it comes to equal scales, they're either BOTH the MOS's that coincide at them, or they're neither, and we should be consistent, and I like the idea of them being both. However, your comment about 7-EDO not being a case of it, I might disagree. You don't HAVE to stop stacking generators once you hit a multiple of the period; for instance, in 12-tET, we have augmented 2nds and diminished 3rds, which seem to imply a more-than-12-note structure. So you could say 12 has a 19-note and 31-note MOS too, it's just that a lot of notes end up being the same pitch. Do you agree?

> > It doesn't? How do you propose to get around it? The same way it's done in H.E.?
>
> You could formulate, I suppose, a mathematical structure in which
> notes aren't represented with exact pitches, but represent a range of
> pitches that goes out by some listener tolerance epsilon in either
> direction. I'm not sure exactly how the math would work, and it
> probably wouldn't have to do with discrete set theory anymore. Feel
> free to try and work it out, it's a little bit over my head. But I
> think if you read Rothenberg's stuff, you'll find he's basically
> worked a lot of it out already, and just dodged around the issue by
> noting that there'd be this listener-dependent tolerance factor. I
> found that to be acceptable.

Well, I suppose that's what we're stuck with. It would be good to at least pin down a range or something. Or I wonder if, instead of a range, we could limit it to a level on the Stern-Brocot tree?

The way I'm skirting the issue in my EDO primer is just limiting my discussion to EDOs up to 37 (blame Andrew Heathwaite for getting me to stop there, I wanted to do up to 36 but he pushed me one greater). If you're only dealing with EDOs, and only up to a certain number, then it's really a non-issue where you draw the line between what sounds more like 5L+2s and what sounds more like 5-EDO. I mean, you have to get pretty high in the EDOs before you can get scales that really straddle the boundaries anyway, and there aren't any optimal temperaments near the boundaries either (in fact, they all seem to be pretty close the middle of whatever MOS family they are in). So, yeah...I guess we can say "boundaries schmoundaries" and get on with our lives.

> In fact, what I really was hoping was to find that this idea I was
> posing had already been worked out, and see how so. And it seems like
> a lot of it is territory that has been stumbled into before anyway. At
> this stage, admitting that "common sense" distinctions do exist and
> fleshing them out is probably key, and then working out retroactively
> what the rigorous pattern in the common sense was can come later.

And what I was trying to do was show you some of the cases that violate the "common sense" distinctions, or at least show that the "common sense" distinctions are hinting at something that is based neither on temperament nor MOS (or maybe on both, and something else to boot). I'm not arguing with you to be a jerk, Mike, I'm just showing you all the problems I've found when trying to pursue this same line of thought. I'm regurgitating to you stuff that was told to me when I asked similar questions. I don't expect you to have answers to any of these questions. When I raise an objection or ask you a question or tell you your usage of a term is inconsistent with the way I've been told it's supposed to be used, I'm trying to be helpful.

> > Any attempt at defining tuning ranges for temperaments beyond the "optimum tuning" is problematic. Maybe this isn't an issue to you, and that as long as a generalization about a temperament is true often enough to be useful, the cases where it fails to be true are irrelevant. I'm holding out for a higher standard of rigor, if I can find one, and I think we all should do the same. If we fail to find a more rigorous and complete taxonomic method, we can always come back and settle for any of the various incomplete possibilities.
>>
> Fair enough, but I still remain hopeful that there's some sort of
> brilliant million dollar idea that will tie it all together. We can,
> of course, still call meantone with a generator of 2380 cents and a
> period of 1234908102398102983 cents a meantone, but it seems prudent
> to simultaneously explore this other space, and perhaps see what
> connections this could lead to later on.

I share the same hope. I just don't think it will come from the theory of linear temperaments/the "regular mapping paradigm". It's not you I'm trying to shoot down, it's the idea of temperament. I think the pattern we're glimpsing will be better described by an as-yet undiscovered paradigm, and trying to relate MOS's back to Temperaments (and vice versa) is a dead end. It's like quantum physics and relativity, or something. They both work in different cases and point toward a "grand unified theory", if we can but uncover it.

> One idea I had was to make it work the other way - for any MOS, see
> which mappings would give the lowest badness. Hence 2+3 would give a
> few mappings with low badness (mavila, meantone, superpyth), and 5+2
> would give a few more (meantone, superpyth), and then 5+7 would best
> be mapped to superpyth and 7+5 best be mapped to meantone, and so on.
> This would have to be done within a certain limit. It's an interesting
> idea I'm going to explore further, I think.

I will certainly be interested to see the correlation between billion-limit optimal generators and the "golden generators" on the scale tree, that's for sure!

> > Right. It's all arbitrary. And it's all problematic. Should we really settle for boundaries that we know are inconsistent or problematic in some way? Is that really any more useful than not bothering to draw boundaries at all and just "winging it"? BTW, I'm not being rhetorical with this question.
>
> No, we shouldn't. But the problem is that music is, unfortunately,
> partially subjective, partially due to cognitive differences in
> listeners (as per Rothenberg's ideas about subjective maps), and
> partially due to how the auditory system is tweaked. We can use math
> to describe things, but not proscribe them. And in this case, we find
> that there is clearly some kind of cognitive or perceptual order to
> uncover when you plow into making some kind of scale taxonomy, and
> that the idea is sound on some level. Most importantly, it seems to
> reflect an organization of a subtle perceptual quality that I think
> deserves some looking into. Perhaps when we delve further into the
> taxonomy, we'll start to see patterns and then the rigor and
> overarching order will become clear. Perhaps not, and it'll be a red
> herring, similar to how the "JI is the foundation for all of music"
> idea was a red herring. Perhaps it's already been figured out.

I doubt it's already been figured out, or someone would have shut us by saying "so-and-so published a paper on this years ago; congratulations you two douche-bags, you've re-invented the wheel!"

-Igs

On Thu, Sep 2, 2010 at 12:31 AM, cityoftheasleep
<igliashon@...> wrote:
>
> Okay Mike, let me stop picking nits. I think it would be a worthwhile endeavor to index the TOP generators for the multitudes of temperaments out there according to their MOS configurations. That way, it would be easy to find the harmonically-optimal tuning for all our favorite MOS scales, or find ways to transform between temperaments using common MOS scales as a bridge.

Damn those nits. But I think we're mostly in agreement now. I'm only
going to respond to the few parts of the first half of your message
that I think are relevant to the actual work forward from here. If I
don't respond to something, it can be assumed that I agree.

> > The octaves can be detuned and 2 can be mapped any way you'd like,
> > with an unlimited amount of error. The point is that you could say
> > that 6-edo is just 12-edo with really stretched octaves. You could map
> > things that way too.
>
> Are we talking about temperaments or MOS here? There's no mapping involved in MOS scales. And this is why I have zero tolerance for deviating the period of MOS scales. But it is interesting to note how MOS's with the same shape but different periods occasionally may have similar properties.

I suppose it depends on how you define it. There are three ways to
define an MOS (or scale) that I see:

1) A pattern of intervals, such as LLsLLLs, with no explicit period
stated - such that LLsLLLs with a period of 1/5 of an octave, and
LLsLLLs with a period of 1/2 of an octave, will count as "the same
MOS." Advantages: these two scales, admittedly, do have the same
mathematical properties - if we're completely ignoring primes.
Disadvantages: perceptually, these two scales couldn't be further
apart.

2) A pattern of intervals, such as LLsLLLs, with an explicit period
stated - such that LLsLLLs with a period of 1200 cents, and LLsLLLs
with a period of 1199.999 cents, will count as two different MOS's.
Advantages: solves the problem of when a detuned octave stops being a
detuned octave by making them all different. Disadvantages: allows for
the generator to be any size while locking the period down to just one
size, which isn't cool. Also seems a bit impractical, since LLsLLLs
could refer to an infinite number of MOS's.

3) A pattern of intervals and a "pseudo-mapping" for the period. That
is, just like with regular temperament, you can map 2 to 300 cents, we
take the same approach with MOS's. That means that LLsLLLs with a
period of 1200 cents could be viewed as a diatonic MOS if you map the
octave to 1200 cents, it could be viewed as a hemi-diatonic or
whatever you'd call it MOS if you map the "half-octave" to 1200 cents.
Or the pajara scales could be looked at as 5-note MOS's if you map the
octave to the 600 cent interval and just say it has really high error.
Note that the period wouldn't have to be a rational number or even a
division of the octave. Regular mapping would then be further applied
later to warp things even more.

Advantages: solves the disadvantages of the above approaches.
Disadvantages: finding it difficult to flesh out exactly how the
mapping would work here.

I'm attracted to either the first or third approaches. I think the
first is best, since it only involves one mapping - this means all
LLsLLLs scales become "the same," whether it's got a period of a
half-octave or not. This might make sense because it's really up to
regular mapping anyway to decide how we interpret things - if we have
a Pajara scale, which we define as ssLss with a period of 600 cents
(symmetric pentachordal major), but we map things so that 2 is 600
cents flat and thus has 600 cents of error, is it really Pajara
anyway, at that point? If we're going to really assume the perspective
that temperament error is unbounded, and if we're mapping 2 to 600
cents, can we say that that's Pajara? Seems like it would be "happy
pentatonic" at that point.

> > You don't have to stop. This is just another way to organize the
> > generator space. In one of my messages, I proposed that anything
> > producing 5+2 scales be called an isotone, that anything producing 7+5
> > scales be called a meantone (and anything producing 5+12 scales be
> > called a superpyth). Then I pointed out that we can further subdivide
> > it and that anything producing 12+7 scales be called a meantone (or a
> > "regular meantone" if you prefer) and anything producing 7+12 scales
> > be called an inframeantone, etc. I basically just stole these names
> > from Margo and we can keep subdividing it as much as we want, and stop
> > when it seems perceptually useless.
>
> Okay, I think I finally get what you're trying to do. You're using "Meantone" not to refer specifically to the linear temperament called "Meantone" but to a category of MOS scales that bears some resemblance to the temperament. Or you're using the term "temperament" in the colloquial sense, rather than the more esoteric mathematical sense that gets used here. Man, stupid WORDS....

Right. And I think that what would happen, to tie this back into
actual regular temperament theory, is that we could come up with some
function to figure out the ideal mapping(s) for a certain MOS.

That is, find the generator range for a certain MOS - for LLsLLLs it's
between 3/5 and 4/7. Although anything outside of that might still be
"mapped" as meantone, it's definitely not going to be in that MOS
family anymore.

Then we experiment with different mappings for that generator space.
We figure out how it looks if use porcupine for it, meantone,
schismatic, whatever. We come up with an expression to calculate the
error relative to the meantone mapping for a generator of cents value
xxx, then integrate it with respect to the cents value from 1200 * 4/7
to 1200 * 3/5. Or something like that.

Then you sort them by how well they match, and voila, you have the
best mappings for each MOS, which is about as rigorous a way as I can
think of to start connecting MOS's to temperaments :)

> Yes, a special type of MOS where L=s. But if you agree that 12 is also a special case of 5+7, then I'm on board. When it comes to equal scales, they're either BOTH the MOS's that coincide at them, or they're neither, and we should be consistent, and I like the idea of them being both.

I agree.

> However, your comment about 7-EDO not being a case of it, I might disagree. You don't HAVE to stop stacking generators once you hit a multiple of the period; for instance, in 12-tET, we have augmented 2nds and diminished 3rds, which seem to imply a more-than-12-note structure. So you could say 12 has a 19-note and 31-note MOS too, it's just that a lot of notes end up being the same pitch. Do you agree?

I see your point, but this is where I think it all gets a bit
dangerous. It seems like if we can say that 7-EDO is a type of 7+5
scale in which the "s" intervals are 0 cents - is the math more
rigorous that way? If so I'm for it. Perhaps Gene could weigh in.

I suppose my reasoning was that it's possible for you to literally
hear 7-equal as a type of 5+2 scale, but not quite as possible for you
to hear 5-equal as a type of 5+2 scale. Maybe I'm wrong. Either way
let's go with whatever makes the math more straightforward.

> Well, I suppose that's what we're stuck with. It would be good to at least pin down a range or something. Or I wonder if, instead of a range, we could limit it to a level on the Stern-Brocot tree?

What do you mean by this?

> The way I'm skirting the issue in my EDO primer is just limiting my discussion to EDOs up to 37 (blame Andrew Heathwaite for getting me to stop there, I wanted to do up to 36 but he pushed me one greater).

No 41? :) You're missing out on derr uber-EDO!

> > In fact, what I really was hoping was to find that this idea I was
> > posing had already been worked out, and see how so. And it seems like
> > a lot of it is territory that has been stumbled into before anyway. At
> > this stage, admitting that "common sense" distinctions do exist and
> > fleshing them out is probably key, and then working out retroactively
> > what the rigorous pattern in the common sense was can come later.
>
> And what I was trying to do was show you some of the cases that violate the "common sense" distinctions, or at least show that the "common sense" distinctions are hinting at something that is based neither on temperament nor MOS (or maybe on both, and something else to boot).

That is what I suspect - that it's all related somehow. But perhaps
I'm wrong. Either way, we are clearly using common sense to
communicate perceptual borders that ARE rigorously defined, mentally.
That is, our algorithm for processing this stuff is rigorously defined
in our heads, for starters. I do agree that we need to be careful
about not writing cultural biases into the theory (with a jazz
background I am especially sensitive to common-practice "rules" that
are really arbitrary).

I think that in this brave new world of music theory that this list
has founded, what is really important is coming up with some kind of
formal, rigorous mathematical foundation (regular mapping). However,
what really interests me personally is to use mathematics to model and
describe perceptual phenomena (harmonic entropy). The first is
probably necessary to fully express ideas about the second one. The
fact that MOS's (or more precisely, rank-order matrixes) really define
something about where the instantation of a temperament perceptually
"shifts" I think means that plowing into that side of things would be
immensely useful.

I also think that exploring this side more fully and trying to connect
it to regular mapping might be accomplished by applying some kind of
psychoacoustic limits about what's "valid," these limits being
modelled by mathematics (harmonic entropy again). Although I'd
probably be happier if I could figure it out by composing first, and
writing about it later... :)

> I'm not arguing with you to be a jerk, Mike, I'm just showing you all the problems I've found when trying to pursue this same line of thought. I'm regurgitating to you stuff that was told to me when I asked similar questions. I don't expect you to have answers to any of these questions. When I raise an objection or ask you a question or tell you your usage of a term is inconsistent with the way I've been told it's supposed to be used, I'm trying to be helpful.

I apologize then. It seemed as though you were simply trying to
nitpick or play devil's advocate. My fault.

> I share the same hope. I just don't think it will come from the theory of linear temperaments/the "regular mapping paradigm". It's not you I'm trying to shoot down, it's the idea of temperament. I think the pattern we're glimpsing will be better described by an as-yet undiscovered paradigm, and trying to relate MOS's back to Temperaments (and vice versa) is a dead end. It's like quantum physics and relativity, or something. They both work in different cases and point toward a "grand unified theory", if we can but uncover it.

I think that Rothenberg has sort of figured something out. But I think
the grand pattern can be seen by comparing the aeolian scales of 1/3
comma meantone, and 1/3 comma superpyth (where the minor thirds are a
pure 7/6). And meditating on it and asking what it really means.

Note that the "mood" of the scale changes when the intonation does,
but note also that something else about the scale stays "the same."
What exactly is it that's staying the same? How every chord and note
"functions," I suppose... But what exactly is a "function," anyway?
It's definitely more than just V resolving to I, since you have chords
like IV that don't necessitate any kind of resolution, they just have
their own unique feeling or mood.

So you have one kind of mood coming from the ~JI intonation, and one
kind of mood coming from the scale map. What the hell does it mean?

Perhaps some delving into the semiology literature is necessary here?
I haven't explored it much.

> I doubt it's already been figured out, or someone would have shut us by saying "so-and-so published a paper on this years ago; congratulations you two douche-bags, you've re-invented the wheel!"

Haha, I hope so. I keep seeing allusions to some paper that Graham has
written on this subject... Graham?

-Mike

> > Valid was a bad choice, so I'll say this. Given timbre mapping, there is a goal: minimize
> > sensory dissonance while retaining a timbre that has the recognizable function of a fully
> > harmonic timbre.
>
> > Though certainly not useless, a timbre of a snare drum playing a 4/5/6 major triad
> > wouldn't do much for me, you know?
>
> Not all inharmonic timbres are "noise".

I realize that, I was only trying to use the extreme as an example. The moment one breaks down the rules determining what allows a timbre to function harmonically, one approaches the extreme where the tendencies of harmony are likely to not apply as consistantly. I am not trying to define what can *not* function musically. I understand that's how it looked, but I know that no one can do that. I do I think, however, you *can* define what is more likely to function musically, compared to other examples. For example, scales with only two steps sizes are more likely to function melodically than those with three. Those with three step sizes are more likely to function melodically than those with four. And harmonically related timbres are morel likely to function as a timbre with the possibility for recognizable harmony than those that are not related harmonically.

> I'm afraid it IS circular logic.
> You said above that you define 5/4 as the ratio between the 4th and 5th partials, but you can't know what that ratio is until you first define the timbre.

Ok.

> To know the timbre, in your case, you need the temperament.

Ok.

> But to get the temperament, you have to know what 3/2 and 5/4 are.

No, to get the temperament I have to give cent values to the generators, or 3/2 and 2/1 (not 5/4), which will then determine both the tuning of the temperament and the tuning of the timbre. I don't think that's circular, is it?

> In "Father" temperament, 16/15 is tempered out, which means, among other things, 5/4=4/3 and 3/2=8/5. Or, in your terms, the distance between the 3rd and 4th partials equals the distance between the 4th and 5th partials. Gene doesn't like calling it a temperament because it has such high error, but it is a valid temperament nonetheless, and it would explicitly violate your conditions.

If I knew how to make the greater-than-or-equal-to symbol, I would have been using it in all my examples. That is, in a 5-limit timbre defined by a 5-limit temperament, for that timbre to function harmonically it must follow that 4/3 [is greater than or equal to] 5/4 [is greater than or equal to] 6/5. This is why, given 5-limit meantone/syntonic, 7-edo and 5-edo we included instead of excluded. Now let's look at your example where 16/15 is tempered out:

5/4 becomes equal to 4/3, and 6/5 becomes equal to 9/8.
Filling those values back into the inequality:
4/3 [is greater than or equal to] 4/3 [is greater than or equal to] 9/8

It still works, it just so happens that the first part of the inequality will *always* be satisfied, and the timbre will only cease to function harmonically where 9/8 (or6/5)>4/3.

> Similarly, if you temper out 25/24, you get 6/5 and 5/4 mapped to the same note, meaning the distance between the 4th and 5th partials should be different from the 5th and 6th partials. Temper out 49/48, same thing happens with partials 6, 7, and 8. And, for the clincher: in Meantone, the difference between 9/8 and 10/9 is tempered out, causing the same thing to happen with partials 8, 9, and 10. So really, it's impossible to make a "harmonically functional" timbre out of a temperament, so far as I can tell, because at some point the ratio between two consecutive harmonics will NOT shrink. And the more commas you temper, the more partials will end up in the same ratios.

Next is your example where 25/24 is tempered out:
5/4 becomes equal to 6/5
Filling those values into the inequality:
4/3 [is greater than or equal to] 5/4 [is greater than or equal to] 5/4

Once again, it still works. It just so happens that the second inequality is *always* satisfied, and the first one breaks down where 5/4>4/3.

> I believe my above example handily disproves this. Sorry, John.

Unless there is a fault in my math, I don't believe it does. You are bring up some very interesting cases though!

John M

Has any such thing been observed in 12ET with the vast spectrum of timbres that have been used in it?

J>M> wrote

I realize that, I was only trying to use the extreme as an example. The moment
one breaks down the rules determining what allows a timbre to function
harmonically, one approaches the extreme where the tendencies of harmony are
likely to not apply as consistently.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:

> > Not all inharmonic timbres are "noise".
>
> I realize that, I was only trying to use the extreme as an example. The moment one
> breaks down the rules determining what allows a timbre to function harmonically, one
> approaches the extreme where the tendencies of harmony are likely to not apply as
> consistantly. I am not trying to define what can *not* function musically. I understand
> that's how it looked, but I know that no one can do that. I do I think, however, you *can*
> define what is more likely to function musically, compared to other examples.

If you can't define what *can't* function musically, what use is there in defining what *can*? The point of defining a concept is to give it limits in what it can describe.

> For example, scales with only two steps sizes are more likely to function melodically than > those with three. Those with three step sizes are more likely to function melodically than > those with four.

I take it you're not a fan of JI, then? Some JI scales, like Centaur for instance, have WAY more than 4 step-sizes, and yet they seem to function just fine to my ears.

> And harmonically related timbres are morel likely to function as a timbre
> with the possibility for recognizable harmony than those that are not related
> harmonically.

Perhaps that case could be made, but this way of looking at timbre still places a perfectly-harmonic timbre (i.e. one whose partials form a harmonic series) as the "ideal" of harmonic functionality, so detuning the partials to match the scale is going to push the timbre further from the ideal. This effect is clearly audible to me in a lot of Sethares' music, where timbres mapped to fit bizarre scales like 8-EDO or 13-EDO sound less harmonically-functional (and less pleasant, at least to my ears) than harmonic timbres playing the same scales.

And at any rate, I've never found diatonic harmony in, e.g., 22-EDO to be any less recognizable than, e.g., 19-EDO harmony with timbre held constant, so I'm not sure that the idea of tempering partials is really very necessary or even helpful.

> No, to get the temperament I have to give cent values to the generators, or 3/2 and 2/1 > (not 5/4), which will then determine both the tuning of the temperament and the tuning > of the timbre. I don't think that's circular, is it?

It is if you maintain that JI ratios are defined as the ratios between partials in a timbre. Are you now saying that 3/2 and 2/1 can be defined in some way that is unrelated to partials in a timbre?

Remember that we can stretch or compress the partials in a timbre as much as we want and still fulfill the condition that the ratios between successive partials decrease as we go up the series. The 2nd and 3rd partial can be as close together or as far apart as we care to make them, suggesting that a 3/2 defined in this way can be of any size.

> If I knew how to make the greater-than-or-equal-to symbol, I would have been using it > in all my examples. That is, in a 5-limit timbre defined by a 5-limit temperament, for
> that timbre to function harmonically it must follow that 4/3 [is greater than or equal to] > 5/4 [is greater than or equal to] 6/5. This is why, given 5-limit meantone/syntonic, 7-
> edo and 5-edo we included instead of excluded.

Well, that changes everything. There is a world of difference between "greater than" and "greater than or equal to". There is an ascii character for it, but Yahoo usually butchers it, so I usually write ">/=" or just "greater than or equal to".

But if you allow for this, then you have to allow for a timbre in which all the partials are equally-spaced. For something like a triangle wave, this is fine, since the ratio between partials is a maximally-consonant 2/1. But what about a timbre where the ratio between any two successive partials is something like 18/17? Or even 6/5? Are you going to tell me, with a straight face, that timbres made like this are actually more harmonically-functional than a timbre where the ratios between successive partials increase as we go up the series? You'd clearly have to stipulate some minimum distance between partials

At any rate, I think it's nonsense to say that a frequency ratio--which is what 5/4 and 3/2 and 2/1 are, they are *ratios between two frequencies*, like 550 Hz:440 Hz--should be defined as ratios between ordinal partials in a timbre, rather between two fundamental frequencies. I mean, you're basically suggesting that in a timbre where the 4th partial is 440 Hz and the 5th partial is 510 Hz, that we could say "510:440 = 5:4".

An example of why this is problematic: say you have a timbre where the 4th and 5th partials are 440 and 510 Hz, respectively, and the rest of the partials all fit your criteria as stated above. Now, you play two notes with this timbre: the first note is 110 Hz, and the 2nd note is 137.5 Hz. You've defined 5/4 as being 510/440, so what do you call 137.5/110?

As I said above, there's no limit to how far apart or close together we can space the partials in a timbre, effectively meaning that there is no boundary to the size of the ratios between them. So we could say 4:5 = 200:201, or we could say it equals 200:276394, presuming the rest of the partials are sized so that the ratios between them decrease as we go up the series.

Furthermore, in order for temperament to make sense at all, we have to be able to define a JI map that is absolute, which we can then use temperament to warp. But you are basically suggesting that we define JI *according to a temperament*. If we defined JI as being the pitch-space defined by the map of Meantone temperament, for example, there is no such thing as the syntonic comma. Since we are no longer allowing for existence of an absolute JI map, "temperaments" as we know it cease to exist, and we can only define different pitch-space maps in relation to each other. At this point, it would be impossible to use frequency ratios in a meaningful way, unless we specified the timbre before hand. And this would mean that a 3/2 on a clarinet would be different than a 3/2 on a flute, and we'd have to do timbral analysis of every instrument we wanted to use before we could even describe a scale on it.

Ludicrous, John. Ludicrous. You don't want to open the Pandora's box of letting frequency ratios be flexibly defined.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Perhaps that case could be made, but this way of looking at timbre still places a perfectly-harmonic timbre (i.e. one whose partials form a harmonic series) as the "ideal" of harmonic functionality, so detuning the partials to match the scale is going to push the timbre further from the ideal. This effect is clearly audible to me in a lot of Sethares' music, where timbres mapped to fit bizarre scales like 8-EDO or 13-EDO sound less harmonically-functional (and less pleasant, at least to my ears) than harmonic timbres playing the same scales.

Years ago when I learned about critical bands I was very interested in the idea of matching timbres to scales. Sadly, when I finally got to try it it reminded me of the Anacin commercial "Why trade a headache for an upset stomach?" When the detuning is slight, it sounds wonderful. But then it sounds fine without detuning. When the detuning is extreme, it sounds kind of grotty. But it sounds out of tune without the detuning. In between, you get results in between. It's a different sound either way, but neither way not without things one might objrct to.

On Thu, Sep 2, 2010 at 9:46 PM, genewardsmith
<genewardsmith@...> wrote:
>
> Years ago when I learned about critical bands I was very interested in the idea of matching timbres to scales. Sadly, when I finally got to try it it reminded me of the Anacin commercial "Why trade a headache for an upset stomach?" When the detuning is slight, it sounds wonderful. But then it sounds fine without detuning. When the detuning is extreme, it sounds kind of grotty. But it sounds out of tune without the detuning. In between, you get results in between. It's a different sound either way, but neither way not without things one might objrct to.

An idea that I had recently, which I'll develop someday when I have an
unlimited amount of time and money and nothing else to do, is to come
up with a clever algorithm that selectively drops beating partials out
when you play a chord.

So if you're playing a 12-equal C major, it'll detect a clash between
5xC and 4xE, realize that those partials are sufficiently close enough
and within the critical bandwidth to be a problem, and get rid of
whichever is softer with that timbre. It will then increase the volume
of the other one to make up the difference. It'll do the same for the
C and the G, although the difference will be more slight. For
C-E-G-Bb, the 7th harmonic of C gets nixed as well.

How close partials can be before they're nixed could be a
user-adjustable setting, and whether or not partials are detuned could
also be an option.

Hence we have no beating, ever, and yet timbres are still perfectly
harmonic. At least they become no less harmonic than the roots
themselves, which doesn't seem to be causing anyone a problem.

Someday...

-Mike

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

> An idea that I had recently, which I'll develop someday when I have an
> unlimited amount of time and money and nothing else to do, is to come
> up with a clever algorithm that selectively drops beating partials out
> when you play a chord.

THAT is absolutely friggin' GENIUS, Mike. I mean, SERIOUSLY. It's like you could achieve the same effect as adaptive-JI but stay in an equal temperament. Too bad I don't have the resources to get you a research grant.

-Igs

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>Years ago when I learned about critical bands I was very interested in the idea of matching
>timbres to scales. Sadly, when I finally got to try it it reminded me of the Anacin commercial
>"Why trade a headache for an upset stomach?" When the detuning is slight, it sounds
>wonderful. But then it sounds fine without detuning. When the detuning is extreme, it
>sounds kind of grotty. But it sounds out of tune without the detuning. In between, you get
>results in between. It's a different sound either way, but neither way not without things one
>might objrct to.

Yeah, I remember listening to some of Sethares' early examples and thinking, "okay, there's no beating, but these timbres sound terrible!" I'll take 8-EDO on an acoustic guitar over whatever timbre he used in his "I made 8-EDO sound in-tune!" example ANY DAY. It takes a rough timbre to make a rough scale sound in-tune, so is it really a worthwhile trade-off?

MikeB>, is to come
> up with a clever algorithm that selectively drops beating partials out
> when you play a chord.
Hmm....but isn't that what adaptive JI already does?
BTW...I already have a program that does that (kinda)...you use a slider to
determine how high-limit the chord can be. So, of course, the higher the limit
the greater the accuracy toward the original tones but the more discordance (on
average)...the lower limit the less accuracy and more "comma-tic drift" but the
more concordance.

The program method is use is pretty crude, though. I loop through all
possible values of, say...w,x,y,z for a 4-tone chord (say w,x,y,z = 1 to 8 for
fairly low limit) looking for the lowest fractional representation that's within
a certain number of cents error for all possible dyads.

Then I simply the fraction by dividing all of w,x,y,z by primes until
reduction can't be performed to get a lowest term chord. Brute force, yes...but
it seems to work. Plus I actually posted an earlier version of it (IE with no
slider) on this list about a month ago.

I'm sure once you get the "lowest form JI chord"...it would be pretty easy to
add a part to the algorithm to see which TET would give the lowest error in
rounding it. Only problem is then you may end up constantly "modulating"
between different TET's depending on which chord you are playing and might want
to limit it to only certain TET's which you like the sound of and/or work well
with each other melodically.

________________________________
From: cityoftheasleep <igliashon@...>
To: tuning@yahoogroups.com
Sent: Thu, September 2, 2010 10:51:12 PM
Subject: [tuning] Re: What other temperaments support these 12-tet scales?

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

> An idea that I had recently, which I'll develop someday when I have an
> unlimited amount of time and money and nothing else to do, is to come
> up with a clever algorithm that selectively drops beating partials out
> when you play a chord.

THAT is absolutely friggin' GENIUS, Mike. I mean, SERIOUSLY. It's like you
could achieve the same effect as adaptive-JI but stay in an equal temperament.
Too bad I don't have the resources to get you a research grant.

-Igs

>"Yeah, I remember listening to some of Sethares' early examples and thinking,
>"okay, there's no beating, but these timbres sound terrible!"

I think there's a fine line where the timbres needed for Sethares' algorithm
are just too warped. The brain can only take a certain degree of lack of
periodicity in either tuning or timbre, or so it seems. What could really help,
IMVHO, is an equation to complement Sethares' algorithm which adds a periodicity
factor to his discordance curve.
That being said, I wonder how 7TET, which doesn't seem all that warped, would
work with a perfectly matched timbre (IE one where each overtone is 2^(1/7)
times the last and timber = tuning)...

________________________________
From: cityoftheasleep <igliashon@...>
To: tuning@yahoogroups.com
Sent: Thu, September 2, 2010 11:00:23 PM
Subject: [tuning] Re: What other temperaments support these 12-tet scales?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>Years ago when I learned about critical bands I was very interested in the idea
>of matching
>
>timbres to scales. Sadly, when I finally got to try it it reminded me of the
>Anacin commercial
>
>"Why trade a headache for an upset stomach?" When the detuning is slight, it
>sounds
>
>wonderful. But then it sounds fine without detuning. When the detuning is
>extreme, it
>
>sounds kind of grotty. But it sounds out of tune without the detuning. In
>between, you get
>
>results in between. It's a different sound either way, but neither way not
>without things one
>
>might objrct to.

Yeah, I remember listening to some of Sethares' early examples and thinking,
"okay, there's no beating, but these timbres sound terrible!" I'll take 8-EDO
on an acoustic guitar over whatever timbre he used in his "I made 8-EDO sound
in-tune!" example ANY DAY. It takes a rough timbre to make a rough scale sound
in-tune, so is it really a worthwhile trade-off?

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

> The program method is use is pretty crude, though. I loop through all
> possible values of, say...w,x,y,z for a 4-tone chord (say w,x,y,z = 1 to 8 for
> fairly low limit) looking for the lowest fractional representation that's within
> a certain number of cents error for all possible dyads.
>
> Then I simply the fraction by dividing all of w,x,y,z by primes until
> reduction can't be performed to get a lowest term chord. Brute force, yes...but
> it seems to work.

I can't figure out what you are saying. Can you provide an example?

>"I can't figure out what you are saying. Can you provide an example?"

Say you input the values 1/1, 1.255 (around 5/4), 4/3, and 7/5 as your chord and
specify 23 as your "limit"

The program loops through w = 1 to 23, x = 1 to 23, y = 1 to 23, z = 1 to 23
(since 23 is the "limit" you gave)

In each step of the loop, when z > y > x > w
A) it compares x/w to 1.255/1.0...then y/w to 1.3333 over 1.0...then z/w to 1.4
and records one error for each of these.

B) Then it compares all four of those errors and finds the highest of them.
C) Finally it sees if that highest error is lower than the lowest error
recorded. If it is...it records the values of w,x,y, and z thus giving a JI
chord (with w being the least common denominator).

Quick example: in the loop when values are w = 4 x = 5 y = 6 z = 7 it gets
5/4 with no error, but finds 6/4 to be far from 4/3 (high error) and 7/5 to be
very far from 7/5 (thus becoming the highest error and getting recorded in part
B...hence the chord 4,5,6,7 gets beaten by higher values of w,x,y, and z. An
example of that is where w = 7 x = 9, y = 12...where 9/7 is a whole lot closer
to 5/4 than 6/4 is to 4/3, which causes those values to overwrite the values
above as "closer answers" in part C while going through the loop.

When all the comparisons are done, the program divides your answer by prime
numbers (3,5,7,11,13) to simplify the answer to lowest terms.

For 1.255, 5/4 is obviously the closest. For 1.3333, 4/3 is the closest, For
1.4, 7/5 is the closest.
BUT since those fractions don't satisfy z > y > x > w (IE w being the LCD) the
lowest value of w,x,y,z that can satisfy the criteria while having 17 as the
limit (which it finds after going through the loops) is
12 15 16 17
(this gets 5/4 and 4/3 dead on, but misses 7/5 by about 20 cents with its value
of 17/12)

Now if you set 40 as the limit you get
27 34 36 38
...which means the dyads (x/w,y/w,z/w) are closer to the original values you
specified and shift less, but are also less "just". This means 34/27 is about
13 cents off 5/4, 36/27 is dead-on perfect, and 38/27 is about 9 cents off...a
better "mini-maxi" result.

Note the greatest dyadic error in the higher limit attempt to match the chord
(13 cents) is less than the 20 cents above....thus complying with the goal of
being able to be "as close or closer to the original notes" per increase in
"limit".

Side note: the reason I don't just set w as always being 1.0 is that your
root tone will not always be 1 IE you can have 1.1 1.1*1.25 1.1*1.333 1.1 * 1.4
as your answer...and get the same chord types as your result.
*********************************************
One major limitation is this program only does strict/in-order harmonic
series JI. It does not compare, say, error for z/x IE between the second and
fourth note of the chord to find the worst case dyad and come up with answers
NOT sharing a common denominator. Do any of you think it would be worthwhile to
add a non-straight-harmonic-series/dyadic chord finding option to the program?

Yesterday I had a couple of hours on the train so I had a chance to make an example (laptops are so handy). Maybe you'll agree from this example (uploaded to files here) that tempering out 81/80 is not necessary.

-Cameron Bobro

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Aug 29, 2010 at 4:14 AM, cameron <misterbobro@...> wrote:
> >
> > And, maintaining near-perfect 7-limit Just interpretation of this scale with full transposability and circulation, the comma to temper out is simply 225/224, the difference between (two pure M3+m3) and (septimal tritone+pure fourth), also the difference between (septimal subminor third + pure fourth) and (two pure M3), etc.
>
> How did you come up with marvel here, exactly? I'm also noticing that
> if the C-F# is intoned 7/5, and if the F#-A-C can both be 6/5's, that
> equates 36/25 with 10/7, which tempers out 126/125 - the starling
> comma, which I haven't really played around with before. Tempering
> both 126/125 and 225/224 means that 81/80 is tempered as well, so
> we're back to septimal meantone. I ended up playing around with this
> in 31-tet, as per both this and Gene's suggestion, and it sounded
> great. So it basically means that it works very well as a subset of
> meantone[12]. As for what else it works as, I'm curious, as the
> decatonic scales can work as a part of meantone[12] too :)
>
> I just discovered that scala's "fit/mode" function, so fitting it to
> the 12-equal version gives 20-equal as having a strictly proper
> version. It's 2 3 2 3 3 2 3 2 for those interested. It is apparently
> CS too.
>
> -Mike
>

nicely dissolving moments taking the forefront. i kept trying to picture it in conjunction with the view out the window and as the polyrhymic background it would make with the incessant upward flams of clicking tracks~

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 4/09/10 3:01 PM, cameron wrote:
>
> Yesterday I had a couple of hours on the train so I had a > chance to make an example (laptops are so handy). Maybe you'll > agree from this example (uploaded to files here) that > tempering out 81/80 is not necessary.
>
> -Cameron Bobro
>
> --- In tuning@yahoogroups.com > <mailto:tuning%40yahoogroups.com>, Mike Battaglia > <battaglia01@...> wrote:
> >
> > On Sun, Aug 29, 2010 at 4:14 AM, cameron <misterbobro@...> > wrote:
> > >
> > > And, maintaining near-perfect 7-limit Just interpretation > of this scale with full transposability and circulation, the > comma to temper out is simply 225/224, the difference between > (two pure M3+m3) and (septimal tritone+pure fourth), also the > difference between (septimal subminor third + pure fourth) and > (two pure M3), etc.
> >
> > How did you come up with marvel here, exactly? I'm also > noticing that
> > if the C-F# is intoned 7/5, and if the F#-A-C can both be > 6/5's, that
> > equates 36/25 with 10/7, which tempers out 126/125 - the > starling
> > comma, which I haven't really played around with before. > Tempering
> > both 126/125 and 225/224 means that 81/80 is tempered as > well, so
> > we're back to septimal meantone. I ended up playing around > with this
> > in 31-tet, as per both this and Gene's suggestion, and it > sounded
> > great. So it basically means that it works very well as a > subset of
> > meantone[12]. As for what else it works as, I'm curious, as the
> > decatonic scales can work as a part of meantone[12] too :)
> >
> > I just discovered that scala's "fit/mode" function, so > fitting it to
> > the 12-equal version gives 20-equal as having a strictly proper
> > version. It's 2 3 2 3 3 2 3 2 for those interested. It is > apparently
> > CS too.
> >
> > -Mike
> >
>
>

Cameron>"Yesterday I had a couple of hours on the train so I had a chance to
make an example (laptops are so handy). Maybe you'll agree from this example
(uploaded to files here) that tempering out 81/80 is not necessary."

Hmm...isn't that saying having an octave (and also the fifth over that) a
bit over 20 cents off is "ok"? I'll see what it sounds like...but I would
think any chord played between C6 and C8 (IE a C5 G5 A5 E6, where the first
overtone would clash....or C5 D#7 F7 D#6, where the second overtone would
clash) would sound quite off. Perhaps the trick is to avoid widely space
chords and playing note an octave above each other at the same time at once (or
avoid widely spaced chords or sustained notes of any sort all together)?

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> nicely dissolving moments taking the forefront. i kept trying
> to picture it in conjunction with the view out the window and as
> the polyrhymic background it would make with the incessant
> upward flams of clicking tracks~
>

Thanks Kraig- maybe I'm just getting cheesey with age :-) but I don't have any problems with program music. Of course outside the 12-tET box program ideas haven't been turned into a standard lexicon of sound-bites.

Hmmm... not tempering out 81/80 usually implies having pure fifths. 1/1, 5/4, 81/64, 3/2. The 81/80 is there between the 5/4 and the 81/64 (which is 3/2*4, modulo2), and not tempering 81/80 out means you've got both the Just 5-limit major third and the 3-limit ditone ("Pythagorean Major 3d").

If you're using a subset of a tuning or temperament that doesn't temper out 81/80 (JI itself for example), you're going to get wolf fifths of course. Which you can either avoid, or use deliberately. In the case of the tuning I used in this piece, wolves are never unavoidable, as the tuning circulates. So if do they appear it is not an accident.

Using a scale of such augmented/diminished nature, C,C#,Eb,E,F#,G#,A,B,C in 12-speak, I think it's dandy to be able to vary and choose between augmented sonorities like 5/4+5/4 and 5/4+81/64, etc., and to use seven-limit. D#/Eb here is 7/6, F# is 7/5 for example, all intervals are very close to pure, as you can hear.

And 81/80 is not tempered out. Not tempering out 81/80 is like the first fundamental step of "detwelvulating".

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Cameron>"Yesterday I had a couple of hours on the train so I had a chance to
> make an example (laptops are so handy). Maybe you'll agree from this example
> (uploaded to files here) that tempering out 81/80 is not necessary."
>
> Hmm...isn't that saying having an octave (and also the fifth over that) a
> bit over 20 cents off is "ok"? I'll see what it sounds like...but I would
> think any chord played between C6 and C8 (IE a C5 G5 A5 E6, where the first
> overtone would clash....or C5 D#7 F7 D#6, where the second overtone would
> clash) would sound quite off. Perhaps the trick is to avoid widely space
> chords and playing note an octave above each other at the same time at once (or
> avoid widely spaced chords or sustained notes of any sort all together)?
>

Gene wrote:

> Years ago when I learned about critical bands I was very
> interested in the idea of matching timbres to scales. Sadly,
> when I finally got to try it it reminded me of the Anacin
> commercial "Why trade a headache for an upset stomach?" When
> the detuning is slight, it sounds wonderful. But then it
> sounds fine without detuning. When the detuning is extreme,
> it sounds kind of grotty. But it sounds out of tune without
> the detuning. In between, you get results in between. It's
> a different sound either way, but neither way not without
> things one might object to.

I agree there's no net benefit at the extremes, but I think
it really can give a little cake both now and later in the
"in between" range, e.g. for errors like those of 12-ET or
pajara. -Carl

On Sun, Sep 5, 2010 at 2:24 AM, cameron <misterbobro@...> wrote:
>
> Hmmm... not tempering out 81/80 usually implies having pure fifths. 1/1, 5/4, 81/64, 3/2. The 81/80 is there between the 5/4 and the 81/64 (which is 3/2*4, modulo2), and not tempering 81/80 out means you've got both the Just 5-limit major third and the 3-limit ditone ("Pythagorean Major 3d").
>
> If you're using a subset of a tuning or temperament that doesn't temper out 81/80 (JI itself for example), you're going to get wolf fifths of course. Which you can either avoid, or use deliberately. In the case of the tuning I used in this piece, wolves are never unavoidable, as the tuning circulates. So if do they appear it is not an accident.
>
> Using a scale of such augmented/diminished nature, C,C#,Eb,E,F#,G#,A,B,C in 12-speak, I think it's dandy to be able to vary and choose between augmented sonorities like 5/4+5/4 and 5/4+81/64, etc., and to use seven-limit. D#/Eb here is 7/6, F# is 7/5 for example, all intervals are very close to pure, as you can hear.
>
> And 81/80 is not tempered out. Not tempering out 81/80 is like the first fundamental step of "detwelvulating".

Hi Cameron,
I just listened to the file. I really like what you did with it - how
did you intone things? Did you use a linear temperament, or stick with
marvel as a planar tuning? Or did you just tune it by ear without
following any specific methodology?

-Mike

MikeB>"If you're using a subset of a tuning or temperament that doesn't temper
out 81/80 (JI itself for example), you're going to get wolf fifths of course.
Which you can either avoid, or use deliberately. In the case of the tuning I
used in this piece, wolves are never unavoidable, as the tuning circulates. So
if do they appear it is not an accident."
I keep bringing this up, but my "mini-max" scale of

1
9/8 (196.198 cents exact value)
6/5
4/3 (504.0937 cents exact)
3/2

5/3 (891.9594 cents exact)
9/5 (1009.8847 cents exact)
2/1

It is "irregularly tempered" and, as I understand it, does not (or at least
does not suddenly) temper out the comma and maintains all possible dyads within
about 8 cents of pure. There is a 10/7 between 5/3 and 6/5 on the next octave
in place of the tri-tave and an 8/5 between 5/3 and 4/3 on the next octave (more
or less) in the one spot where the fifth is "missing".
And since it has the semi-tones between 9/8 and 6/5...and again between 9/5
and 5/3...it feel a lot like JI diatonic...minus that one wolf fifth. So far
I've found (at least concerning this type of 7-tone scale)...it's about the
closest thing I can find to having your cake and eating it too...even when
compared to something much like it, such as the 7-tone diatonic subset of 1/4
comma mean-tone.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
So far
> I've found (at least concerning this type of 7-tone scale)...it's about the
> closest thing I can find to having your cake and eating it too...even when
> compared to something much like it, such as the 7-tone diatonic subset of 1/4
> comma mean-tone.
>

It's an irregularly tempered version of meantone diatonic. What would you say is its advantage over 1/4 comma?

By way of comparison, here is a least-squares optimization in the same mode:

! dialeastsquares.scl
!
Least squares diatonic
7
!
192.22706
309.53714
504.94761
695.05239
890.46286
1007.77294
1200.00000

Gene>"It's an irregularly tempered version of meantone diatonic. What would you
say is its advantage over 1/4 comma?"

At least according to my computer, it does better at "minimax" IE it's
highest error from JI-diatonic type intervals is lower than 1/4 comma meantone's
(all dyads under 8 cents error)...at least according to a program I build. Ah
yes...and that analysis includes having a perfect 2/1 octave...without any sort
of major tempering on the last note to force it to fit the octave.

>"By way of comparison, here is a least-squares optimization in the same mode:
! dialeastsquares.scl
!
Least squares diatonic
7
!
192.22706
309.53714
504.94761
695.05239
890.46286
1007.77294
1200.00000"

What exactly does "least squares" mean?
I'm going to try plugging this in to my program at home later and see what I
get...
It seems incredibly close to my scale at first glance and I'm guessing a tad
more accurate (since my program only calculates down to an accuracy of a few
cents or so and I'm guessing your version may generate a cent or so better
"mini-max").
But I'm wondering how (IE via what formula) you used to manage to come up
with it?...especially since I did it via a program that tries tons of
possibilities and takes the best one given a list of desirable dyads, rather
than using a firm/"single" mathematical equation.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Gene>"It's an irregularly tempered version of meantone diatonic. What would you
> say is its advantage over 1/4 comma?"

> What exactly does "least squares" mean?

Instead of minimizing the maximum error, it minimizes the sum of th squares of the errors. This has several features which can be useful:

(1) It is less sensitive to outliers

(2) The solution is unique

(3) It is very easy to compute if you are equipped to solve systems of linear equations

> But I'm wondering how (IE via what formula) you used to manage to come up
> with it?...

The same old one Carl Friedrich Gauss used; you take the sum of the squares of the errors, with unknown/indeterminate quantities for each of the scale steps aside from unity. You then differentiate it with respect to each unknown, and end up with n linear equations in n unknowns which you then solve. The unknowns in this case would be the values in cents of all of the scale steps aside from unity.

Mike, I just figured that the "tritone" in the scale is a sine qua non. I think we understood that the C is the tonic and we want all 8 tones without a heavy hierarchy, otherwise the scale is hardly challenging, for you could just take it as c# minor/harmonic minor for example). And we wanted 7-limit. So, either 7/5 or 10/7 (its merely a question as to which voicing you're going to tend toward as to which one is more concordant in practice, of course). The aug2/M3 ( or m3/dim4, 12-tET letting us choose) combo is obviously important, so I went for 7/6 and 5/4, which both sound great and contrast with each other. The m2/supertonic I gave to 7, so, 21/20 (4/3 below 7/5). The lower hemioctachord kind of writes itself. The upper hemioctachord though raises the question of tempering. Should the aug5/m6 belong to 7 (14/9, 4/3 above 7/6) or to 5 (25/16, two 5-limit M3s)? Should the M7 belong to 5 (15/8) or to 7 (28/15, 4/3 above 7/5)? These choices lie very close to each other, less than 8 cents apart, 225/224 to be exact, quite good natural puns. So I just tempered out 225/224 by using 41 equal divisions of the octave. I guess that means the temperament would be called "miracle" here.

The complement of this scale, in 12-tone set theory jargon, is the material of a Gm7 chord. Its a very nice place to go I think, there's a few moments of G dorian toward the end there, with 7/4 as the 7-limit minor third and G,D,F Pythagorean from the C (the rest of the dorian scale/13th chord is an intersection with the first set, in set-speak, shared tones in normal human lingo :-) ) I didn't comma-adjust anything by using a step of 41 in this piece, no need, but I would do so for various different modulations. The resulting scale I used turned out to almost identical to Kraig's Centaur scale, which was fun (I did Compare Scale in Scala), in fact it's the same as Centaur with 225/224 tempered out via 41-edo. For some stuff that comes to ear, as far as this piece, I'd need a few more tones, about 16 tones altogether, in order to shift some wolves. (But of course you could spend years just with the Centaur scale).

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Sep 5, 2010 at 2:24 AM, cameron <misterbobro@...> wrote:
> >
> > Hmmm... not tempering out 81/80 usually implies having pure fifths. 1/1, 5/4, 81/64, 3/2. The 81/80 is there between the 5/4 and the 81/64 (which is 3/2*4, modulo2), and not tempering 81/80 out means you've got both the Just 5-limit major third and the 3-limit ditone ("Pythagorean Major 3d").
> >
> > If you're using a subset of a tuning or temperament that doesn't temper out 81/80 (JI itself for example), you're going to get wolf fifths of course. Which you can either avoid, or use deliberately. In the case of the tuning I used in this piece, wolves are never unavoidable, as the tuning circulates. So if do they appear it is not an accident.
> >
> > Using a scale of such augmented/diminished nature, C,C#,Eb,E,F#,G#,A,B,C in 12-speak, I think it's dandy to be able to vary and choose between augmented sonorities like 5/4+5/4 and 5/4+81/64, etc., and to use seven-limit. D#/Eb here is 7/6, F# is 7/5 for example, all intervals are very close to pure, as you can hear.
> >
> > And 81/80 is not tempered out. Not tempering out 81/80 is like the first fundamental step of "detwelvulating".
>
> Hi Cameron,
> I just listened to the file. I really like what you did with it - how
> did you intone things? Did you use a linear temperament, or stick with
> marvel as a planar tuning? Or did you just tune it by ear without
> following any specific methodology?
>
> -Mike
>

Sorry to rehash this after it's been gone for so long, it just took me a while to get to everything.

> If you can't define what *can't* function musically, what use is there in defining what *can*? The point of defining a concept is to give it limits in what it can describe.

Again, I'm not even defining what can or can't at all. I'm defining what is more likely to function than something else. Music, obviously, is partial subjective, and partial cognitive/mathematical/what have you. What I'm trying to say is you can't define what is or isn't musically functional because to someone a rainforest ambient track might be music. But I think you can determine that a 4/5/6 triad played by three trombones is *more likely* to be interpreted as "musical" than the sound of a car crash.

> > For example, scales with only two steps sizes are more likely to function melodically than > those with three. Those with three step sizes are more likely to function melodically than > those with four.
>
> I take it you're not a fan of JI, then? Some JI scales, like Centaur for instance, have WAY more than 4 step-sizes, and yet they seem to function just fine to my ears.

I love myself some good JI. But have you noticed that so many JI scales imitate MOS scales, specifically the diatonic scale? Melodic stability is usually just as important in designing scales as harmonic stability, and just diatonic scales are *approximating* the diatonic scale, a scale with two *perceptual* step sizes.

> > And harmonically related timbres are morel likely to function as a timbre
> > with the possibility for recognizable harmony than those that are not related
> > harmonically.
>
> Perhaps that case could be made, but this way of looking at timbre still places a perfectly-harmonic timbre (i.e. one whose partials form a harmonic series) as the "ideal" of harmonic functionality, so detuning the partials to match the scale is going to push the timbre further from the ideal. This effect is clearly audible to me in a lot of Sethares' music, where timbres mapped to fit bizarre scales like 8-EDO or 13-EDO sound less harmonically-functional (and less pleasant, at least to my ears) than harmonic timbres playing the same scales.

Absolutely. Similar to the way tempering just ratios to fit melodic purposes and transposition has its payoffs and downturns, so does the tempering of a timbre. You might end up with a dirty sounding timbre with consonant triads, as opposed to a pure tone with triads that make you cringe. Sometimes I prefer one to the other, and sometimes it's switched.

> > No, to get the temperament I have to give cent values to the generators, or 3/2 and 2/1 > (not 5/4), which will then determine both the tuning of the temperament and the tuning > of the timbre. I don't think that's circular, is it?
>
> It is if you maintain that JI ratios are defined as the ratios between partials in a timbre. Are you now saying that 3/2 and 2/1 can be defined in some way that is unrelated to partials in a timbre?

Before I said something along the lines of:
I define 3/2 as the interval between the second and third partials, regardless of what they may be.

To put it simply, I shouldn't have said that. It is too general to fit what I meant, and I think it's gotten me into trouble (x_X)

I mean to say that just intervals (like 3/2) are a "perceptual identity" or "function" represented by the musical connotation of the interval between the second and third partial of a harmonic timbre. We can't define over which values of cents the human race as a whole will still perceive this identity, but we can define over which values of cents it makes sense to define a just interval when we are defining other just intervals in terms of it.

It makes sense to define a just interval in terms of others (defining 5/4 in a stack of fifths and octaves) only when you can still sensibly relate each back to the perceptual identity's representation in the harmonic series. If you define 5/4 as (3/2)^4 * (2/1)^-2, then 4/3 >= 5/4 >= 6/5 when 686<3/2<720, and this important because, when relating it back to the harmonic series and it's functions, the function of 5/4 is not smaller than that of a 6/5, and the function of a 4/3 is not smaller than that of a 5/4.
Think about a major triad. Where 6/5>5/4, won't 10:12:15 be major, and 4:5:6 be minor? But we define 4:5:6 as the function of a major triad, and it only stays major above a fifth of 786 cents.
Similarly, above 720 cents, the 4:5:6 triad will overlap with 3:4:5, and this will make the definition of the 4:5:6 as a major triad irrelevant, so we don't go above 720.

Now, I realize this reasoning is slightly different than the angle I've been taking about mapping a timbres' partials, but I think they revolve around the same central idea. It is perceptually meaningful to, when tempering just ratios, not allow for a/b<b/c>c/d… when a<b, b<c, and c<d, because of the way one can define harmonic function by just ratios' relationships to the functions of the intervals between harmonic partials. Tempering the timbre or not.

> Remember that we can stretch or compress the partials in a timbre as much as we want and still fulfill the condition that the ratios between successive partials decrease as we go up the series. The 2nd and 3rd partial can be as close together or as far apart as we care to make them, suggesting that a 3/2 defined in this way can be of any size.

That is a good point, and one I've struggled with before. What if you set 2/1 = 400 cents? Right now, I don't know.
Before we had an issue of which generators are those over which we should define a temperament. We couldn't just say, "A 3/2 stops being a 3/2 right…. Here!", and the same is true for the period (which is what would cause the compression). Obviously right now I'm proposing an objective way to define generator ranges, but as far as the period is concerned I don't know if there is an objective way to define a range.
I would suggest, however, that given that extremely stretched or compressed temperament of the timbre, it would still be more likely to function harmonically if it fits those conditions than if it didn't. I don't have many example of this recorded, but I'll have to work up a few and put them on the list. I've heard some pretty extreme stretching (2/1=1300 cents for instance), and it can still function very recognizably as meantone, with mapped timbres.

> But if you allow for this, then you have to allow for a timbre in which all the partials are equally-spaced.

That's correct, that would be a temperament where 2/1=3/2=4/3 etc. A very very low accuracy temperament mind you (where 4/3=1!), but still a temperament which could define a (very low accuracy) timbre that would most allow for the temperament's definitions to funciton. 4/3 would be most likely to be interpreted as 1/1 given a timbre of equally spaced partials.

> For something like a triangle wave, this is fine, since the ratio between partials is a maximally-consonant 2/1.

I don't understand this. Doesn't a triangle wave simply contain odd harmonics? f, 3f, 5f, etc as opposed to f, 2f, 4f, 8f which it seems like what it is you are describing.

> But what about a timbre where the ratio between any two successive partials is something like 18/17? Or even 6/5? Are you going to tell me, with a straight face, that timbres made like this are actually more harmonically-functional than a timbre where the ratios between successive partials increase as we go up the series? You'd clearly have to stipulate some minimum distance between partials.

Given that you are using the temperament where 2/1=3/2=4/3 etc, and you assign 2/1=something like 215 cents, then I suggest that though it may be a HUGELY inaccurate temperament, a timbre defined by this temperament would be mostly likely to function as the temperament describes, with that given period size. How desirable that would be for me, I don't know, but I'm not the one asking for such a low accuracy temperament with such a mangled period! =P

> As I said above, there's no limit to how far apart or close together we can space the partials in a timbre, effectively meaning that there is no boundary to the size of the ratios between them. So we could say 4:5 = 200:201, or we could say it equals 200:276394, presuming the rest of the partials are sized so that the ratios between them decrease as we go up the series.

Yes. Again, a very low accuracy temperament with an extremely tempered period, but given that one would expect a very low accuracy timbre.

> Furthermore, in order for temperament to make sense at all, we have to be able to define a JI map that is absolute, which we can then use temperament to warp. But you are basically suggesting that we define JI *according to a temperament*.

Again, I was overzealous in saying that "5/4 is the relationship between the fifth and fourth partials, whatever that may be". I think I've addressed that above.

John M