11tet as JI

Here are the progressively better just approximations for
22tet (since it is discussed a lot), 11 is represented
by the even scale degrees.

1 : =25/24 =26/25 =27/26 =28/27
=29/28 =30/29 =31/30 =32/31 =97/94
2 : =15/14 =16/15 =33/31 =49/46 =82/77
3 : =11/10 =67/61 =78/71 =89/81 =100/91
4 : =8/7 =17/15 =42/37 =59/52 =76/67
5 : =7/6 =27/23 =34/29 =41/35 =48/41
6 : =6/5 =17/14 =23/19 =29/24
7 : =5/4 =51/41 =56/45 =61/49 =66/53
=71/57 =76/61 =81/65 =86/69 =91/73 =96/77
8 : =9/7 =103/80 =112/87 =121/94
9 : =41/31 =45/34 =49/37 =53/40 =57/43
=61/46 =65/49 =69/52 =73/55 =77/58 =81/61
10 : =11/8 =26/19 =37/27
11 : =17/12 =24/17 =41/29 =99/70 =140/99
12 : =16/11 =19/13 =35/24 =54/37
13 : =92/61 =95/63 =98/65 =101/67 =104/69
=107/71 =110/73 =113/75 =116/77 =119/79 =122/81
14 : =14/9 =87/56 =101/65 =115/74 =129/83 =143/92
15 : =8/5 =45/28 =53/33 =61/38 =69/43 =77/48
16 : =28/17 =33/20 =38/23 =43/26 =48/29 =101/61 =149/90
17 : =12/7 =29/17 =41/24
18 : =23/13 =30/17 =37/21 =67/38
19 : =20/11 =71/39 =91/50 =111/61
20 : =15/8 =47/25 =62/33 =77/41 =169/90
21 : =27/14 =29/15 =31/16 =157/81 =188/97

with an arbitrary limit at 100 in the denominator and ".01" as
the initial error criteria to be sought out and improved upon.

Relaxing the initial error term to .05 allows makes these
changes in two entries (rest omitted).

10 : =4/3 =7/5 =11/8 =26/19 =37/27

12 : =3/2 =10/7 =13/9 =16/11 =19/13 =35/24 =54/37

Now I have a question regarding limits. Some have described 22tet as
giving good 7-limit intervals, and it does differentiate between
6/5 and 7/6, which seems to be fundamental to make the claim. But the
3-limits are sitting on top of the 7-limits? If I tune the error
term, I can make it contain all the 5-limit intervals unambiguously,
but not the 7-limit.

Do you compositionally differntiate between the two interpretations
of an interval (like the augmented second / minor third in 12tet)?

I'd been thinking about this in a different context. If you take
a somewhat (but not too) complex et, you can approximate sections
of the harmonic series within it. For instance in 53 tet

et scale
degree 0 7 15 22 29 33

harmonic
degree 9 10 11 12 13 14

But I could also go

et scale
degree -2 7 15 22 29 35

harmonic
degree 8 9 10 11 12 13

You can see that by changing the (unshown but implied)
root, and moving the outer voices, the same 4 inner
voices are being a different portion of the overtone
series. Again, its just punning, breaking 'limits'
as it were, but do you play with these things while
composing (as most of us are all familiar with in 12tet
composing)?

Bob Valentine

Robert!
If the 22 tone step is 54.54 cents. the maximum tolerance anyone could
allow is 27 cents. At this point you would run into the next tone. Now if
you allow 15 cents away as in the 25/24 you are more than half way to the
27 cent maximum. This means that any ratio would have more than a 50%
chance of fitting. Don't you think this is a little all-inclusive?

Robert C Valentine wrote:

>
> Here are the progressively better just approximations for
> 22tet (since it is discussed a lot), 11 is represented
> by the even scale degrees.
>
> 1 : =25/24 =26/25 =27/26 =28/27
> =29/28 =30/29 =31/30 =32/31 =97/94
> 2 : =15/14 =16/15 =33/31 =49/46 =82/77
> 3 : =11/10 =67/61 =78/71 =89/81 =100/91
> 4 : =8/7 =17/15 =42/37 =59/52 =76/67
> 5 : =7/6 =27/23 =34/29 =41/35 =48/41
> 6 : =6/5 =17/14 =23/19 =29/24
> 7 : =5/4 =51/41 =56/45 =61/49 =66/53
> =71/57 =76/61 =81/65 =86/69 =91/73 =96/77
> 8 : =9/7 =103/80 =112/87 =121/94
> 9 : =41/31 =45/34 =49/37 =53/40 =57/43
> =61/46 =65/49 =69/52 =73/55 =77/58 =81/61
> 10 : =11/8 =26/19 =37/27
> 11 : =17/12 =24/17 =41/29 =99/70 =140/99
> 12 : =16/11 =19/13 =35/24 =54/37
> 13 : =92/61 =95/63 =98/65 =101/67 =104/69
> =107/71 =110/73 =113/75 =116/77 =119/79 =122/81
> 14 : =14/9 =87/56 =101/65 =115/74 =129/83 =143/92
> 15 : =8/5 =45/28 =53/33 =61/38 =69/43 =77/48
> 16 : =28/17 =33/20 =38/23 =43/26 =48/29 =101/61 =149/90
> 17 : =12/7 =29/17 =41/24
> 18 : =23/13 =30/17 =37/21 =67/38
> 19 : =20/11 =71/39 =91/50 =111/61
> 20 : =15/8 =47/25 =62/33 =77/41 =169/90
> 21 : =27/14 =29/15 =31/16 =157/81 =188/97
>
> with an arbitrary limit at 100 in the denominator and ".01" as
> the initial error criteria to be sought out and improved upon.
>
> Relaxing the initial error term to .05 allows makes these
> changes in two entries (rest omitted).
>
> 10 : =4/3 =7/5 =11/8 =26/19 =37/27
>
> 12 : =3/2 =10/7 =13/9 =16/11 =19/13 =35/24 =54/37
>
> Now I have a question regarding limits. Some have described 22tet as
> giving good 7-limit intervals, and it does differentiate between
> 6/5 and 7/6, which seems to be fundamental to make the claim. But the
> 3-limits are sitting on top of the 7-limits? If I tune the error
> term, I can make it contain all the 5-limit intervals unambiguously,
> but not the 7-limit.
>
> Do you compositionally differntiate between the two interpretations
> of an interval (like the augmented second / minor third in 12tet)?
>
> I'd been thinking about this in a different context. If you take
> a somewhat (but not too) complex et, you can approximate sections
> of the harmonic series within it. For instance in 53 tet
>
> et scale
> degree 0 7 15 22 29 33
>
> harmonic
> degree 9 10 11 12 13 14
>
> But I could also go
>
> et scale
> degree -2 7 15 22 29 35
>
> harmonic
> degree 8 9 10 11 12 13
>
> You can see that by changing the (unshown but implied)
> root, and moving the outer voices, the same 4 inner
> voices are being a different portion of the overtone
> series. Again, its just punning, breaking 'limits'
> as it were, but do you play with these things while
> composing (as most of us are all familiar with in 12tet
> composing)?
>
> Bob Valentine
>
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-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

Robert C Valentine wrote:

>
>
> Now I have a question regarding limits. Some have described 22tet as
> giving good 7-limit intervals, and it does differentiate between
> 6/5 and 7/6, which seems to be fundamental to make the claim. But the
> 3-limits are sitting on top of the 7-limits? If I tune the error
> term, I can make it contain all the 5-limit intervals unambiguously,
> but not the 7-limit.

The 7/6 is good, but then again only because the terms are out of tune in
the same direction. The 7/4 is 13+ cents off with the 16/9 15 cents in the
other direction. This is about as ambiguous as you can get. The 5 though
is out of tune in the opposite direction which when compared to the 3 is as
out of tune as a 5/4 in 12ET. With tolerances like this why not call the
major third in 12ET 11/7. I am having a really hard time understanding all
the brooch. There is nothing like a good 9/8. If 22ET has merit which I
assume it does, it is not because of its relationships to these ratios. As
someone who knows many of them I would never mistake one for the other.

>
>
> Do you compositionally differntiate between the two interpretations
> of an interval (like the augmented second / minor third in 12tet)?
>
> I'd been thinking about this in a different context. If you take
> a somewhat (but not too) complex et, you can approximate sections
> of the harmonic series within it. For instance in 53 tet
>
> et scale
> degree 0 7 15 22 29 33
>
> harmonic
> degree 9 10 11 12 13 14
>
> But I could also go
>
> et scale
> degree -2 7 15 22 29 35
>
> harmonic
> degree 8 9 10 11 12 13
>
> You can see that by changing the (unshown but implied)
> root, and moving the outer voices, the same 4 inner
> voices are being a different portion of the overtone
> series. Again, its just punning, breaking 'limits'
> as it were, but do you play with these things while
> composing (as most of us are all familiar with in 12tet
> composing)?
>
> Bob Valentine
>

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

Robert C Valentine wrote,

>Now I have a question regarding limits. Some have described 22tet as
>giving good 7-limit intervals, and it does differentiate between
>6/5 and 7/6, which seems to be fundamental to make the claim. But the
>3-limits are sitting on top of the 7-limits?

I don't know what you mean.

>If I tune the error
>term, I can make it contain all the 5-limit intervals unambiguously,
>but not the 7-limit.

That's true, since 7/5 and 10/7 are approximated by the same interval.

>Do you compositionally differntiate between the two interpretations
>of an interval (like the augmented second / minor third in 12tet)?

In this case, the other notes forming the 7-limit tetrad (or even just a
triad) will determine whether 7/5 or 10/7 is meant.

>Again, its just punning, breaking 'limits'
>as it were, but do you play with these things while
>composing (as most of us are all familiar with in 12tet
>composing)?

I do!

Kraig Grady wrote,

>Robert!
> If the 22 tone step is 54.54 cents. the maximum tolerance anyone could
>allow is 27 cents. At this point you would run into the next tone. Now if
>you allow 15 cents away as in the 25/24 you are more than half way to the
>27 cent maximum. This means that any ratio would have more than a 50%
>chance of fitting. Don't you think this is a little all-inclusive?

Oddly enough, Robert's first table does not include 3/2, even though it's
only 7 cents away from 13/22 oct.

Kraig Grady wrote,

>The 7/6 is good, but then again only because the terms are out of tune in
>the same direction.

7/6 is a basic 7-limit consonance and should not be "broken down" into its
"terms".

>The 7/4 is 13+ cents off with the 16/9 15 cents in the
>other direction. This is about as ambiguous as you can get.

Not at all -- the numbers in 7/4 are more than twice as small as those in
16/9 and so its "pull" is over twice as strong. For 8/7 vs. 9/8, though, the
ambiguity is significant unless additional notes are there to clear up the
context (I advocate tetradic harmony as the norm for 22-tET).

>The 5 though
>is out of tune in the opposite direction which when compared to the 3 is as
>out of tune as a 5/4 in 12ET.

That's a very convoluted statement, and I don't know what it means. 5/4 is
only 4 cents flat in 22-tET, and 3/2 is 7 cents sharp.

>With tolerances like this why not call the
>major third in 12ET 11/7.

14/11? The 12-tET major third is 14 cents sharp of 5/4 and 18 cents flat of
14/11. Since the numbers of the former are almost three times smaller than
those of the latter, 5/4 is about three times stronger than 14/11 in the
interpretation of the 12-tET major third.

>There is nothing like a good 9/8.

True, but 9/4 is acceptable, especially with other notes clearing up the
context. 12-tET has nothing like a good 8/7 or a good 10/9, and yet these
intervals can be clearly implied within certain chords in 12-tET.

>If 22ET has merit which I
>assume it does, it is not because of its relationships to these ratios. As
>someone who knows many of them I would never mistake one for the other.

Two-member ratios are not the full story. Chords of three or four notes can
resolve ambiguities inherent in the individual intervals. Common-practice
Western music since the abandonment of meantone temperament considers dyads
to be incomplete harmonies, and the 5-limit triad is the norm. My idea of
music in 22-tET uses the 7-limit tetrad as the norm. Each ratio within the
7-limit tetrad is clarified by its position relative to the others. The
otonal tetrad in 22-tET is much closer to just than in 12-tET, though
musicians have been trying to use the latter for at least a century. 22-tET
can distinguish the utonal tetrad from a 5:6:7:9 chord, which 12-tET can't
do. To my ears, the distinction made in 22-tET is qualitatively the same as
that made in JI.

Robert,

I see what you mean now. There's something weird about your algorithm,
especially the "relaxed" version. 10/22 oct. and 12/22 oct. are not good
approximations of 4:3, 7:5, 10:7, or 3:2. 4:3 is 7 cents off 9/22 oct., 3:2
is 7 cents off 13/22 oct. and 7:5 and 10:7 are each 17 cents off 11/22 oct.
I don't know why your algorithm assigned these ratios to the wrong scale
degrees.

-Paul

"Paul H. Erlich" wrote:

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Kraig Grady wrote,
>
> >The 7/6 is good, but then again only because the terms are out of tune in
> >the same direction.
>
> 7/6 is a basic 7-limit consonance and should not be "broken down" into its
> "terms".

Basically I was saying they are both out of tune in the same direction.

>
>
> >The 7/4 is 13+ cents off with the 16/9 15 cents in the
> >other direction. This is about as ambiguous as you can get.
>
> Not at all -- the numbers in 7/4 are more than twice as small as those in
> 16/9 and so its "pull" is over twice as strong.

Since the 9 has been implied in Western music for hundred of years, I can't be
convinced of the above statement

> For 8/7 vs. 9/8, though, the
> ambiguity is significant unless additional notes are there to clear up the
> context (I advocate tetradic harmony as the norm for 22-tET).
>
> >The 5 though
> >is out of tune in the opposite direction which when compared to the 3 is as
> >out of tune as a 5/4 in 12ET.

my mistake. I was referring to the 5 in relationship to the 7 where the
combined out of tuneness 13+ and the 4 cents flat get us up to 17 cents out of
tune. this is how far 11/7 is away from a 12ET major third. I can't take things
like this seriously.

>
> >There is nothing like a good 9/8.
>
> True, but 9/4 is acceptable, especially with other notes clearing up the
> context. 12-tET has nothing like a good 8/7 or a good 10/9, and yet these
> intervals can be clearly implied within certain chords in 12-tET.

They may be implied, but this out of tuneness is exactly what got many of us
here. If you can accept pitches that extend 17 cents in both directions, you
are up to a 34 cent band width. this is so all incluvise as to almost be
meaningless.

>
>
> >If 22ET has merit which I
> >assume it does, it is not because of its relationships to these ratios. As
> >someone who knows many of them I would never mistake one for the other.
>
> Two-member ratios are not the full story. Chords of three or four notes can
> resolve ambiguities inherent in the individual intervals.

That means that this music is only useful when using full block chords?

> Common-practice
> Western music since the abandonment of meantone temperament considers dyads
> to be incomplete harmonies, and the 5-limit triad is the norm.

Western music has not been concerned with triads for at least 100 years.

> My idea of
> music in 22-tET uses the 7-limit tetrad as the norm. Each ratio within the
> 7-limit tetrad is clarified by its position relative to the others.

How can I tell a 1-3-5-7 from a 1-3-5 with a 9/8 below the 1. One is a
consonant the other is not. If the context determines it all, the composer is
constantly hemmed in just making clear what it is he means. Why not use 31 or
41?

> The
> otonal tetrad in 22-tET is much closer to just than in 12-tET, though
> musicians have been trying to use the latter for at least a century. 22-tET
> can distinguish the utonal tetrad from a 5:6:7:9 chord, which 12-tET can't
> do. To my ears, the distinction made in 22-tET is qualitatively the same as
> that made in JI.

12et is no standard to build any new upon. Otherwise we keep delaying the
problem till later.

>
>

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

"Paul H. Erlich" wrote:

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Kraig Grady wrote,
>
> >The 7/6 is good, but then again only because the terms are out of tune in
> >the same direction.
>
> 7/6 is a basic 7-limit consonance and should not be "broken down" into its
> "terms".

Basically I was saying they are both out of tune in the same direction.

>
>
> >The 7/4 is 13+ cents off with the 16/9 15 cents in the
> >other direction. This is about as ambiguous as you can get.
>
> Not at all -- the numbers in 7/4 are more than twice as small as those in
> 16/9 and so its "pull" is over twice as strong.

Since the 9 has been implied in Western music for hundred of years, I
can't be
convinced of the above statement

> For 8/7 vs. 9/8, though, the
> ambiguity is significant unless additional notes are there to clear up the
> context (I advocate tetradic harmony as the norm for 22-tET).
>
> >The 5 though
> >is out of tune in the opposite direction which when compared to the 3 is as
> >out of tune as a 5/4 in 12ET.

my mistake. I was referring to the 5 in relationship to the 7 where the
combined out of tuneness 13+ and the 4 cents flat get us up to 17 cents
out of
tune. this is how far 11/7 is away from a 12ET major third. I can't take things
like this seriously.

>
> >There is nothing like a good 9/8.
>
> True, but 9/4 is acceptable, especially with other notes clearing up the
> context. 12-tET has nothing like a good 8/7 or a good 10/9, and yet these
> intervals can be clearly implied within certain chords in 12-tET.

They may be implied, but this out of tuneness is exactly what got many
of us
here. If you can accept pitches that extend 17 cents in both directions, you
are up to a 34 cent band width. this is so all inclusive as to almost be
meaningless.

>
>
> >If 22ET has merit which I
> >assume it does, it is not because of its relationships to these ratios. As
> >someone who knows many of them I would never mistake one for the other.
>
> Two-member ratios are not the full story. Chords of three or four notes can
> resolve ambiguities inherent in the individual intervals.

That means that this music is only useful when using full block chords?

> Common-practice
> Western music since the abandonment of meantone temperament considers dyads
> to be incomplete harmonies, and the 5-limit triad is the norm.

Western music has not been concerned with triads for at least 100 years.

> My idea of
> music in 22-tET uses the 7-limit tetrad as the norm. Each ratio within the
> 7-limit tetrad is clarified by its position relative to the others.

How can I tell a 1-3-5-7 from a 1-3-5 with a 9/8 below the 1. One is a
consonant the other is not. If the context determines it all, the
composer is
constantly hemmed in just making clear what it is he means. Why not use
31 or
41?

> The
> otonal tetrad in 22-tET is much closer to just than in 12-tET, though
> musicians have been trying to use the latter for at least a century. 22-tET
> can distinguish the utonal tetrad from a 5:6:7:9 chord, which 12-tET can't
> do. To my ears, the distinction made in 22-tET is qualitatively the same as
> that made in JI.

12et is no standard to build any new upon. Otherwise we keep delaying the
problem till later.

>

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

Kraig Grady wrote,

>They may be implied, but this out of tuneness is exactly what got many of
us
>here. If you can accept pitches that extend 17 cents in both directions,
you
>are up to a 34 cent band width. this is so all incluvise as to almost be
>meaningless.

Yet many professional singers in the West easily extend 30 cents in either
direction, giving a 60 cent band width.

>> >If 22ET has merit which I
>> >assume it does, it is not because of its relationships to these ratios.
As
>> >someone who knows many of them I would never mistake one for the other.
>
>> Two-member ratios are not the full story. Chords of three or four notes
can
>> resolve ambiguities inherent in the individual intervals.

>That means that this music is only useful when using full block chords?

No, but it can certainly function more like 7-limit JI that way.

>> Common-practice
>> Western music since the abandonment of meantone temperament considers
dyads
>> to be incomplete harmonies, and the 5-limit triad is the norm.

>Western music has not been concerned with triads for at least 100 years.

That's an odd statement that Ezra Sims also made. I don't know when Western
music reverted to two-part counterpoint. But I do know that Western pop
music in the 20th century is largely triadic, while Schoenberg and his ilk
have had little influence outside academia.

>> My idea of
>> music in 22-tET uses the 7-limit tetrad as the norm. Each ratio within
the
>> 7-limit tetrad is clarified by its position relative to the others.

>How can I tell a 1-3-5-7 from a 1-3-5 with a 9/8 below the 1. One is a
>consonant the other is not. If the context determines it all, the composer
is
>constantly hemmed in just making clear what it is he means.

If the 7 is not in the bass then the chord will certainly be heard as
1-3-5-7.

>Why not use 31 or
>41?

Because my understanding of music is based on the idea that the harmony is
formed from patterns in the underlying melodic scale. The decatonic scale in
22 forms a pseudo-diatonic set from which 7-limit harmonies may be drawn. No
one has yet found such scales in 31 or 41 (though we've certainly been
looking!). The closest things are 19-out-of-31 (Dave Keenan), an MOS
generated by fifths, and 22-out-of-41 (me), an MOS generated by major
thirds. However, these scales are very large and would be difficult to
project compositionally, not to mention some other properties they don't
share with diatonic and decaonic scales.

Once again, I'm more than happy to advocate dynamic retuning so that each
member of the 22-tone tuning may fluctuate slightly to better the harmony at
each moment. However, I suspect many of us will be working with fixed-pitch
instruments for some time to come, so 22-tET has some importance.

>12et is no standard to build any new upon. Otherwise we keep delaying the
>problem till later.

That seems to imply that JI is some kind of ultimate goal in musical
evolution. I strongly disagree.