Bosenquet order and class

I've been looking at Bosanquet's "On the Hindu Division of the Octave"
with an eye to improving the Wikpedia article on 22 equal. Another
thing in it I found interesting is that Bosanquet proposes classifying
equal temperaments according to "order" and "class". The order of an
equal temperament is what the corresponding val maps the Pythagorean
comma to. For example, 12-et is of order 0, 53-et of order 1, and
22-et of order 2. Presumably also 19-et would be of order -1, and so
forth. The class of an equal temperament is where 81/80 gets mapped
to. Hence giving (order, class) goes a long way towards characterizing
5-limit equal temperaments completely; choosing one more suitable
5-limit comma (say, 3125/3072, or 15625/15552) would do so. Anyway,
Bosanquet's order and class may be worth a Tonalsoft entry.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> I've been looking at Bosanquet's "On the Hindu Division of the
Octave"
> with an eye to improving the Wikpedia article on 22 equal. Another
> thing in it I found interesting is that Bosanquet proposes
classifying
> equal temperaments according to "order" and "class". The order of an
> equal temperament is what the corresponding val maps the Pythagorean
> comma to. For example, 12-et is of order 0, 53-et of order 1, and
> 22-et of order 2. Presumably also 19-et would be of order -1, and so
> forth.

From "On the Theory of the Division of the Octave":
Order 1 positive: 17, 29, 41, 53, 65
Order 2 positive: 118
Order 1 negative: 19, 31
Order 2 negative: 50

>The class of an equal temperament is where 81/80 gets mapped
> to. Hence giving (order, class) goes a long way towards
characterizing
> 5-limit equal temperaments completely; choosing one more suitable
> 5-limit comma (say, 3125/3072, or 15625/15552) would do so. Anyway,
> Bosanquet's order and class may be worth a Tonalsoft entry.
>

--- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@N...> wrote:

> From "On the Theory of the Division of the Octave":
> Order 1 positive: 17, 29, 41, 53, 65
> Order 2 positive: 118
> Order 1 negative: 19, 31
> Order 2 negative: 50

He has a similar chart in this article, but it doesn't go negative:

Order 1: 17 29 41 53
Order 2: 22 34 .. 118
Order 3: 15 27 39 ..

He also gives a few examples of order-class pairings: 34 is (2,1), and
87 is (3, 2). If you extend this pairing to a triple showing how
the pythagorean comma, 81/80, and 3125/3072 get mapped, you get the
following:

12: <0 0 1|
15: <3 1 1|
19: <-1 0 0|
22: <2 1 0|
31: <-1 0 1|
34: <2 1 1|
41: <1 1 0|
46: <2 1 2|
53: <1 1 1|
65: <1 1 2|
72: <0 1 1|
84: <0 1 2|
87: <3 2 2|
99: <3 2 3|
118: <2 2 3|

I'm writing it like this since it represents a coordinate change on
the prime mapping.

Gene Ward Smith wrote:
> He has a similar chart in this article, but it doesn't go negative:
> > Order 1: 17 29 41 53
> Order 2: 22 34 .. 118
> Order 3: 15 27 39 ..
> > He also gives a few examples of order-class pairings: 34 is (2,1), and
> 87 is (3, 2). If you extend this pairing to a triple showing how
> the pythagorean comma, 81/80, and 3125/3072 get mapped, you get the
> following:
> > 12: <0 0 1|
> 15: <3 1 1|
> 19: <-1 0 0|
> 22: <2 1 0|
> 31: <-1 0 1|
> 34: <2 1 1|
> 41: <1 1 0|
> 46: <2 1 2|
> 53: <1 1 1|
> 65: <1 1 2|
> 72: <0 1 1|
> 84: <0 1 2|
> 87: <3 2 2|
> 99: <3 2 3|
> 118: <2 2 3|
> > I'm writing it like this since it represents a coordinate change on
> the prime mapping.

I can see where the 81/80 mapping would be of interest. Where it's zero, you get the typical meantone/diatonic tunings and multiples of those (5, 7, 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, 55, 57, 62, 67, 69, 74, 76, 81, 86, 88, 93, 98, ...). Where it's 1, a step of the ET functions as a comma (3, 8, 10, 15, 17, 22, 27, 29, 34, 39, 41, 46, 48, 53, 58, 60, 65, 70, 72, 77, 79, 84, 89, 91, 96, ...) Not that 3-ET is likely to attract much interest, but technically there it is. Other ET's map the 81/80 to a larger number of steps.

But there's a small class of temperaments that map 81/80 to -1 steps, which has an interesting consequence: the "wolf" fifth ends up larger than the ordinary fifth, instead of smaller. This class includes 2, 4, 9, 14, 16, 21, 23, 28, 33, 35, 40, 47, 52, and 64-ET, which all have fifths flatter than just, so the "wolf" fifth is off by less than one step. (A similar situation arises in ET's with sharp fifths, like 39-ET and 58-ET, but the effect is less dramatic.) Starting with 35-ET, the "wolf" fifth (the 5-ET fifth) is marginally acceptable; in 47-ET, the "wolf" fifth is almost as good as the regular fifth. So you can use these ET's to play meantone-based music with interesting color variations in the fifths.

I can imagine that a similar sort of feature would be useful for other commas.

> Another thing in it I found interesting is that Bosanquet proposes
> classifying equal temperaments according to "order" and "class".

You're just learning this??

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > Another thing in it I found interesting is that Bosanquet proposes
> > classifying equal temperaments according to "order" and "class".
>
> You're just learning this??

I knew he had positive and negative systems. Is it well-known he
classified ets according to where they mapped both commas? I can't
recall hearing it here.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> I've been looking at Bosanquet's "On the Hindu Division
> of the Octave" with an eye to improving the Wikpedia
> article on 22 equal. Another thing in it I found interesting
> is that Bosanquet proposes classifying equal temperaments
> according to "order" and "class". The order of an equal
> temperament is what the corresponding val maps the
> Pythagorean comma to. For example, 12-et is of order 0,
> 53-et of order 1, and 22-et of order 2. Presumably also
> 19-et would be of order -1, and so forth. The class of
> an equal temperament is where 81/80 gets mapped
> to. Hence giving (order, class) goes a long way towards
> characterizing 5-limit equal temperaments completely;
> choosing one more suitable 5-limit comma (say, 3125/3072,
> or 15625/15552) would do so. Anyway, Bosanquet's order
> and class may be worth a Tonalsoft entry.

Thanks, Gene. Actually, i remember putting something
about order in the Encyclopedia somewhere a long time ago,
back when it was the Dictionary. Can't remember now, but
i do recall John Chalmers explaining it to me.

I'd be most appreciative if you would just write up
a pair of Encyclopedia entries for these terms, and just
post them here. I can simply copy what you write into
the text of the webpages. Care to invent a term for
that third comma?

BTW, my old webpage on Indian tuning:

http://tonalsoft.com/monzo/indian/indian.htm

got its start from data in David Lentz's book, wherein
he uses 5-limit ratios to describe the basis of the
22 srutis, but before that he explains its origin as
a pythagorean chain of 4ths and 5ths. Does Bosanquet
say anything about that?

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

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

> BTW, my old webpage on Indian tuning:
>
> http://tonalsoft.com/monzo/indian/indian.htm
>
> got its start from data in David Lentz's book, wherein
> he uses 5-limit ratios to describe the basis of the
> 22 srutis, but before that he explains its origin as
> a pythagorean chain of 4ths and 5ths. Does Bosanquet
> say anything about that?

It's interesting that this thread is headed in the
same direction as what i just posted in response to
Neil regarding pythagorean schismic near-equivalents
of 5-limit JI ratios. Lentz ignores the skhisma
difference is changing his discussion of srutis away
from really large pythagorean ratios to their nearly
equivalent but much simpler 5-limit cousins.

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

> > > Another thing in it I found interesting is that Bosanquet
> > > proposes classifying equal temperaments according to "order"
> > > and "class".
> >
> > You're just learning this??
>
> I knew he had positive and negative systems. Is it well-known he
> classified ets according to where they mapped both commas? I can't
> recall hearing it here.

It's been discussed on this list numerous times, though perhaps
before you joined. But I thought I quoted it for you once.

-Carl

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

> I'd be most appreciative if you would just write up
> a pair of Encyclopedia entries for these terms, and just
> post them here. I can simply copy what you write into
> the text of the webpages. Care to invent a term for
> that third comma?

How about this:

Bosanquet order and class

"Order" and "class" are terms invented by R. H. M. Bosanquet, and used
by him to classify equal temperaments. The order of an equal
temperament is the number of scale steps it assigns to the Pythagoream
comma 531441/524288, or |-19 12>. The class is the number of scale
steps assigned to the syntonic comma 81/80, or |-4 4 -1>. Hence 12-et
has order 0 and class 0, whereas 53-et has order 1 and class 1.

> BTW, my old webpage on Indian tuning:
>
> http://tonalsoft.com/monzo/indian/indian.htm
>
> got its start from data in David Lentz's book, wherein
> he uses 5-limit ratios to describe the basis of the
> 22 srutis, but before that he explains its origin as
> a pythagorean chain of 4ths and 5ths. Does Bosanquet
> say anything about that?

Bosanquet simply guesses that the srutis were probably "intended to be
equal in a general sort of way, probably without any very great
precision". In spite of its title, Nosanquet's article is really more
an article about 22-et, and classifying equal temperaments in general.

The comma you want me to name is the magic comma, and already on the
site. It's not the only choice possible; I picked it because it's
about the same size as a comma, and because 12-et maps it to 1 step.
If you toss in the septimal comma, 64/63, you can extend it further,
but of course that's not a unique choice either.

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

> It's interesting that this thread is headed in the
> same direction as what i just posted in response to
> Neil regarding pythagorean schismic near-equivalents
> of 5-limit JI ratios. Lentz ignores the skhisma
> difference is changing his discussion of srutis away
> from really large pythagorean ratios to their nearly
> equivalent but much simpler 5-limit cousins.

The trouble is, Pythagorean or schismatic tuning is *not* always a
feature of the sruti scales that people put forth. I think the
presence of the Sa-grama scale, and having 4 srutis for 9/8, 3 for
10/9, and 2 for 16/15, is a much better way to get a handle on what is
going on. On your page, Mahadeven Ramesh gives the indian3 scale as
his version of srutis, and this seems to be pretty typical. It isn't
Pythagorean, but does satisfy the above conditions. Consequently, I
think Magic[22] or Shrutar[22] should probably be regarded as more
authentic possibilities for a rationalized sruti system than
Pythagorean or schismic approaches.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > It's interesting that this thread is headed in the
> > same direction as what i just posted in response to
> > Neil regarding pythagorean schismic near-equivalents
> > of 5-limit JI ratios. Lentz ignores the skhisma
> > difference is changing his discussion of srutis away
> > from really large pythagorean ratios to their nearly
> > equivalent but much simpler 5-limit cousins.
>
> The trouble is, Pythagorean or schismatic tuning is *not*
> always a feature of the sruti scales that people put forth.
> I think the presence of the Sa-grama scale, and having
> 4 srutis for 9/8, 3 for 10/9, and 2 for 16/15, is a much
> better way to get a handle on what is going on.

I agree with that. However ...

> On your page, Mahadeven Ramesh gives the indian3 scale as
> his version of srutis, and this seems to be pretty typical.
> It isn't Pythagorean, but does satisfy the above conditions.
> Consequently, I think Magic[22] or Shrutar[22] should
> probably be regarded as more authentic possibilities for
> a rationalized sruti system than Pythagorean or schismic
> approaches.

That may be true according to various theoretical criteria,
but i'd suspect that the pythagorean chain of 11 5ths and
10 4ths is either the origin of the 22-sruti system, or
a very early systematization of a haphazardly-organized
original tuning system.

The fact that 5-limit JI has chains of equivalents only
a skhisma away means that the sruti system can be modeled
in a compact form on a 5-limit lattice, but based on what
i know of tuning history, it makes a lot of sense to me
to think that the pythagorean tuning is the older one.

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

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

> The fact that 5-limit JI has chains of equivalents only
> a skhisma away means that the sruti system can be modeled
> in a compact form on a 5-limit lattice, but based on what
> i know of tuning history, it makes a lot of sense to me
> to think that the pythagorean tuning is the older one.

Possibly. By adjusting by a schisma, we can get the Pythagorean
version of Sa-grama: 9/8-8152/6561-4/3-3/2-27/16-4096/2087-2. If we
locate that in your Pythagorean scale, we find 4 srutis from 1 to 9/8,
3 stutis from 9/8 to 8152/6561, 2 srutis from 8152/6561 to 4/3, 4
srutis from 4/3 to 3/2, 4 strutis from 3/2 to 27/16, 3 srutis from
27/16 to 4096/2087, and 2 srutis from 4096/2087 to 2. This gives the
correct 4324432 pattern of srutis for Sa-grama. In 53-et, we get the
pattern 4131-413-14-1314-4131-413-14 for the subdivision of these
steps, which adds up to 9-8-5-9-9-8-5 for Sa-grama, which has now been
identified with the 5-limit version. Hence Schismatic[22] will work as
a sruti system.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

Hence Schismatic[22] will work as
> a sruti system.

We might use the ratio of what 256/243 maps to with what the
Pythgorean comma maps to as a measure of the irregularity of this;
large sruti pver small sruti. For pure Pythagorean, it is 3.846. For
flat fifths, it becomes more irregular: 4 for 53-et, 4.5 for 118-et.
For sharp fifths it becomes less irregular: 3.5 for 94-et, 3 for 41-et.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> I've been looking at Bosanquet's "On the Hindu Division of the Octave"
> with an eye to improving the Wikpedia article on 22 equal. Another
> thing in it I found interesting is that Bosanquet proposes classifying
> equal temperaments according to "order" and "class". The order of an
> equal temperament is what the corresponding val maps the Pythagorean
> comma to. For example, 12-et is of order 0, 53-et of order 1, and
> 22-et of order 2. Presumably also 19-et would be of order -1, and so
> forth. The class of an equal temperament is where 81/80 gets mapped
> to. Hence giving (order, class) goes a long way towards characterizing
> 5-limit equal temperaments completely; choosing one more suitable
> 5-limit comma (say, 3125/3072, or 15625/15552) would do so. Anyway,
> Bosanquet's order and class may be worth a Tonalsoft entry.

They should be linked to, and link from, the terms "positive"
and "negative", which were Bosanquet terms.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > It's interesting that this thread is headed in the
> > same direction as what i just posted in response to
> > Neil regarding pythagorean schismic near-equivalents
> > of 5-limit JI ratios. Lentz ignores the skhisma
> > difference is changing his discussion of srutis away
> > from really large pythagorean ratios to their nearly
> > equivalent but much simpler 5-limit cousins.
>
> The trouble is, Pythagorean or schismatic tuning is *not* always a
> feature of the sruti scales that people put forth. I think the
> presence of the Sa-grama scale, and having 4 srutis for 9/8, 3 for
> 10/9, and 2 for 16/15, is a much better way to get a handle on what
is
> going on. On your page, Mahadeven Ramesh gives the indian3 scale as
> his version of srutis, and this seems to be pretty typical. It isn't
> Pythagorean, but does satisfy the above conditions. Consequently, I
> think Magic[22] or Shrutar[22] should probably be regarded as more
> authentic possibilities for a rationalized sruti system than
> Pythagorean or schismic approaches.

I don't think Magic[22] has any chance; does it even support the
Modern Indian Gamut?