The world according to Rothenberg (re Kraig Grady)

>I appreciate you going through all this as I have never seen Rothenberg
>work. I seem to be missing somerthing as I don't see where any thing like
>MOS is spelt out. It seems to be barely implied.

Rothenberg's theory does not address MOS as such. However, I do believe
that MOS can be explained with only Rothenberg's model plus extended
reference. That is, if we take Rothenberg and extended reference to be our
requirements for human melodic enjoyment, then MOS shouldn't miss many good
scales. I doubt tetrachordality produces significantly different results
here, provided we don't take the "tetra" literally, and that we don't
insist on 3/2's or fifths (that is, we consider tetrachordality of the
generator).

>What are musical application of anything having to do with equivalence
>classes

This is arguably the most useful part of the model. It predicts that
different scales can sometimes sound like re-tunings of eachother.
Actually, it predicts when this will be the case. How well does it work?
This is one of the most important parts of the model to test, and
Rothenberg has designed ingenious experiments to test it. In the 60's he
commissioned Moog to built him a microtonal synth for the experiments, but
Moog's design was a dud, the air-force decided to pull funding, and the
experiments are unperformed to this day.

However, Rothenberg claims to have achieved good results with
ethno-musicology. Apparently, many of the instruments in the Indonesian
gamelan change their timbre as they age, and the gamelan is periodically
re-tuned to accommodate this. Eventually there is a point at which,
suddenly, the basic scale seems to have changed. Rothenberg claims this is
the point where the equivalence class has changed.

>or what is gain by the labeling scales proper and improper.

The difference is in how they tend to be used. Improper scales are usually
sectioned into principle and ornamental tones, the tonic is often kept by
some device like a drone, or is static, and composition is much less likely
to use modal transposition of themes. Instead, rhythms trace out the
boundaries of the principle tones, and the principle/ornamental sectioning
can be changed on the fly.

If a scale is proper, a melody can be played in any mode of the scale and
it will still be recognized as the same melody. Scale degree fun may or
may not be used, depending on the efficiency (see below).

>I really don't know what to do with
>
>>n!
>>Sigma(S_i) / n!
>>i=1
>>
>>F(P) may be interpreted as the average number of elements in a non-
>>repeating sequence of n elements of P(x) required to determine the
>>key, x. Efficiency, E, is defined as F(P)/n and Redundancy, R, as
>>1 - F(P)/n. Both numbers lie between 0 and 1.

If my language here is accurate, Sigma represents a function which sums the
values of one variable as another is varied over some range, which is given
above and below the Sigma sign (which I couldn't figure out how to do in
ASCI).

In this case, we are considering all non-repeating strings of tones
possible (what is called a peal in change ringing, I believe) in a scale
with n tones per octave (there are n! of them). Imagine I play one for
you, starting on some random frequency. Your job is to shout "stop!" as
soon as you can name the tonic frequency. When you yell stop, I stop, and
the number of tones you heard that far is called S_i (the "i" subscript
tells which string the number S_i came from -- since there are n! possible
strings, we vary the "i" from 1 to n!). Now suppose you want to find the
average number of tones you had to hear for all strings. You'd sum them
all up (Sigma) and divide by the total number of strings (n!).

That's the equation. Efficiency is actually the answer of that equation
divided by n. So if you have a scale with 22 tones per octave, and the
average number of tones you need to hear to identify the key is 3, you've
got 19 tones that aren't doing anything to help you identify the key, which
means low efficiency.

The important thing about efficiency is that it measures the difficulty of
finding the tonic in a segment of music. For improper scales, where scale
degrees (and therefore the whereabouts of the tonic) is all-important,
efficiency must be low. For proper scales, efficiency can be high or low,
but it may be better high (Paul, do you think your preference for the
pentachordal decatonics can be explained by the lower efficiency of the
symmetric ones?).

Does that help? Comments, corrections (as always, it should go without
saying) wanted.

-C.

Carl!
Thanks for doing this. comments below if I am correct!

Carl Lumma wrote:

> From: Carl Lumma <clumma@nni.com>
>

So equivalence could be found on the scale tree as the range between
different factors on the same level .
http://www.anaphoria.com/scaletree.html example-so a scale that could be
defined as a 3/7 scale would be equivalent. with the limits being those
defined as 3/8 and 4/7.

>
>
> >What are musical application of anything having to do with equivalence
> >classes
>
> This is arguably the most useful part of the model. It predicts that
> different scales can sometimes sound like re-tunings of eachother.
> Actually, it predicts when this will be the case. How well does it work?
> This is one of the most important parts of the model to test, and
> Rothenberg has designed ingenious experiments to test it. In the 60's he
> commissioned Moog to built him a microtonal synth for the experiments, but
> Moog's design was a dud, the air-force decided to pull funding, and the
> experiments are unperformed to this day.
>
> However, Rothenberg claims to have achieved good results with
> ethno-musicology. Apparently, many of the instruments in the Indonesian
> gamelan change their timbre as they age, and the gamelan is periodically
> re-tuned to accommodate this. Eventually there is a point at which,
> suddenly, the basic scale seems to have changed. Rothenberg claims this is
> the point where the equivalence class has changed.
>
> >or what is gain by the labeling scales proper and improper.

Are not the improper scales 2nd level MOS and proper 1st level?

>
>
> The difference is in how they tend to be used. Improper scales are usually
> sectioned into principle and ornamental tones, the tonic is often kept by
> some device like a drone, or is static, and composition is much less likely
> to use modal transposition of themes. Instead, rhythms trace out the
> boundaries of the principle tones, and the principle/ornamental sectioning
> can be changed on the fly.
>
> If a scale is proper, a melody can be played in any mode of the scale and
> it will still be recognized as the same melody. Scale degree fun may or
> may not be used, depending on the efficiency (see below).

>
>
> >I really don't know what to do with
> >
> >>n!
> >>Sigma(S_i) / n!
> >>i=1
> >>
> >>F(P) may be interpreted as the average number of elements in a non-
> >>repeating sequence of n elements of P(x) required to determine the
> >>key, x. Efficiency, E, is defined as F(P)/n and Redundancy, R, as
> >>1 - F(P)/n. Both numbers lie between 0 and 1.
>
> If my language here is accurate, Sigma represents a function which sums the
> values of one variable as another is varied over some range, which is given
> above and below the Sigma sign (which I couldn't figure out how to do in
> ASCI).

I am not sure that the nature of tonic can be taken as being a
universal. so
should efficiency involve intervals instead or with. Example C D F Gb an
all-interval set of Walter O'Connell. all 12 tone intervals can be found in
this set.

>
>
> In this case, we are considering all non-repeating strings of tones
> possible (what is called a peal in change ringing, I believe) in a scale
> with n tones per octave (there are n! of them). Imagine I play one for
> you, starting on some random frequency. Your job is to shout "stop!" as
> soon as you can name the tonic frequency. When you yell stop, I stop, and
> the number of tones you heard that far is called S_i (the "i" subscript
> tells which string the number S_i came from -- since there are n! possible
> strings, we vary the "i" from 1 to n!). Now suppose you want to find the
> average number of tones you had to hear for all strings. You'd sum them
> all up (Sigma) and divide by the total number of strings (n!).
>
> That's the equation. Efficiency is actually the answer of that equation
> divided by n. So if you have a scale with 22 tones per octave, and the
> average number of tones you need to hear to identify the key is 3, you've
> got 19 tones that aren't doing anything to help you identify the key, which
> means low efficiency.
>
> The important thing about efficiency is that it measures the difficulty of
> finding the tonic in a segment of music. For improper scales, where scale
> degrees (and therefore the whereabouts of the tonic) is all-important,
> efficiency must be low. For proper scales, efficiency can be high or low,
> but it may be better high (Paul, do you think your preference for the
> pentachordal decatonics can be explained by the lower efficiency of the
> symmetric ones?).
>
> Does that help? Comments, corrections (as always, it should go without
> saying) wanted.
>
> -C.
>
> --------------------------- ONElist Sponsor ----------------------------
>
> ONElist: your connection to people who share your interests.
>
> ------------------------------------------------------------------------
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@onelist.com - subscribe to the tuning list.
> tuning-unsubscribe@onelist.com - unsubscribe from the tuning list.
> tuning-digest@onelist.com - switch your subscription to digest mode.
> tuning-normal@onelist.com - switch your subscription to normal mode.

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

>So equivalence could be found on the scale tree as the range between
>different factors on the same level.
>
>http://www.anaphoria.com/scaletree.html
>
>so a scale that could be defined as a 3/7 scale would be equivalent with the
>limits being those defined as 3/8 and 4/7.

Now the position is reversed- could you explain this to me? As I
understand it, these fractions refer to generator size when applied to
log-frequency (ie 3/7 of an octave, etc). But what do you mean by limits
here? Examples please.

>Are not the improper scales 2nd level MOS and proper 1st level?

Proper scales are not necessarily MOS, and MOS scales are not necessarily
proper (take the recent 11-tone chain-of-minor-thirds scale, for example).

I haven't looked much at 2nd-level MOS yet; my initial reaction to these
was that they seemed like a 'bit of a stretch'.

>I am not sure that the nature of tonic can be taken as being a universal.

Some confusion may result from multiple meanings of "tonic". There is the
sense of the virtual fundamental in harmony, which is, owing to the design
of the human nervous system, a factor in all harmonic music (and probably
in much solo melodic music as well). This is what is commonly meant by
'tonic of the chord', and it is not the meaning of tonic I was using in
this thread.

The other (main) meaning for tonic is: the note that is the 1st degree of a
scale. This is what is commonly meant by 'tonic of the song' or phrase,
and it was the meaning I was using in my last message. It is only
important in music that uses the tracking of scale degrees to convey
information, but it is universal in the sense that music which does not
expect the listener to track scale degrees will lack certain properties.

If, in listening to a melody, you can name with _certainty_ the scale
degree represented by any single tone, or the tone represented by any scale
degree, then you can do it for all tones and all scale degrees. In the
change-ringing example I asked for the tonic, but I could have asked for
the 2nd or the 3rd (keep in mind these are scale degrees here, not intervals).

So what Rothenberg is really trying to measure with Efficiency is: how soon
can one begin to associate tones with scale degree numbers when listening?
As I said, this is not important in all music. But Rothenberg wants to
know if it isn't. Kapeesh?

>so should efficiency involve intervals instead or with. Example C D F Gb an
>all-interval set of Walter O'Connell. all 12 tone intervals can be found in
>this set.

Hmmm. I'm not sure if this is related to Efficiency, but it sounds
interesting. Could you elaborate on where you were going here?

-C.

P.S. This nitty-gritty stuff takes a lot of time, and you've got it ---
I'm leaving in a few hours for New England, and I will probably not check
my mail until Monday or Tuesday.

P.P.S. This is not an 'inside' thread just because it addresses someone in
particular!

Carl Lumma wrote:

> From: Carl Lumma <clumma@nni.com>
>
> >So equivalence could be found on the scale tree as the range between
> >different factors on the same level.
> >
> >http://www.anaphoria.com/scaletree.html
> >
> >so a scale that could be defined as a 3/7 scale would be equivalent with the
> >limits being those defined as 3/8 and 4/7.
>
> Now the position is reversed- could you explain this to me? As I
> understand it, these fractions refer to generator size when applied to
> log-frequency (ie 3/7 of an octave, etc). But what do you mean by limits
> here? Examples please.

I wasn't completely clear here, sorry. First of all the scale tree is not limited
to log. It can also be used acoustically. But lets look at
http://www.anaphoria.com/key.html . the keyboard designs. Those scales that would
fit on the 3/7 keyboard would be equivalent if carried out to the same number of
places. This would include Contant Structures. A simpler example would be
Centaur, which is a 12-tone tuning which fits over a 12 tone keyboard. This
scale, I am asuming would be eqiuvalent to 12et in that one could sound as a
retuning of the other. Anyway those scales that would fall on a 3/8 keyboard
would not be equivalent because of the underlying structure. By limits, I mean
the scales that are contained beneath and included below lets say 3/7 as opposed
to 3/8. Going back to the 5/12 scales (12ET or centuar) and look at the scale
tree section that contains them http://www.anaphoria.com/ST07.html . The scales
below will all contain 12 substets that I believe would fall under Rothenberg
equivalence? Maybe Chamers can help us here

>
>
> >Are not the improper scales 2nd level MOS and proper 1st level?
>
> Proper scales are not necessarily MOS, and MOS scales are not necessarily
> proper (take the recent 11-tone chain-of-minor-thirds scale, for example).

I believe it is an MOS! I realize that Erv fomula for finding the MOS of a given
Generator is not up so I am going to put it up in a seperate post so it does'nt
get lost in the middle of this!

>
>
> I haven't looked much at 2nd-level MOS yet; my initial reaction to these
> was that they seemed like a 'bit of a stretch'.

They have historical examples. Take the examples e-f-a-b-c-e or g-b-c-d-f. They
are bedrock scales.

>
>
> >I am not sure that the nature of tonic can be taken as being a universal.
>
> Some confusion may result from multiple meanings of "tonic". There is the
> sense of the virtual fundamental in harmony, which is, owing to the design
> of the human nervous system, a factor in all harmonic music (and probably
> in much solo melodic music as well). This is what is commonly meant by
> 'tonic of the chord', and it is not the meaning of tonic I was using in
> this thread.
>
> The other (main) meaning for tonic is: the note that is the 1st degree of a
> scale. This is what is commonly meant by 'tonic of the song' or phrase,
> and it was the meaning I was using in my last message. It is only
> important in music that uses the tracking of scale degrees to convey
> information, but it is universal in the sense that music which does not
> expect the listener to track scale degrees will lack certain properties.
>
> If, in listening to a melody, you can name with _certainty_ the scale
> degree represented by any single tone, or the tone represented by any scale
> degree, then you can do it for all tones and all scale degrees. In the
> change-ringing example I asked for the tonic, but I could have asked for
> the 2nd or the 3rd (keep in mind these are scale degrees here, not intervals).
>
> So what Rothenberg is really trying to measure with Efficiency is: how soon
> can one begin to associate tones with scale degree numbers when listening?
> As I said, this is not important in all music. But Rothenberg wants to
> know if it isn't. Kapeesh?

ok

>
>
> >so should efficiency involve intervals instead or with. Example C D F Gb an
> >all-interval set of Walter O'Connell. all 12 tone intervals can be found in
> >this set.
>
> Hmmm. I'm not sure if this is related to Efficiency, but it sounds
> interesting. Could you elaborate on where you were going here?

No its not really related. O'Connell found the 4 4note chord that contain all the
intervals (by inversion also) These are c d f f#, inv.c c# e f#, c e f#g, inv. c
c# d# g. Alot of my pre intonational music was based on these sets!

>
>
> -C.
>
>
> P.P.S. This is not an 'inside' thread just because it addresses someone in
> particular!

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

Carl Lumma wrote,

>> Proper scales are not necessarily MOS, and MOS scales are not necessarily
>> proper (take the recent 11-tone chain-of-minor-thirds scale, for
example).

Kraig Grady wrote

>I believe it is an MOS!

Yes it is, but it's not proper.

"Paul H. Erlich" wrote:

> Carl Lumma wrote,
>
> >> Proper scales are not necessarily MOS, and MOS scales are not necessarily
> >> proper (take the recent 11-tone chain-of-minor-thirds scale, for
> example).
>
> Kraig Grady wrote
>
> >I believe it is an MOS!
>
> Yes it is, but it's not proper.

then I am confused about what is Proper still. Why isn't it proper.

>
>

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