new lattice diagram on my Robert Johnson page

Because numbers are infinite, there are infinitely many
arbitrary ways to define the tuning of any given musical
performance. This even results in the paradox that
a tuning can be described in numbers more precisely than
what is actually audible to a real ear.

The lattice diagram I've just added my analysis of the rational
pitches which I found to be closest to the vocal sung by
Robert Johnson on the first verse of _Drunken Hearted Man_:
http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
can be interpreted as the periodicity-block _gestalt_
of rational categorical pitches/intervals analyzed in
the tuning of that performance.

This was just a thought that I had, while looking at
that lattice and pondering the concept of finity.
Brain McLaren sent me some very perceptive arguments
against the use of ratio-space in musical tunings,
but I continue to believe that there's important
information being displayed on lattice diagrams.
And Paul Erlich's been sending interesting stuff in
to the List, on periodicity blocks.

I've added to this lattice a number of vectors that
go to missing pitches, so that the planes representing
the various primes can be sketched in greater detail.
I thinks this helps the eye to grasp the placement
of all the elements within the complete set (i.e.,
all of the individual notes in the context of the
entire verse).

I invite alternate interpretations of these pitches,
either new ratios for my measured cents-values, or
altogether new measurements of the pitches off the CD.
Also, I'd be interested in any alternate methods of
measuring this tuning, instead of rational; i.e., equal,
meantone, etc. I want to get as accurate a representation
as possible of the tuning of Johnson's actual vocal
(and guitar playing).

In particular, I'm not sure about that 23:16. My rationale
here was to find the smallest-integer ratio that fit
within a more-or-less 'imperceptible' interval of the
pitch-bend that sounded in tune to me. The 23:16 (~628 cents)
is pretty close to 10:7 (~617 cents), but I hear it as
distinct from the 10:7 that Johnson uses. Any other ideas?

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------
___________________________________________________________________
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Try Juno Web: http://dl.www.juno.com/dynoget/tagj.

>The lattice diagram I've just added my analysis of the rational
>pitches which I found to be closest to the vocal sung by
>Robert Johnson on the first verse of _Drunken Hearted Man_:
>http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
>can be interpreted as the periodicity-block _gestalt_
>of rational categorical pitches/intervals analyzed in
>the tuning of that performance.

No offense, Joe, but this practically sounds like you're pulling terms out of
your dictionary at random and stringing them together. Of course I'm
exaggerating, but are you really prepared to back up this statement?

>Also, I'd be interested in any alternate methods of
>measuring this tuning, instead of rational; i.e., equal,
>meantone, etc. I want to get as accurate a representation
>as possible of the tuning of Johnson's actual vocal

I should really get the recording, but it seems to me that it's wholly
inappropriate to analyze the vocal in terms of a JI lattice constructed on one
and only one pitch on Johnson's guitar. Is he holding that pitch throughout the
tune? What about the other 11 pitches on his guitar?

>(and guitar playing).

Is there a lot of bending and/or use of a slide on this recording?

>In particular, I'm not sure about that 23:16. My rationale
>here was to find the smallest-integer ratio that fit
>within a more-or-less 'imperceptible' interval of the
>pitch-bend that sounded in tune to me. The 23:16 (~628 cents)
>is pretty close to 10:7 (~617 cents), but I hear it as
>distinct from the 10:7 that Johnson uses. Any other ideas?

Perhaps it's closer to a simple JI interval constructed on one of the other ET
pitches on Johnson's guitar. For example, if Johnson was playing the 900 cent
major sixth on his guitar at this moment, I might consider the vocal pitch to
be a 7:6 below that. Just an example.

But the important point is not to
confuse pitches with intervals. You should really be saying 23/16 and 10/7
rather than 23:16 and 10:7 since you are referring to pitches and not
intervals. However, it's the _intervals_ in the music, not the pitches, which
should be analyzed in terms of small-integer ratios. This is a mistake that I
see made all the time, both by supporters and detractors of the idea that simple
ratios have an important role in music. For example, detractors are fond of
pointing out that very few of the chords or scales in use resemble the harmonic
series, while they miss the fact that most chords and scales are constructed of
the _intervals_ found in the very beginning of the harmonic series.

Ironically, lattice diagrams, particularly the triangular ones, are well suited
to bringing out the _intervallic_ construction of pitch sets. No other
representation allows one to see all the important intervals at a glance. If
one is solely interested in the simplicity of the ratios representing the
_pitches_, one need go no further than to simple write them down (and possibly
perform some operation such as removing all powers of 2).

Response to Paul Erlich:

> [Paul Erlich, TD 332.6]
>
>> [me, monz]
>> The lattice diagram I've just added to my analysis of the
>> rational pitches which I found to be closest to the vocal sung
>> by Robert Johnson on the first verse of _Drunken Hearted Man_:
>> http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
>> can be interpreted as the periodicity-block _gestalt_
>> of rational categorical pitches/intervals analyzed in
>> the tuning of that performance.
>
> No offense, Joe, but this practically sounds like you're
> pulling terms out of your dictionary at random and stringing
> them together. Of course I'm exaggerating, but are you really
> prepared to back up this statement?

Well, I did say they *can* be interpreted as... etc. etc., not
that they *are*... etc. etc.

And I further mitigated the strength of this statement
a few paragraphs down, by my call for alternate
interpretations.

I was merely trying to say here that if one analyzes the
pitches sung by Johnson as ratios, this is the simplest
rational interpretation possible in the 13...23-limit (that
is, mainly 13-limit but including the one ratio of 23 that I
included) which comes within a few cents of the actual
pitches sung, assuming my measurements of the pitches are
correct (if they're not, they are *very* close).

So there's the answer to your question: not only am I prepared
to back up what I said; that *is* the argument supporting my
statements, right there.

>> Also, I'd be interested in any alternate methods of
>> measuring this tuning, instead of rational; i.e., equal,
>> meantone, etc. I want to get as accurate a representation
>> as possible of the tuning of Johnson's actual vocal
>
> I should really get the recording,

Of course you should, if only to have in your possession
one of the great documents of American musical history.
(Especially since you are a fan of the blues: it doesn't
get any better than this.)

However, while I would never say that my MIDI sequence on my
webpage can substitute for the actual recording of Johnson's
performance, as far as tempo and intonation, it's damn
close to what he sang and played.

Rather than my arguing that point, just listen to the CD
and tell me how far off you think my MIDI sequence is,
because I'm *quite* confident of the job I did on that.

If it's possible to do a spectrogram analysis of Johnson's
vocal on this tune, I'd love to see it, and correct my
webpage with that info. (...Dante?)

> but it seems to me that
> it's wholly inappropriate to analyze the vocal in terms of
> a JI lattice constructed on one and only one pitch on Johnson's
> guitar. Is he holding that pitch throughout the tune? What
> about the other 11 pitches on his guitar?

Well, I'd say that perhaps it's *somewhat* inappropriate to base
a JI analysis of this vocal on one pitch, but I definitely
would not say 'wholly inappropriate'.

I can certainly grant the possibility that an analysis should
consider the relationships of the pitches in the vocal to the
pitches being played on the guitar *at that moment*, especially
at the points where the guitar notes imply a different harmonic
series (i.e., the IV and V chords, G7 and A7) than that over 1/1
(in this case, 1/1 is 'D').

But to my ears, the vocal in this tune seems firmly 'rooted',
melodically, to that 1/1 'D' key. An argument in favor of
this is the fact that the repeated lines where he sings 'my
life seems so misery' both occur over G7 chords but clearly
outline a resolution onto the tonic D7, which is exactly what
he plays on the guitar as soon as he's finished singing those
lines. So, to my way of thinking, the vocal sort of anticipates
the chord changes.

My webpage gives the notation for the vocal, with the pitches
as analyzed by me. To see the guitar part, it's a simple
matter of opening the MIDI file with any sequencer or notation
program and looking at the guitar track.

I'd be glad to add the notation of the guitar part to the
webpage if that helps anyone understand or dispute what
I've done here. I'm particularly interested in any alternate
rational interpretation you have of this tune, Paul.
(or anyone else, but particularly Paul)

>> (and guitar playing).
>
> Is there a lot of bending and/or use of a slide on this
> recording?

As I made clear in the text of my webpage, on this song, no.

Johnson was a master of many different styles of blues guitar
playing, and in this particular tune (and his others which
resemble it), the guitar part sounds to me like an imitation
of banjo playing. (That is, the 'old' style of banjo playing,
before 'frailing', and the more modern styles of finger-picking
used by Earl Scruggs and widely copied). Other guitar styles in
Johnson's other songs make far more use of bending and sliding.

So as I said above, since the guitar part is almost entirely
12-eq, it can be read by any software that will open MIDI
files and provide staff notation.

The rest of your response to me, concerning the error of
confusing the measurement of pitches with the measurement
of intervals, is a point well taken. However, I would still
argue that lattice diagrams provide the easiest way to
grasp the numeric information contained in a rational
measurement of either pitches *or* intervals.

You, Paul, are an accomplished mathematician, a person to
whom numbers readily reveal their secrets even when notated
*as* numbers. To me, on the other hand, numbers are mostly
just another batch of 'Greek' symbols, and it's not until
I put that information onto a graph (which is exactly what
lattice diagrams are) that I can see patterns that would
otherwise elude me.

Thanks (as always with you) for the very useful criticism.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

___________________________________________________________________
Get the Internet just the way you want it.
Free software, free e-mail, and free Internet access for a month!
Try Juno Web: http://dl.www.juno.com/dynoget/tagj.

Response to Paul Erlich:

> [Paul Erlich, TD 332.6]
>
>> [me, monz]
>> The lattice diagram I've just added to my analysis of the
>> rational pitches which I found to be closest to the vocal sung
>> by Robert Johnson on the first verse of _Drunken Hearted Man_:
>> http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
>> can be interpreted as the periodicity-block _gestalt_
>> of rational categorical pitches/intervals analyzed in
>> the tuning of that performance.
>
> No offense, Joe, but this practically sounds like you're
> pulling terms out of your dictionary at random and stringing
> them together. Of course I'm exaggerating, but are you really
> prepared to back up this statement?

Well, I did say they *can* be interpreted as... etc. etc., not
that they *are*... etc. etc.

And I further mitigated the strength of this statement
a few paragraphs down, by my call for alternate
interpretations.

I was merely trying to say here that if one analyzes the
pitches sung by Johnson as ratios, this is the simplest
rational interpretation possible in the 13...23-limit (that
is, mainly 13-limit but including the one ratio of 23 that I
included) which comes within a few cents of the actual
pitches sung, assuming my measurements of the pitches are
correct (if they're not, they are *very* close).

So there's the answer to your question: not only am I prepared
to back up what I said; that *is* the argument supporting my
statements, right there.

>> Also, I'd be interested in any alternate methods of
>> measuring this tuning, instead of rational; i.e., equal,
>> meantone, etc. I want to get as accurate a representation
>> as possible of the tuning of Johnson's actual vocal
>
> I should really get the recording,

Of course you should, if only to have in your possession
one of the great documents of American musical history.
(Especially since you are a fan of the blues: it doesn't
get any better than this.)

However, while I would never say that my MIDI sequence on my
webpage can substitute for the actual recording of Johnson's
performance, as far as tempo and intonation, it's damn
close to what he sang and played.

Rather than my arguing that point, just listen to the CD
and tell me how far off you think my MIDI sequence is,
because I'm *quite* confident of the job I did on that.

If it's possible to do a spectrogram analysis of Johnson's
vocal on this tune, I'd love to see it, and correct my
webpage with that info. (...Dante?)

> but it seems to me that
> it's wholly inappropriate to analyze the vocal in terms of
> a JI lattice constructed on one and only one pitch on Johnson's
> guitar. Is he holding that pitch throughout the tune? What
> about the other 11 pitches on his guitar?

Well, I'd say that perhaps it's *somewhat* inappropriate to base
a JI analysis of this vocal on one pitch, but I definitely
would not say 'wholly inappropriate'.

I can certainly grant the possibility that an analysis should
consider the relationships of the pitches in the vocal to the
pitches being played on the guitar *at that moment*, especially
at the points where the guitar notes imply a different harmonic
series (i.e., the IV and V chords, G7 and A7) than that over 1/1
(in this case, 1/1 is 'D').

But to my ears, the vocal in this tune seems firmly 'rooted',
melodically, to that 1/1 'D' key. An argument in favor of
this is the fact that the repeated lines where he sings 'my
life seems so misery' both occur over G7 chords but clearly
outline a resolution onto the tonic D7, which is exactly what
he plays on the guitar as soon as he's finished singing those
lines. So, to my way of thinking, the vocal sort of anticipates
the chord changes.

My webpage gives the notation for the vocal, with the pitches
as analyzed by me. To see the guitar part, it's a simple
matter of opening the MIDI file with any sequencer or notation
program and looking at the guitar track.

I'd be glad to add the notation of the guitar part to the
webpage if that helps anyone understand or dispute what
I've done here. I'm particularly interested in any alternate
rational interpretation you have of this tune, Paul.
(or anyone else, but particularly Paul)

>> (and guitar playing).
>
> Is there a lot of bending and/or use of a slide on this
> recording?

As I made clear in the text of my webpage, on this song, no.

Johnson was a master of many different styles of blues guitar
playing, and in this particular tune (and his others which
resemble it), the guitar part sounds to me like an imitation
of banjo playing. (That is, the 'old' style of banjo playing,
before 'frailing', and the more modern styles of finger-picking
used by Earl Scruggs and widely copied). Other guitar styles in
Johnson's other songs make far more use of bending and sliding.

So as I said above, since the guitar part is almost entirely
12-eq, it can be read by any software that will open MIDI
files and provide staff notation.

The rest of your response to me, concerning the error of
confusing the measurement of pitches with the measurement
of intervals, is a point well taken. However, I would still
argue that lattice diagrams provide the easiest way to
grasp the numeric information contained in a rational
measurement of either pitches *or* intervals.

You, Paul, are an accomplished mathematician, a person to
whom numbers readily reveal their secrets even when notated
*as* numbers. To me, on the other hand, numbers are mostly
just another batch of 'Greek' symbols, and it's not until
I put that information onto a graph (which is exactly what
lattice diagrams are) that I can see patterns that would
otherwise elude me.

Thanks (as always with you) for the very useful criticism.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

___________________________________________________________________
Get the Internet just the way you want it.
Free software, free e-mail, and free Internet access for a month!
Try Juno Web: http://dl.www.juno.com/dynoget/tagj.

Response to Paul Erlich:

> [Paul Erlich, TD 332.6]
>
>> [me, monz]
>> The lattice diagram I've just added to my analysis of the
>> rational pitches which I found to be closest to the vocal sung
>> by Robert Johnson on the first verse of _Drunken Hearted Man_:
>> http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
>> can be interpreted as the periodicity-block _gestalt_
>> of rational categorical pitches/intervals analyzed in
>> the tuning of that performance.
>
> No offense, Joe, but this practically sounds like you're
> pulling terms out of your dictionary at random and stringing
> them together. Of course I'm exaggerating, but are you really
> prepared to back up this statement?

Well, I did say they *can* be interpreted as... etc. etc., not
that they *are*... etc. etc.

And I further mitigated the strength of this statement
a few paragraphs down, by my call for alternate
interpretations.

I was merely trying to say here that if one analyzes the
pitches sung by Johnson as ratios, this is the simplest
rational interpretation possible in the 13...23-limit (that
is, mainly 13-limit but including the one ratio of 23 that I
included) which comes within a few cents of the actual
pitches sung, assuming my measurements of the pitches are
correct (if they're not, they are *very* close).

So there's the answer to your question: not only am I prepared
to back up what I said; that *is* the argument supporting my
statements, right there.

>> Also, I'd be interested in any alternate methods of
>> measuring this tuning, instead of rational; i.e., equal,
>> meantone, etc. I want to get as accurate a representation
>> as possible of the tuning of Johnson's actual vocal
>
> I should really get the recording,

Of course you should, if only to have in your possession
one of the great documents of American musical history.
(Especially since you are a fan of the blues: it doesn't
get any better than this.)

However, while I would never say that my MIDI sequence on my
webpage can substitute for the actual recording of Johnson's
performance, as far as tempo and intonation, it's damn
close to what he sang and played.

Rather than my arguing that point, just listen to the CD
and tell me how far off you think my MIDI sequence is,
because I'm *quite* confident of the job I did on that.

If it's possible to do a spectrogram analysis of Johnson's
vocal on this tune, I'd love to see it, and correct my
webpage with that info. (...Dante?)

> but it seems to me that
> it's wholly inappropriate to analyze the vocal in terms of
> a JI lattice constructed on one and only one pitch on Johnson's
> guitar. Is he holding that pitch throughout the tune? What
> about the other 11 pitches on his guitar?

Well, I'd say that perhaps it's *somewhat* inappropriate to base
a JI analysis of this vocal on one pitch, but I definitely
would not say 'wholly inappropriate'.

I can certainly grant the possibility that an analysis should
consider the relationships of the pitches in the vocal to the
pitches being played on the guitar *at that moment*, especially
at the points where the guitar notes imply a different harmonic
series (i.e., the IV and V chords, G7 and A7) than that over 1/1
(in this case, 1/1 is 'D').

But to my ears, the vocal in this tune seems firmly 'rooted',
melodically, to that 1/1 'D' key. An argument in favor of
this is the fact that the repeated lines where he sings 'my
life seems so misery' both occur over G7 chords but clearly
outline a resolution onto the tonic D7, which is exactly what
he plays on the guitar as soon as he's finished singing those
lines. So, to my way of thinking, the vocal sort of anticipates
the chord changes.

My webpage gives the notation for the vocal, with the pitches
as analyzed by me. To see the guitar part, it's a simple
matter of opening the MIDI file with any sequencer or notation
program and looking at the guitar track.

I'd be glad to add the notation of the guitar part to the
webpage if that helps anyone understand or dispute what
I've done here. I'm particularly interested in any alternate
rational interpretation you have of this tune, Paul.
(or anyone else, but particularly Paul)

>> (and guitar playing).
>
> Is there a lot of bending and/or use of a slide on this
> recording?

As I made clear in the text of my webpage, on this song, no.

Johnson was a master of many different styles of blues guitar
playing, and in this particular tune (and his others which
resemble it), the guitar part sounds to me like an imitation
of banjo playing. (That is, the 'old' style of banjo playing,
before 'frailing', and the more modern styles of finger-picking
used by Earl Scruggs and widely copied). Other guitar styles in
Johnson's other songs make far more use of bending and sliding.

So as I said above, since the guitar part is almost entirely
12-eq, it can be read by any software that will open MIDI
files and provide staff notation.

The rest of your response to me, concerning the error of
confusing the measurement of pitches with the measurement
of intervals, is a point well taken. However, I would still
argue that lattice diagrams provide the easiest way to
grasp the numeric information contained in a rational
measurement of either pitches *or* intervals.

You, Paul, are an accomplished mathematician, a person to
whom numbers readily reveal their secrets even when notated
*as* numbers. To me, on the other hand, numbers are mostly
just another batch of 'Greek' symbols, and it's not until
I put that information onto a graph (which is exactly what
lattice diagrams are) that I can see patterns that would
otherwise elude me.

Thanks (as always with you) for the very useful criticism.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

___________________________________________________________________
Get the Internet just the way you want it.
Free software, free e-mail, and free Internet access for a month!
Try Juno Web: http://dl.www.juno.com/dynoget/tagj.

Response to Paul Erlich:

> [Paul Erlich, TD 332.6]
>
>> [me, monz]
>> The lattice diagram I've just added to my analysis of the
>> rational pitches which I found to be closest to the vocal sung
>> by Robert Johnson on the first verse of _Drunken Hearted Man_:
>> http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
>> can be interpreted as the periodicity-block _gestalt_
>> of rational categorical pitches/intervals analyzed in
>> the tuning of that performance.
>
> No offense, Joe, but this practically sounds like you're
> pulling terms out of your dictionary at random and stringing
> them together. Of course I'm exaggerating, but are you really
> prepared to back up this statement?

Well, I did say they *can* be interpreted as... etc. etc., not
that they *are*... etc. etc.

And I further mitigated the strength of this statement
a few paragraphs down, by my call for alternate
interpretations.

I was merely trying to say here that if one analyzes the
pitches sung by Johnson as ratios, this is the simplest
rational interpretation possible in the 13...23-limit (that
is, mainly 13-limit but including the one ratio of 23 that I
included) which comes within a few cents of the actual
pitches sung, assuming my measurements of the pitches are
correct (if they're not, they are *very* close).

So there's the answer to your question: not only am I prepared
to back up what I said; that *is* the argument supporting my
statements, right there.

>> Also, I'd be interested in any alternate methods of
>> measuring this tuning, instead of rational; i.e., equal,
>> meantone, etc. I want to get as accurate a representation
>> as possible of the tuning of Johnson's actual vocal
>
> I should really get the recording,

Of course you should, if only to have in your possession
one of the great documents of American musical history.
(Especially since you are a fan of the blues: it doesn't
get any better than this.)

However, while I would never say that my MIDI sequence on my
webpage can substitute for the actual recording of Johnson's
performance, as far as tempo and intonation, it's damn
close to what he sang and played.

Rather than my arguing that point, just listen to the CD
and tell me how far off you think my MIDI sequence is,
because I'm *quite* confident of the job I did on that.

If it's possible to do a spectrogram analysis of Johnson's
vocal on this tune, I'd love to see it, and correct my
webpage with that info. (...Dante?)

> but it seems to me that
> it's wholly inappropriate to analyze the vocal in terms of
> a JI lattice constructed on one and only one pitch on Johnson's
> guitar. Is he holding that pitch throughout the tune? What
> about the other 11 pitches on his guitar?

Well, I'd say that perhaps it's *somewhat* inappropriate to base
a JI analysis of this vocal on one pitch, but I definitely
would not say 'wholly inappropriate'.

I can certainly grant the possibility that an analysis should
consider the relationships of the pitches in the vocal to the
pitches being played on the guitar *at that moment*, especially
at the points where the guitar notes imply a different harmonic
series (i.e., the IV and V chords, G7 and A7) than that over 1/1
(in this case, 1/1 is 'D').

But to my ears, the vocal in this tune seems firmly 'rooted',
melodically, to that 1/1 'D' key. An argument in favor of
this is the fact that the repeated lines where he sings 'my
life seems so misery' both occur over G7 chords but clearly
outline a resolution onto the tonic D7, which is exactly what
he plays on the guitar as soon as he's finished singing those
lines. So, to my way of thinking, the vocal sort of anticipates
the chord changes.

My webpage gives the notation for the vocal, with the pitches
as analyzed by me. To see the guitar part, it's a simple
matter of opening the MIDI file with any sequencer or notation
program and looking at the guitar track.

I'd be glad to add the notation of the guitar part to the
webpage if that helps anyone understand or dispute what
I've done here. I'm particularly interested in any alternate
rational interpretation you have of this tune, Paul.
(or anyone else, but particularly Paul)

>> (and guitar playing).
>
> Is there a lot of bending and/or use of a slide on this
> recording?

As I made clear in the text of my webpage, on this song, no.

Johnson was a master of many different styles of blues guitar
playing, and in this particular tune (and his others which
resemble it), the guitar part sounds to me like an imitation
of banjo playing. (That is, the 'old' style of banjo playing,
before 'frailing', and the more modern styles of finger-picking
used by Earl Scruggs and widely copied). Other guitar styles in
Johnson's other songs make far more use of bending and sliding.

So as I said above, since the guitar part is almost entirely
12-eq, it can be read by any software that will open MIDI
files and provide staff notation.

The rest of your response to me, concerning the error of
confusing the measurement of pitches with the measurement
of intervals, is a point well taken. However, I would still
argue that lattice diagrams provide the easiest way to
grasp the numeric information contained in a rational
measurement of either pitches *or* intervals.

You, Paul, are an accomplished mathematician, a person to
whom numbers readily reveal their secrets even when notated
*as* numbers. To me, on the other hand, numbers are mostly
just another batch of 'Greek' symbols, and it's not until
I put that information onto a graph (which is exactly what
lattice diagrams are) that I can see patterns that would
otherwise elude me.

Thanks (as always with you) for the very useful criticism.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

___________________________________________________________________
Get the Internet just the way you want it.
Free software, free e-mail, and free Internet access for a month!
Try Juno Web: http://dl.www.juno.com/dynoget/tagj.

Response to Paul Erlich:

> [Paul Erlich, TD 332.6]
>
>> [me, monz]
>> The lattice diagram I've just added to my analysis of the
>> rational pitches which I found to be closest to the vocal sung
>> by Robert Johnson on the first verse of _Drunken Hearted Man_:
>> http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
>> can be interpreted as the periodicity-block _gestalt_
>> of rational categorical pitches/intervals analyzed in
>> the tuning of that performance.
>
> No offense, Joe, but this practically sounds like you're
> pulling terms out of your dictionary at random and stringing
> them together. Of course I'm exaggerating, but are you really
> prepared to back up this statement?

Well, I did say they *can* be interpreted as... etc. etc., not
that they *are*... etc. etc.

And I further mitigated the strength of this statement
a few paragraphs down, by my call for alternate
interpretations.

I was merely trying to say here that if one analyzes the
pitches sung by Johnson as ratios, this is the simplest
rational interpretation possible in the 13...23-limit (that
is, mainly 13-limit but including the one ratio of 23 that I
included) which comes within a few cents of the actual
pitches sung, assuming my measurements of the pitches are
correct (if they're not, they are *very* close).

So there's the answer to your question: not only am I prepared
to back up what I said; that *is* the argument supporting my
statements, right there.

>> Also, I'd be interested in any alternate methods of
>> measuring this tuning, instead of rational; i.e., equal,
>> meantone, etc. I want to get as accurate a representation
>> as possible of the tuning of Johnson's actual vocal
>
> I should really get the recording,

Of course you should, if only to have in your possession
one of the great documents of American musical history.
(Especially since you are a fan of the blues: it doesn't
get any better than this.)

However, while I would never say that my MIDI sequence on my
webpage can substitute for the actual recording of Johnson's
performance, as far as tempo and intonation, it's damn
close to what he sang and played.

Rather than my arguing that point, just listen to the CD
and tell me how far off you think my MIDI sequence is,
because I'm *quite* confident of the job I did on that.

If it's possible to do a spectrogram analysis of Johnson's
vocal on this tune, I'd love to see it, and correct my
webpage with that info. (...Dante?)

> but it seems to me that
> it's wholly inappropriate to analyze the vocal in terms of
> a JI lattice constructed on one and only one pitch on Johnson's
> guitar. Is he holding that pitch throughout the tune? What
> about the other 11 pitches on his guitar?

Well, I'd say that perhaps it's *somewhat* inappropriate to base
a JI analysis of this vocal on one pitch, but I definitely
would not say 'wholly inappropriate'.

I can certainly grant the possibility that an analysis should
consider the relationships of the pitches in the vocal to the
pitches being played on the guitar *at that moment*, especially
at the points where the guitar notes imply a different harmonic
series (i.e., the IV and V chords, G7 and A7) than that over 1/1
(in this case, 1/1 is 'D').

But to my ears, the vocal in this tune seems firmly 'rooted',
melodically, to that 1/1 'D' key. An argument in favor of
this is the fact that the repeated lines where he sings 'my
life seems so misery' both occur over G7 chords but clearly
outline a resolution onto the tonic D7, which is exactly what
he plays on the guitar as soon as he's finished singing those
lines. So, to my way of thinking, the vocal sort of anticipates
the chord changes.

My webpage gives the notation for the vocal, with the pitches
as analyzed by me. To see the guitar part, it's a simple
matter of opening the MIDI file with any sequencer or notation
program and looking at the guitar track.

I'd be glad to add the notation of the guitar part to the
webpage if that helps anyone understand or dispute what
I've done here. I'm particularly interested in any alternate
rational interpretation you have of this tune, Paul.
(or anyone else, but particularly Paul)

>> (and guitar playing).
>
> Is there a lot of bending and/or use of a slide on this
> recording?

As I made clear in the text of my webpage, on this song, no.

Johnson was a master of many different styles of blues guitar
playing, and in this particular tune (and his others which
resemble it), the guitar part sounds to me like an imitation
of banjo playing. (That is, the 'old' style of banjo playing,
before 'frailing', and the more modern styles of finger-picking
used by Earl Scruggs and widely copied). Other guitar styles in
Johnson's other songs make far more use of bending and sliding.

So as I said above, since the guitar part is almost entirely
12-eq, it can be read by any software that will open MIDI
files and provide staff notation.

The rest of your response to me, concerning the error of
confusing the measurement of pitches with the measurement
of intervals, is a point well taken. However, I would still
argue that lattice diagrams provide the easiest way to
grasp the numeric information contained in a rational
measurement of either pitches *or* intervals.

You, Paul, are an accomplished mathematician, a person to
whom numbers readily reveal their secrets even when notated
*as* numbers. To me, on the other hand, numbers are mostly
just another batch of 'Greek' symbols, and it's not until
I put that information onto a graph (which is exactly what
lattice diagrams are) that I can see patterns that would
otherwise elude me.

Thanks (as always with you) for the very useful criticism.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

___________________________________________________________________
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Response to Paul Erlich:

> [Paul Erlich, TD 332.6]
>
>> [me, monz]
>> The lattice diagram I've just added to my analysis of the
>> rational pitches which I found to be closest to the vocal sung
>> by Robert Johnson on the first verse of _Drunken Hearted Man_:
>> http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
>> can be interpreted as the periodicity-block _gestalt_
>> of rational categorical pitches/intervals analyzed in
>> the tuning of that performance.
>
> No offense, Joe, but this practically sounds like you're
> pulling terms out of your dictionary at random and stringing
> them together. Of course I'm exaggerating, but are you really
> prepared to back up this statement?

Well, I did say they *can* be interpreted as... etc. etc., not
that they *are*... etc. etc.

And I further mitigated the strength of this statement
a few paragraphs down, by my call for alternate
interpretations.

I was merely trying to say here that if one analyzes the
pitches sung by Johnson as ratios, this is the simplest
rational interpretation possible in the 13...23-limit (that
is, mainly 13-limit but including the one ratio of 23 that I
included) which comes within a few cents of the actual
pitches sung, assuming my measurements of the pitches are
correct (if they're not, they are *very* close).

So there's the answer to your question: not only am I prepared
to back up what I said; that *is* the argument supporting my
statements, right there.

>> Also, I'd be interested in any alternate methods of
>> measuring this tuning, instead of rational; i.e., equal,
>> meantone, etc. I want to get as accurate a representation
>> as possible of the tuning of Johnson's actual vocal
>
> I should really get the recording,

Of course you should, if only to have in your possession
one of the great documents of American musical history.
(Especially since you are a fan of the blues: it doesn't
get any better than this.)

However, while I would never say that my MIDI sequence on my
webpage can substitute for the actual recording of Johnson's
performance, as far as tempo and intonation, it's damn
close to what he sang and played.

Rather than my arguing that point, just listen to the CD
and tell me how far off you think my MIDI sequence is,
because I'm *quite* confident of the job I did on that.

If it's possible to do a spectrogram analysis of Johnson's
vocal on this tune, I'd love to see it, and correct my
webpage with that info. (...Dante?)

> but it seems to me that
> it's wholly inappropriate to analyze the vocal in terms of
> a JI lattice constructed on one and only one pitch on Johnson's
> guitar. Is he holding that pitch throughout the tune? What
> about the other 11 pitches on his guitar?

Well, I'd say that perhaps it's *somewhat* inappropriate to base
a JI analysis of this vocal on one pitch, but I definitely
would not say 'wholly inappropriate'.

I can certainly grant the possibility that an analysis should
consider the relationships of the pitches in the vocal to the
pitches being played on the guitar *at that moment*, especially
at the points where the guitar notes imply a different harmonic
series (i.e., the IV and V chords, G7 and A7) than that over 1/1
(in this case, 1/1 is 'D').

But to my ears, the vocal in this tune seems firmly 'rooted',
melodically, to that 1/1 'D' key. An argument in favor of
this is the fact that the repeated lines where he sings 'my
life seems so misery' both occur over G7 chords but clearly
outline a resolution onto the tonic D7, which is exactly what
he plays on the guitar as soon as he's finished singing those
lines. So, to my way of thinking, the vocal sort of anticipates
the chord changes.

My webpage gives the notation for the vocal, with the pitches
as analyzed by me. To see the guitar part, it's a simple
matter of opening the MIDI file with any sequencer or notation
program and looking at the guitar track.

I'd be glad to add the notation of the guitar part to the
webpage if that helps anyone understand or dispute what
I've done here. I'm particularly interested in any alternate
rational interpretation you have of this tune, Paul.
(or anyone else, but particularly Paul)

>> (and guitar playing).
>
> Is there a lot of bending and/or use of a slide on this
> recording?

As I made clear in the text of my webpage, on this song, no.

Johnson was a master of many different styles of blues guitar
playing, and in this particular tune (and his others which
resemble it), the guitar part sounds to me like an imitation
of banjo playing. (That is, the 'old' style of banjo playing,
before 'frailing', and the more modern styles of finger-picking
used by Earl Scruggs and widely copied). Other guitar styles in
Johnson's other songs make far more use of bending and sliding.

So as I said above, since the guitar part is almost entirely
12-eq, it can be read by any software that will open MIDI
files and provide staff notation.

The rest of your response to me, concerning the error of
confusing the measurement of pitches with the measurement
of intervals, is a point well taken. However, I would still
argue that lattice diagrams provide the easiest way to
grasp the numeric information contained in a rational
measurement of either pitches *or* intervals.

You, Paul, are an accomplished mathematician, a person to
whom numbers readily reveal their secrets even when notated
*as* numbers. To me, on the other hand, numbers are mostly
just another batch of 'Greek' symbols, and it's not until
I put that information onto a graph (which is exactly what
lattice diagrams are) that I can see patterns that would
otherwise elude me.

Thanks (as always with you) for the very useful criticism.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

___________________________________________________________________
Get the Internet just the way you want it.
Free software, free e-mail, and free Internet access for a month!
Try Juno Web: http://dl.www.juno.com/dynoget/tagj.

Joe wrote,

>>> The lattice diagram I've just added to my analysis of the
>>> rational pitches which I found to be closest to the vocal sung
>>> by Robert Johnson on the first verse of _Drunken Hearted Man_:
>>> http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
>>> can be interpreted as the periodicity-block _gestalt_
>>> of rational categorical pitches/intervals analyzed in
>>> the tuning of that performance.

I wrote,

>> No offense, Joe, but this practically sounds like you're
>> pulling terms out of your dictionary at random and stringing
>> them together. Of course I'm exaggerating, but are you really
>> prepared to back up this statement?

>Well, I did say they *can* be interpreted as... etc. etc., not
>that they *are*... etc. etc.

>And I further mitigated the strength of this statement
>a few paragraphs down, by my call for alternate
>interpretations.

>I was merely trying to say here that if one analyzes the
>pitches sung by Johnson as ratios, this is the simplest
>rational interpretation possible in the 13...23-limit (that
>is, mainly 13-limit but including the one ratio of 23 that I
>included) which comes within a few cents of the actual
>pitches sung, assuming my measurements of the pitches are
>correct (if they're not, they are *very* close).

>So there's the answer to your question: not only am I prepared
>to back up what I said; that *is* the argument supporting my
>statements, right there.

OK, first of all, what is a "periodicity-block _gestalt_" and how does it
apply to your example?

>If it's possible to do a spectrogram analysis of Johnson's
>vocal on this tune, I'd love to see it, and correct my
>webpage with that info. (...Dante?)

That seems essential, though a spectogram analyzer would not be a powerful
enough tool -- one would need to isolate the vocals . . .

>But to my ears, the vocal in this tune seems firmly 'rooted',
>melodically, to that 1/1 'D' key. An argument in favor of
>this is the fact that the repeated lines where he sings 'my
>life seems so misery' both occur over G7 chords but clearly
>outline a resolution onto the tonic D7, which is exactly what
>he plays on the guitar

In 12-equal! And yet you'd claim the guitar exerts no influence on his vocal
intonation, other than determining the 1/1?

>You, Paul, are [ . . . ] a person to
>whom numbers readily reveal their secrets even when notated
>*as* numbers. To me, on the other hand, numbers are mostly
>just another batch of 'Greek' symbols, and it's not until
>I put that information onto a graph (which is exactly what
>lattice diagrams are) that I can see patterns that would
>otherwise elude me.

On the contrary, I am more like you. I draw tons of lattice diagrams on this
list, especially when someone proposes a new scale using only numbers. I do
it so that I can see at a glance what consonant intervals and chords (at
least by Partchian definitions or their tempered analogs) the scale
contains.

>>But to my ears, the vocal in this tune seems firmly 'rooted',
>>melodically, to that 1/1 'D' key. An argument in favor of
>>this is the fact that the repeated lines where he sings 'my
>>life seems so misery' both occur over G7 chords but clearly
>>outline a resolution onto the tonic D7, which is exactly what
>>he plays on the guitar
>
>In 12-equal! And yet you'd claim the guitar exerts no influence on his vocal
>intonation, other than determining the 1/1?

Joe, I haven't looked at your Robert Johnson page recently, and Paul, I
vehemently agree that confusion between vertical and horizontal JI has
caused many problems (in fact I have criticized you for it in the past).

The point I wish to make here is that rhythm is a factor that can not be
ignored when considering tuning interaction between parts. I have heard
vocals sung atop piano and guitar that are completely un-influenced by the
fixed instrument's intonation. When the rhythm of the parts is related,
however, the model Paul wants here is definitely the way to go.

-C.

> [Paul Erlich, TD 334.14]
>
>>>> Joe wrote,
>>>>
>>>> The lattice diagram I've just added to my analysis of the
>>>> rational pitches which I found to be closest to the vocal sung
>>>> by Robert Johnson on the first verse of _Drunken Hearted Man_:
>>>> http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm
>>>> can be interpreted as the periodicity-block _gestalt_
>>>> of rational categorical pitches/intervals analyzed in
>>>> the tuning of that performance.

>>> [Paul's reply]
>>>
>>> No offense, Joe, but this practically sounds like you're
>>> pulling terms out of your dictionary at random and stringing
>>> them together. Of course I'm exaggerating, but are you really
>>> prepared to back up this statement?

>> [monz's reply]
>>
>> Well, I did say they *can* be interpreted as... etc. etc., not
>> that they *are*... etc. etc.
>>
>> And I further mitigated the strength of this statement
>> a few paragraphs down, by my call for alternate
>> interpretations.
>>
>> I was merely trying to say here that if one analyzes the
>> pitches sung by Johnson as ratios, this is the simplest
>> rational interpretation possible in the 13...23-limit (that
>> is, mainly 13-limit but including the one ratio of 23 that I
>> included) which comes within a few cents of the actual
>> pitches sung, assuming my measurements of the pitches are
>> correct (if they're not, they are *very* close).
>>
>> So there's the answer to your question: not only am I prepared
>> to back up what I said; that *is* the argument supporting my
>> statements, right there.

> [Paul's reply]
>
> OK, first of all, what is a "periodicity-block _gestalt_" and
> how does it apply to your example?

[monz's reply:]

Well, first of all, I'm not even sure if 'periodicity-block
_gestalt_ is the proper term, and also not sure if I'm being
redundant in coining it, if there is already another way to
describe what I'm getting at.

My point is that there is an infinite number of ways to describe
the tuning of Johnson's vocal, just as any musical performance
can be described infinitely many different ways, each tuning
being a more-or-less arbitrarily-chosen decision on the part
of the person describing the tuning. Unless, that is, the
instruments used in the performance are scrupulously tuned
a particular way (as with Partch, for instance).

My decision here was to assume that Johnson's vocal more-or-less
monophonically (in Partch's definition of the term: see my
online Tuning Dictionary if you're not Paul and you don't know
what I mean) implies a JI tuning 'rooted' on 'D' using the
ratios I present on my webpage.

My criteria for describing these pitches as ratios was that
any ratio used in the description would be no more than a few
(not sure now what my largest error was, but definitely less
than 10) cents within my measurement of the actual cent-value
of the pitches sung by Johnson, and that they would all fall
within the 13-limit, with the single exception of the 23/16
I use to describe a pitch which occurs in the first two lines
of every verse.

Of course, there are infinitely many other ratios which could
be used to describe these pitches. There's also a margin of
error in my actual cents measurements, since they were not
done with a spectrogram, but rather were all done by ear (mine).

So based on the criteria I specified in the above paragraphs,
the lattice diagram I've recently added to this webpage can
be thought of as a 'periodicity-block _gestalt_'.

Perhaps the crucial distinction is that right from the beginning
I should have said *a*, rather than *the*, periodicity-block
_gestalt_. But to use 'the' is still correct, provided that
the criteria are stipulated.

Of course, it is well to bear in mind that there are also
infinitely many *other* ways to describe the tuning of this
vocal beside using JI ratios. Perhaps some sort of well-
temperament or meantone would work... I haven't investigated
any of those possibilities, but hope that others will.

The reason for including the word _gestalt_ in my phrase
is simply to acknowledge the infinite number of possible
ways to interpret the tuning of this vocal. I've picked
one specific periodicity-block that I think provides the
simplest accurate description of what Johnson sang. I am
ready to admit the possibility that I could be partially or
totally wrong.

>> [monz]
>>
>> If it's possible to do a spectrogram analysis of Johnson's
>> vocal on this tune, I'd love to see it, and correct my
>> webpage with that info. (...Dante?)
>
> [Paul]
>
> That seems essential, though a spectrogram analyzer would not be
> a powerful enough tool -- one would need to isolate the vocals
> . . .

Yes, you are correct about that, the vocal would have to be
isolated. But I believe there are devices or software
available that can do that.

>> [monz]
>> But to my ears, the vocal in this tune seems firmly 'rooted',
>> melodically, to that 1/1 'D' key. An argument in favor of
>> this is the fact that the repeated lines where he sings 'my
>> life seems so misery' both occur over G7 chords but clearly
>> outline a resolution onto the tonic D7, which is exactly what
>> he plays on the guitar

> [Paul]
> In 12-equal! And yet you'd claim the guitar exerts no influence
> on his vocal intonation, other than determining the 1/1?

Yes, you are correct that the guitar part is almost entirely
12-equal, and no, I didn't exactly make the firm statement that
'the guitar exerts no influence on his vocal intonation, other
than determining the 1/1'. But I haven't considered the ratios
from the point of view of how they relate to the guitar part
when Johnson is playing the G7 chords and singing a melody
which seems (to my ears) to outline the higher harmonics of D.
Give me some alternate possibilities... anyone...

>> [monz]
>>
>> You, Paul, are [ . . . ] a person to
>> whom numbers readily reveal their secrets even when notated
>> *as* numbers. To me, on the other hand, numbers are mostly
>> just another batch of 'Greek' symbols, and it's not until
>> I put that information onto a graph (which is exactly what
>> lattice diagrams are) that I can see patterns that would
>> otherwise elude me.

> [Paul]
>
> On the contrary, I am more like you. I draw tons of lattice
> diagrams on this list, especially when someone proposes a new
> scale using only numbers. I do it so that I can see at a glance
> what consonant intervals and chords (at least by Partchian
> definitions or their tempered analogs) the scale contains.

OK, sorry - didn't mean to put words in your mouth. I know
you contribute *many* lattice diagrams to the list, any many
of them are quite good (from an aesthetic point of view).
In my opinion, patterns are *always* easier to see on graphs
than they are in tables of numbers.

A couple of weeks back I sent in a posting saying that my
book is a sort 'history of finity in tuning'. What I'm trying
to communicate by that statement is that by using the *same*
type of graph coordinates to portray all the different tunings
described in the theoretical literature, one is able to see
concretely what the periodicity-blocks (or, more simply,
the tuning systems) look like throughout musical history,
how they've changed and what elements they all may have
in common.

Of course, my book is particularly slanted towards a just-
intonation perspective. This simply reflects my own interests
and the amount of time I've spent researching various different
kinds of tuning systems.

I'm in the process of establishing a new draft of the book now,
and would be more than happy to flesh it out with descriptions
of non-JI tunings, but it would require an immense amount of
further work that I simply don't have time for right now.
Any contributions from others are welcome, and will be properly
credited.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

___________________________________________________________________
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Try Juno Web: http://dl.www.juno.com/dynoget/tagj.

Joe Monzo wrote,

>So based on the criteria I specified in the above paragraphs,
>the lattice diagram I've recently added to this webpage can
>be thought of as a 'periodicity-block _gestalt_'.

Those paragraphs don't mention periodicity blocks or any of the elements of
their definition.

>Perhaps the crucial distinction is that right from the beginning
>I should have said *a*, rather than *the*, periodicity-block
>_gestalt_. But to use 'the' is still correct, provided that
>the criteria are stipulated.

>Of course, it is well to bear in mind that there are also
>infinitely many *other* ways to describe the tuning of this
>vocal beside using JI ratios. Perhaps some sort of well-
>temperament or meantone would work... I haven't investigated
>any of those possibilities, but hope that others will.

Those possibilities would be pretty nonsensical in this context. Simply
slapping a tuning system on Johnson's vocal, without deriving it from
considerations of the psychoacoutical parameters relevant in the
performance, would be completely unenlightening. You'd be better off just
specifying the cents values and be done with it. As you point out, there are
an inifinite number of mathematical structures which can approximate a given
pitch set, but if those structures can't be justified in terms of how the
ear/brain system works, then the analysis is devoid of explanatory content,
and therefore meaningless.

>The reason for including the word _gestalt_ in my phrase
>is simply to acknowledge the infinite number of possible
>ways to interpret the tuning of this vocal. I've picked
>one specific periodicity-block that I think provides the
>simplest accurate description of what Johnson sang. I am
>ready to admit the possibility that I could be partially or
>totally wrong.

So you're claiming the pitch-set you describe is a periodicity block? That's
what I'm having trouble believing. What is its Fokker matrix? Or are you, as
I suspected, just throwing around terminology?

>What I'm trying
>to communicate by that statement is that by using the *same*
>type of graph coordinates to portray all the different tunings
>described in the theoretical literature, one is able to see
>concretely what the periodicity-blocks (or, more simply,
>the tuning systems) look like throughout musical history,
>how they've changed and what elements they all may have
>in common.

Joe, this makes me suspect even more strongly that you are using "perodicity
block" much too loosely. Perhaps this also explains why you did not react to
my posting of the periodicity block derivation of the Indian sruti system.
It takes a pretty special set of JI pitches to qualify as a periodicity
block, and the fact that the Hindu system does so (or does so with an
infinitesimal change that is totally consistent with the important
properties of periodicity blocks) is quite striking. I discovered that only
because you asked the question of how historical tuning systems could be
interpreted as periodicity blocks. I expected you to react and perhaps read
more into this than I even would, but it appears you are willing to consider
any set of JI pitches a periodicity block, in which case I can understand
your lack of reaction!

I once thought your definition of "finity" was equivalent to, or some sort
of generalization of, the periodicity block idea. Then you claimed that
Partch's tonality diamonds were examples of finity. Well, I was very
troubled by that, but of course you're free to define your own terms however
you wish. But "periodicity block" was defined by Fokker, and a pitch set
whose cardinality is not quantified by the determinant of a matrix of unison
vectors is most certainly not a periodicity block.

> [Paul Erlich, TD 335.15]
>
> So you're claiming the pitch-set you describe is a
> periodicity block? That's what I'm having trouble believing.
> What is its Fokker matrix? Or are you, as I suspected, just
> throwing around terminology?

> [Paul Erlich, TD 335.16]
>
> Joe, this makes me suspect even more strongly that you are
> using "perodicity block" much too loosely. Perhaps this
> also explains why you did not react to my posting of the
> periodicity block derivation of the Indian sruti system.
> It takes a pretty special set of JI pitches to qualify as
> a periodicity block, and the fact that the Hindu system does
> so (or does so with an infinitesimal change that is totally
> consistent with the important properties of periodicity blocks)
> is quite striking. I discovered that only because you asked
> the question of how historical tuning systems could be
> interpreted as periodicity blocks. I expected you to react
> and perhaps read more into this than I even would, but it
> appears you are willing to consider any set of JI pitches a
> periodicity block, in which case I can understand your lack
> of reaction!
>
> I once thought your definition of "finity" was equivalent to,
> or some sort of generalization of, the periodicity block idea.
> Then you claimed that Partch's tonality diamonds were examples
> of finity. Well, I was very troubled by that, but of course
> you're free to define your own terms however you wish. But
> "periodicity block" was defined by Fokker, and a pitch set
> whose cardinality is not quantified by the determinant of a
> matrix of unison vectors is most certainly not a periodicity
> block.

[monz's reply:]

OK, Paul, as usual when you argue with me, at the end of the
day I must concede that you are right and I was wrong.

After considering what you wrote here about using the term
'periodicity-block' only in conjunction with the attributes
which are part of a rigorous definition of it, I must admit
that I *have* been throwing it around pretty loosely.

I suppose I have to retract my contention that the pitches
in my Robert Johnson lattice are a periodicity-block,
especially since I made no effort whatsoever to define
the matrix numerically.

But still, isn't there a way to illustrate what I'm saying
here - about the reduction of an infinite number of possible
descriptions of a tuning to one specific set of pitches
- in connection with use of the periodicity-block concept?
Help!

(BTW, the only reason I haven't responded to your postings
about the Indian sruti system is because I've only scanned
them, and haven't yet devoted the time to reading them
properly. Busy working on the latest draft of my book...
but this is good stuff that I'd like to include in it.)

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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Joe Monzo wrote,

>But still, isn't there a way to illustrate what I'm saying
>here - about the reduction of an infinite number of possible
>descriptions of a tuning to one specific set of pitches
>- in connection with use of the periodicity-block concept?
>Help!

The periodicity block concept is really about reducing an infinite number of
_pitches_ to one specific set (or, if you don't specify the exact set, the
determinant at least tells you the number of pitches). It's not about
reducing an infinite number of possible descriptions of a tuning to one
specific one. In fact, Paul Hahn and Graham Breed using Fokker matrices in
connection with ETs indicates that periodicity blocks may also be thought of
in an infinite number of possible descriptions of a tuning.

Joe, I must credit you for my interest in Fokker periodicity blocks. When
you came over to my house, and we discussed your concept of finity, I
mentioned that the Fokker stuff was the closest thing to what I understood
of your concept. Since then, I've learned almost everything I know about
that stuff.

>(BTW, the only reason I haven't responded to your postings
>about the Indian sruti system is because I've only scanned
>them, and haven't yet devoted the time to reading them
>properly. Busy working on the latest draft of my book...
>but this is good stuff that I'd like to include in it.)

Well, I hope that posting did a good job of answering the question of yours
it was intended to answer.

-Paul