I've finished reading "A Derivation of the Rules of Voice-leading from Perceptual Principles". It seems to be good sense in so far as it goes. Not really a derivation, but a good place for scientists to start when learning counterpoint.

Janata et al isn't so interesting. All it says is that notes closer to the key center are closer to the key center, or something. And there are some brain images I don't understand.

That "Pitch Schemata" link I gave might have the background details. The first time I looked at it I recognized a lot of Rothenberg's ideas, but they were credited to Balzano. I'll have another look sometime.

Graham

>That "Pitch Schemata" link I gave might have the background details.

>The first time I looked at it I recognized a lot of Rothenberg's ideas,

>but they were credited to Balzano. I'll have another look sometime.

Balzano independently came up with a lot of Rothenberg's ideas, years

later, and along with some other mush.

-Carl

Carl Lumma wrote:

> Balzano independently came up with a lot of Rothenberg's ideas, years

> later, and along with some other mush.

So why doesn't Rothenberg get any credit for them? And it's not only Balzano -- Browne's given some of them as well.

Another weird thing is that a Forte 1973 paper is mentioned as a "formulation" of Balzano's 1982. So what was Forte formulating? It must be peculiar if major and minor triads are considered identical.

http://www-ext.mus.cam.ac.uk/~ic108/PCM/

It says that Balzano's coherence is different to Rothenberg's propriety because it only considers adjacent pairs of intervals. So a Pythagorean diatonic is still coherent because the conflicting sizes of tritones aren't considered. Is that right?

Graham

[I wrote...]

>>That "Pitch Schemata" link I gave might have the background details.

>>The first time I looked at it I recognized a lot of Rothenberg's ideas,

>>but they were credited to Balzano. I'll have another look sometime.

>

>Balzano independently came up with a lot of Rothenberg's ideas, years

>later, and along with some other mush.

While many of their criteria were the same, many were not. So it must

be considered a strange coincidence that R. and B. wind up recommending

the same scale, R in 31-tET, and B in 20-tET. B missed the 31-tET

version because it didn't have all the other nonsense properties that

he was so fascinated with (such as the product of the sizes of the 3rds

giving the number of notes in the embedding et). Dan Stearns further

independently suggested this scale in 20-tET for his own reasons. But

AFAIK I'm the first to notice its excellent approximations to 5:3 and

7:4 nicely interleaved on its 8ths. The 31-tET version gets closer to

JI, but the 20-tET version has higher Lumma stability and already gets

you closer than 12-tET to these intervals.

The lower stability of Rothenberg's scale kicks it down to position 6

on my gd spreadsheet -- Balzano's version is at position 3, just below

the pentatonic and diatonic scales...

All the scales are available as scala files...

http://lumma.org/stuff/gd-scl.zip

-Carl

>http://www-ext.mus.cam.ac.uk/~ic108/PCM/

>

>It says that Balzano's coherence is different to Rothenberg's propriety

>because it only considers adjacent pairs of intervals. So a Pythagorean

>diatonic is still coherent because the conflicting sizes of tritones

>aren't considered. Is that right?

I think that's wrong. ;)

No, I remember there being small differences like that. Here's a bit of

a message I sent to tuning some years ago...

"""

Balzano wants // the scale to be covered with three-note chords that fall

on every-other degree of the scale. Which means they'll be made of thirds

and fifths. It is actually a huge mistake to consider them chords, tho,

since Balzano hasn't given any property that defines "chords" (he's

deliberately thrown out the usual one: harmony). Which means that his

whole idea amounts to a lot of nothing.

Well, not quite. He does require that the same interval appears as a

fifth in exactly n-1 modes of the scale, when the scale has n notes per

2:1. Which is no more and no less than MOS when the generator turns out

to be a fifth and the interval of equivalence a 2:1. So the final list

of stuff is now...

(1) // "coherence" of scale degrees across modes.

Search list archives for "Rothenberg".

(2) tuning coverage; rank order matrix and interval matrix are the same.

I don't think this has much to do with anything. //

(3) linear connectivity; transposing by generator changes one note.

But not less than one --- closed chains are not allowed. Makes

for symmetry at the generator, and prevents the (Rothenberg)

efficiency from becoming very low (see TD 262.14).

(4) fifths; the generator is a fifth in exactly n-1 modes of the scale.

// Choice of fifths is completely arbitrary.

"""

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >http://www-ext.mus.cam.ac.uk/~ic108/PCM/

> >

> >It says that Balzano's coherence is different to Rothenberg's

propriety

> >because it only considers adjacent pairs of intervals. So a

Pythagorean

> >diatonic is still coherent because the conflicting sizes of

tritones

> >aren't considered. Is that right?

>

> I think that's wrong. ;)

>

it is wrong, because balzano, like clough and too many academic

theorists, considers all scales to be subsets of some discrete

cyclic "universe": 12-tone, 20-tone, etc. balzano defines coherence

in terms of units of the "universe set". which means that it, like

maximal evenness, is undefined for a random scale given in cents (or

ratios, or whatever), without any assumed "universe" in which it is

embedded. this is a serious weakness of academic scale theory, in my

opinion.

>>>It says that Balzano's coherence is different to Rothenberg's

>>>propriety because it only considers adjacent pairs of intervals.

//

>>I think that's wrong. ;)

>

>it is wrong, because balzano, like clough and too many academic

>theorists, considers all scales to be subsets of some discrete

>cyclic "universe": 12-tone, 20-tone, etc. balzano defines coherence

>in terms of units of the "universe set". which means that it, like

>maximal evenness, is undefined for a random scale given in cents (or

>ratios, or whatever), without any assumed "universe" in which it is

>embedded. this is a serious weakness of academic scale theory, in my

>opinion.

For sure.

But further, it's wrong unless Balzano has a model that justifies

coherence (which I don't remember him having). Rothenberg assumes

that listeners can rank intervals by size, and gives the condition

required for them to be able to repeatably map what they hear to a

fixed scale. It's obvious, it's simple, and it's probably true.

I can't imagine how one could doctor this so that only adjacent

intervals matter...

-Carl

>So why doesn't Rothenberg get any credit for them?

Presumably because "Mathematical Systems Theory" isn't one

of the journals that the exclusive circle of music theorists

watch.

>And it's not only Balzano -- Browne's given some of them as well.

From the details available in the schemata paper, Browne

rediscovers not only something like propriety ("pattern matchng"),

but also something like efficiency ("position finding").

-Carl

Carl Lumma wrote:

> But further, it's wrong unless Balzano has a model that justifies

> coherence (which I don't remember him having). Rothenberg assumes

> that listeners can rank intervals by size, and gives the condition

> required for them to be able to repeatably map what they hear to a

> fixed scale. It's obvious, it's simple, and it's probably true.

> I can't imagine how one could doctor this so that only adjacent

> intervals matter...

Then did Balzano so doctor it? So that the Pythagorean diatonic embedded in 53-equal would still be coherent? I was only asking if the Pitch Schemata paper had Balzano correct.

Graham

>> But further, it's wrong unless Balzano has a model that justifies

>> coherence (which I don't remember him having). Rothenberg assumes

>> that listeners can rank intervals by size, and gives the condition

>> required for them to be able to repeatably map what they hear to a

>> fixed scale. It's obvious, it's simple, and it's probably true.

>> I can't imagine how one could doctor this so that only adjacent

>> intervals matter...

>

>Then did Balzano so doctor it? So that the Pythagorean diatonic

>embedded in 53-equal would still be coherent? I was only asking if

>the Pitch Schemata paper had Balzano correct.

I was assuming so. I don't remember that from the Balzano paper,

and my copy is in Montana, so...

-Carl

Carl Lumma wrote:

> While many of their criteria were the same, many were not. So it

must

> be considered a strange coincidence that R. and B. wind up

recommending

> the same scale, R in 31-tET, and B in 20-tET. B missed the 31-tET

> version because it didn't have all the other nonsense properties

that

> he was so fascinated with (such as the product of the sizes of the

3rds

> giving the number of notes in the embedding et). Dan Stearns

further

> independently suggested this scale in 20-tET for his own reasons.

But

> AFAIK I'm the first to notice its excellent approximations to 5:3

and

> 7:4 nicely interleaved on its 8ths. The 31-tET version gets closer

to

> JI, but the 20-tET version has higher Lumma stability and already

gets

> you closer than 12-tET to these intervals.

3 3 5 3 3 3 5 3 3 =31

2 2 3 2 2 2 3 2 2 =20

Where does Rothenberg give this scale? I've got the three

Mathematical Systems Theory papers in Manuel's biblography, and can't

find it.

The generator approximates 11:8. That ties in with the 5:3 and 7:4 if

you temper out 385:384. I can get a linear temperament from the best

(not nearest prime) approximations of 31- and 20-equal:

23/51, 541.9 cent generator

basis:

(1.0, 0.45158026468779938)

mapping by period and generator:

[(1, 0), (7, -12), (10, -17), (1, 4), (3, 1)]

mapping by steps:

[(31, 20), (49, 32), (72, 47), (87, 56), (107, 69)]

highest interval width: 28

complexity measure: 28 (31 for smallest MOS)

highest error: 0.009059 (10.871 cents)

unique

It should come up in a search for pairs of the simplest 30 ETs with a

consistency cutoff of 0.8. There are lots of other consonances you

get in the 31-equal version of the scale that don't match this

> The lower stability of Rothenberg's scale kicks it down to position

6

> on my gd spreadsheet -- Balzano's version is at position 3, just

below

> the pentatonic and diatonic scales...

>

> http://lumma.org/stuff/gd.xls

For those who can't read Excel, here's the ranking:

05_pentatonic

07_diatonic

09_balzano-20

08_octatonic

10_subset-13

09_rothenberg

10_blackwood

10_pent-major

06_hexatonic

09_trichordal-433

10_sym-major

08_nova

08_star

07_qm(2)

09_quadrafourths

07_hungarian-minor

06_super7

07_porcupine

09_trichordal-334

05_sss9-31

10_qm(3)

07_harmonic-minor

07_melodic-minor

10_quadrafourths

10_lumma-x2-1

06_sss9-22

08_kleismic

07_neutral-thirds-b-31

10_miracle

09_orwell

07_hungarian-major

10_gamelion

09_tetra-major

06_sss9-31

10_lumma-x2-2

06_mode6

> All the scales are available as scala files...

>

> http://lumma.org/stuff/gd-scl.zip

Graham

>Where does Rothenberg give this scale? I've got the three

>Mathematical Systems Theory papers in Manuel's biblography, and can't

>find it.

I don't actually know, but John Chalmers told me about it, and it's

in the Scala scale archive.

>It should come up in a search for pairs of the simplest 30 ETs with a

>consistency cutoff of 0.8. There are lots of other consonances you

>get in the 31-equal version of the scale that don't match this

Must not be the right linear temperament then?

-Carl

Carl Lumma wrote:

> Must not be the right linear temperament then?

I think it works with Orwell, but the Orwell generator is half the generator of the Rothenberg diatonic. Otherwise, you can't get 6:5, 8:7, 11:8 and 11:9 all at the same time. It's like the diatonic Mark proposed which is half-Magic. In both cases 3:2 isn't produced by the diatonic generator.

Graham