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Message: 4600 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 15:35:27

Subject: Re: My Approach Generalized Diatonicity

From: Carl Lumma

>> >he interval between 5:4 and 6:5 is very high in harmonic
>> entropy.  I don't think any single interval can reasonably
>> be said to serve as a 5:4 and 6:5.
 >

 >if you use an s of 1.2% or 1.5% or so, 11:9 shows up as more 
 >consonant than 9:7 -- but at 11:9 the two most likely interpretations 
 >are '5:4' and '6:5'. this in fact seems to account for the interval 
 >preferences of correspondents such as dan stearns and brian mclaren.


What prefs are those?


>> Phooey!
 >

 >does that 'phooey' apply to 22-equal scales, where the same interval 
 >approximates both 7:5 and 10:7?


It means phooey to this being important:


 >Or that 6:5 can be consonant, but only if it isn't in the root, like
 >fourths are treated in traditional counterpoint.  I don't know how
 >it's going to end up, but other scales might do the same kind of
 >things.

-Carl

Message: 4601 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 03:23:51

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

Hi George, please see:

Yahoo groups: /tuning-math/files/Dave/Sagittal... * [with cont.] 
C2DK.bmp
Yahoo groups: /tuning-math/files/Dave/Sagittal... * [with cont.] 
2DK.bmp

They show my implementation of sagittal notation plan C2.

They show respectively single-symbol and multi-symbol (i.e. with 
conventional sharps and flats) notation of all the steps of 217-ET 
from a double-flat to a double-sharp.

Here's the association of symbols with prime commas again. I'm using 
"w" for "wavy" now instead of "c" for "concavoconvex" (which was too 
much of a mouthful).

Legend:
x convex
s straight
w wavy
v concave

 1     |v      +19
 2    w|     17+
 3     |w    23+
 4    s|      5+
 5     |x      +7
 6     |s      +(11-5)
 7    w|x    17+7
 8    w|s    17+(11-5)
 9    s|x     5+7
10    s|s     5+(11-5)
11    x|x    (11'-7)+7
12     ||v [single]  x|s [multi] (11'-7)+5   ~=  (13'-(11-5))+(11-5)
13    w||    
14     ||w   
15    s||    
16     ||x   
17     ||s   
18    w||x   
19    w||s   
20    s||x   
21    s||s   

The flag apotome-complementation rules are:

s  <->   
w  <-> w
vR <-> xR
except that |x <-> ||x

All criticism and suggestions gratefully received. Feel free to edit 
these bitmaps for your own purposes.

Message: 4602 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 21:22:05

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Hi George, please see:
> 
> Yahoo groups: /tuning- * [with cont.] 
math/files/Dave/SagittalSingle217
> C2DK.bmp
> Yahoo groups: /tuning- * [with cont.] 
math/files/Dave/SagittalMulti217C
> 2DK.bmp
> 
> They show my implementation of sagittal notation plan C2. ...
> 
> All criticism and suggestions gratefully received. Feel free to 
edit 
> these bitmaps for your own purposes.


You have really been busy over the weekend!  I am going to have to 
give all of this a bit of study before replying, so please have 
patience.

However, in reply to what you have done with the symbols, I have 
posted a file here:

Yahoo groups: /tuning- * [with cont.] 
math/files/secor/notation/symbols1.bmp

(The URL broke into two lines, so you'll have to rejoin it before 
using it to access the file.  Also, only you have rights to write to 
a file in a folder that you created, so you'll have to write to your 
own files, while I write to mine.)

I noticed that what I was previously using for conventional symbols 
was a bit different from what is commonly used, so I also made an 
attempt over the weekend to improve on that.  I prefer what you have 
for the conventional sharp and flat symbols, but I suggest a wider 
natural symbol (as shown in the 4th chord, first staff).  You have 
two different double sharp symbols (for the 6th chord); I also came 
up with the same as your second one, which looks good on both a line 
and a space (7th chord).

Per Ted Mook's criticism (found in your own message #24012 of May 30, 
2001 on the main tuning list) about reading new symbols in poor 
lighting at music stand distance, I made the symbols bolder in both 
dimensions, and you will find (for the straight flag symbols only) 
yours (on the second staff) compared with my latest version (on the 
third staff).  I saw no need to make the vertical strokes as long as 
yours, which enables two new symbols altering notes a fifth apart to 
be placed one above the other (first chord on the third staff).  I 
also put the symbols above the staves, making it easier to isolate 
them for study.

Notice that I tried adding some nubs to the right flags to alleviate 
the lateral confusibility problem.  This could also be done with the 
other flags for the larger alteration in each pair.

Another altitude consideration is at the upper right: up-arrows used 
with flats use up a lot of vertical space when the new symbols have 
long vertical lines.

Let's see what agreement we can come to about how the straight-flag 
symbols should look before doing any more with the rest of them.

--George

Message: 4604 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 15:46:10

Subject: Re: My Approach Generalized Diatonicity

From: Carl Lumma

>> >t's two big to claim it as diatonic, in my book.  You're going to
>> hear melodies as a subset of the 41 whether you like it or not.
 >

 >this is where i depart from both mark and carl, maybe. i don't think 
 >11 notes will be enough for the listener to extrapolate a 41-note 
 >universe. and (directed to balzano and his followers), for similar 
 >reasons, i don't think the scale's important properties should depend 
 >in any way upon its tuning as a subset of 41. the scale should be 
 >viewable in its own terms, comprising intervals with some allowed 
 >ranges of values, etc. . . . the reference to a 41-universe, aside 
 >from convenience, should not play any role in evaluating the scale's 
 >suitability or unsuitability. (i make a similar argument about the 
 >diatonic scale vis-a-vis the 12-tone "universe").


That doesn't depart from me at all.  Did you see:


 >I'll mention now that implicit in all of this is the assumption that
 >we want the scale to function as an autonomous thing.  The diatonic
 >scale has interesting subsets, such as the pentatonic scale.  But for
 >our purposes, if we were interested in the pent. scale, we would put
 >it through the list separately.


I see I said "hear melodies as a subset of the 41", which isn't
what I meant.  I meant "the melodies you hear will be in subsets
of the 41; you won't know there are 41.".


 >the wholetone scale, whether used contrapuntally or not, 
 >sounds "floaty" because there's no built-in preference for any tonal 
 >center, or possibility of establishing such a preference through 
 >some 'grammatical' style. however, over a drone, the wholetone scale 
 >becomes just another mode, with about as many opportunities for 
 >tension and release as your typical hindu raga.

Agree 100%.

-Carl

Message: 4605 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 00:08:14

Subject: Re: My Approach Generalized Diatonicity

From: Carl Lumma

>> >ut in general no more than 9, 'cause you'll start dropping stuff, and
>> no fewer than 5, because that would too simple.  You want to have to
>> work a little bit.  Mind-expanding, dude.
 >

 >Except you're also allowing 10?


Yes; my idea is that tetrachordality and chord coverage could buy you
something, but 11 just seems too many.


>> I'll mention now that implicit in all of this is the assumption that
>> we want the scale to function as an autonomous thing.  The diatonic
>> scale has interesting subsets, such as the pentatonic scale.  But for
>> our purposes, if we were interested in the pent. scale, we would put
>> it through the list separately.
 >

 >Does it have to be a fixed set of notes, or are we allowed statistical 
 >definitions?


The entire model assumes a fixed set of notes, but could probably be
successfully extended with statistical definitions.  I don't see how
it would buy you anything when searching for scales, though.


 >You could also say that the flexibility in tuning of a pitch should be 
 >smaller than the difference between pitches.


Sounds like a good idea.  Lumma stability should catch some of this
where flexibility is defined as not breaking propriety.  What did
you have in mind?


>> () series of relative pitches, in particular,
>> the ability to change the absolute pitches
>> but keep the relationships the same.
>> 
>> This is what I call modal transposition.  You can play happy birthday
>> in the relative minor, and it's still happy birthday.  This is what
>> propriety is all about -- when is it no longer happy birthday?
>> Rothenberg says it becomes not happy birthday rather suddenly with
>> respect to smooth, random changes in the pitches of the scale.  Top
>> on my list of Experiments That Should Be Funded are the ones that R.
>> designed to test this.
 >

 >And both stability definitions imply propriety.  Is that right?

Yes.


>> Modal transposition is made up of items 2 and 3 in the Legend.  Item
>> 2, "Modes of the scale function independently".  Efficiency is how
>> many notes of a scale you need to hear before you know what mode
>> you're in.  When it's low, it's bad for mode independence, because
>> the composer has to work harder to get you to accept a New Scale
>> Degree #1, because you still know where the old #1 is.  You want to
>> be a little lost in it.  Similarly, humans can recognize consonant
>> intervals.  If every mode contains a 5:4 as a type of 3rd, then when
>> you hear a 5:4, you know the :4 could be a New Scale Degree #1.  Item
>> 3 in the Legend is propriety, which I won't spill more ink about now.
 >

 >Do you have code for calculating efficiency?  I never got the hang of
 >it.  I'd rather use some measure involving the minimal sufficient sets.


That is efficiency.  R. claims to have trick of enumerating the sets
easily.  Manuel might know more.  I forget if I ever was able to get
him R.'s code, but he obviously can do it in scala.


 >The classic diatonic has lots of 3 note MSSs (any 3 notes including a 
 >tritone) which means it's easy to establish the key when you want to.
 >But it also has two 6 note insufficient sets.  That is, take out either
 >note not involving the tritone and you don't know what key you're in.
 >That means you can also leave the key ambiguous when you want to, which
 >is good for modulation.


Agree, if you meant "either note involving the tritone".


 >The "in a majority of a scale's modes" means it belongs to the majority
 >of intervals of that diatonic class, does it?
 >

>> Notice that I don't generalize things about tonal music, try to pick
>> which modes are tonal, etc., as Erlich has done.  It's all fine and
>> good, but to insist on stuff like characteristic dissonances is to
>> put too much stock in the historical evolution of Western music, in
>> my opinion.
 >

 >So you're really after modality rather than tonality?


Yes.  Once you can do modality, I suspect tonality or something
equally interesting could follow in any number of ways, and I don't
want to cut anything good out by mistake.  But it is possible I
just don't understand tonality.


>> () the simultaneity of two or more series
>> that do one or both of the above.
>> 
>> This is the big one, which I call the Diatonic property, because it
>> weeds the large number of proper, efficient, 5-10 tone scales down
>> to very few indeed.  Pay close attention.
 >

 >I still don't understand what that means, in the light of the
 >explanation.


I might even say that I see diatonicity as making very precise the
difference between additive synthesis with fancy envelopes and music.
We change the fundamental to get melody.  That's the first break.

If we want to play two melodies at once, we have to make sure the
virtual fundamentals don't move in parallel with one of them or the
voices will timbre-fuse.  Power chords on electric guitars lead to
a melodic style, for example.  One might say the diatonic scale
prevents the virtual fundamental mechanism from being used to chunk
the two voices, by scrambling the melody there in the time domain
a bit.


>> If you harmonize in 6ths, however, you get consonance, but not
>> synthesis.  I call it "series-breaking" or "timbre-breaking" harmony.
>> It's the last item in the Legend.
 >

 >This is one thing Mark Gould's process leads to, when the alternating 
 >intervals are consonant.  But it's also stricter because it means all
 >the intervals have to be consonant, and they have to alternate.


Only the majority must be consonant, and they don't have to strictly
alternate -- just trade places once in a while, in some fashion.


 >And it doesn't have the criterion about the consonances not being
 >ambiguous.


In the propriety-breaking sense?  I don't have Mark's paper, but
I just remembered Gene implying he found it on-line.  Anyway, I
deal with ambiguities when determining the stability of the scale,
but can't think what they would hurt by occurring on the interval
class(es) responsible for the diatonic property.


 >So his 11 from 41 note scale may be worth a look.  It can be based
 >on a 7:8:9 chord.  It's outside the Miller limit, but I couldn't find
 >anything better in the "diatonics".  There may be something in the
 >"pentatonics".


It's two big to claim it as diatonic, in my book.  You're going to
hear melodies as a subset of the 41 whether you like it or not.
If some stable subsets have the diatonic property (unstable ones
wouldn't be the subsets you'd wind up hearing), then they would be
candidates.  Else it goes into the category of making a diatonic
harmony with pitches outside the core melodic scale, which damages
something in my opinion, but is still interesting.


 >The consonances of this class don't have to be the only ones adding
 >up to the criterion (5) consonance, do they?  So the generalized fifth
 >doesn't have to be an odd number of diatonic scale steps.

God, no.


>> Could we sing one voice in the scale, and create a timbre-breaking
>> harmony for it _outside_ the scale?  Yes, we could.  And I'd love
>> us to.  But this probably would interfere with following series
>> of absolute and relative pitches.
 >

 >But how about using contrapuntal harmonisation?  This one looks like
 >it isn't so useful in that context.


Either you're hearing a lot of the same relationship or you're not.
Try writing contrapuntal music in the wholetone scale and get back
to me.

-Carl

Message: 4606 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 16:03:32

Subject: the harmonic series segment test

From: Carl Lumma

>I can't find anything else promising by this method.  6:7:8 works with
 >the classic pentatonic.  7:11 is only an even number of steps in
 >3-equal, below 43-equal.  5:7 gives a 9 from 11 scale, but it's nearly
 >50 cents from 5:6:7 with the optimal tuning.  The ideal defining chord
 >is large and simple enough to be comprehensible, and has internal
 >intervals close enough to be the same diatonic class, but also
 >different enough to have a good approximation in the chromatic.


Let's run 8:9:10:11:12:14 through my list.
 * [with cont.]  (Wayb.)

1. yes

2. yes; 5 < 6 < 10

3. 0.36 according to Scala 1.8; diatonic scale in 12-et is 0.77
This is failing.

4. R. stability 0.87 by Scala 1.8; diatonic scale in 12 is 0.95.
This passes.

5. Fails.

6. Passes.

7. Fails.

8. Passes.

The scale has merit, but without 3 and 5 fails to be diatonic.
Since the vast majority of subsets of this scale very strongly
invoke the same root, modality can't happen.  I didn't look for
this specifically, but 3 & 5 caught it, as they always will.

6:7:8:9:10:11:12, another scale with merit, should fail in a
similar way.

-Carl

Message: 4607 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 00:21:57

Subject: Re: My Approach Generalized Diatonicity

From: Carl Lumma

>> >ttp://lumma.org/gd.txt
 >

 >The rules on this page are fairly useless unless there are definitions
 >to go with them. Maybe you and Monzo could get together and define
 >Rothenberg efficient, and so forth?


I thought he had.  Anyway, turnabout is fair play when it comes to
posting things dependent on external definitions.  :)

Message # 4044 on the big list may help.

-Carl

Message: 4608 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 00:22:40

Subject: Re: Fairy Cirlces

From: Carl Lumma

>Ford Circle -- from MathWorld * [with cont.] 


Oh, Ford Circles.  I've seen these.

-Carl

Message: 4609 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 17:40:36

Subject: one from the archives

From: Carl Lumma

Anybody have anything to say about this tuning?  How's
the connectivity, Gene?

-C.

!
  "Stairs", 7-limit tuning, 1998 Carl Lumma.
  12
!
  36/35
  8/7
  6/5
  5/4
  48/35
  10/7
  3/2
  5/3
  12/7
  9/5
  40/21
  2/1
!

Message: 4610 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 17:44:53

Subject: Re: My Approach Generalized Diatonicity

From: Carl Lumma

>> >t means phooey to this being important:
>> 

>>> Or that 6:5 can be consonant, but only if it isn't in the root, 
>>> like fourths are treated in traditional counterpoint.  I don't
>>> know how it's going to end up, but other scales might do the same
>>> kind of things.
 >

 >hmm . . . this seems to agree with some of your own comments, that 
 >whether 5:4 or 6:5 is implied can depend on chordal context or 
 >such . . . no?


Maybe I'm not reading Graham correctly here.

-C.

Message: 4611 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 17:43:10

Subject: Re: the harmonic series segment test

From: Carl Lumma

>> >:7:8:9:10:11:12, another scale with merit, should fail in a
>> similar way.
 >

 >isn't that the same scale, just a different mode?


Um, yes.  :)


 >btw, to hear some wonderful music in this scale, ask prent rodgers 
 >for a copy of his cd. he actually uses the entire tonality diamond, 
 >but at any given point in time, the scale is usually this one.


I've got Prent's cd, and have long held his music as an example of
the workability of series segment scales.  Also: Jules Siegel, which
I believe you have samples of.

-Carl

Message: 4612 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 18:41:00

Subject: Re: My Approach Generalized Diatonicity

From: Carl Lumma

>the wholetone scale, whether used contrapuntally or not, 
 >sounds "floaty" because there's no built-in preference for any tonal 
 >center, or possibility of establishing such a preference through 
 >some 'grammatical' style. however, over a drone, the wholetone scale 
 >becomes just another mode, with about as many opportunities for 
 >tension and release as your typical hindu raga.


Here's one attempt at contrapuntal music in the wholetone scale:

A hexatonic fugue * [with cont.]  (Wayb.)

-Carl

Message: 4613 - Contents - Hide Contents

Date: Mon, 15 Apr 2002 22:39:30

Subject: Re: A common notation for JI and ETs

From: David C Keenan

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

> Judging from your message #4008, you found this one also:
> 
> symb  lft-flgs  rt-flgs
> ------------------------
> 19' =      23 + 19
> 
> but I don't think you mentioned it specifically.


Only recently. I also pointed out that the 19' symbol therefore consists of
two flags on the same side.


> The one thing that still bothers me is that there are two useful 17-
> commas, 2176:2187 and 4096:4131, and neither one is arrived at by a 
> schisma.  Would you consider adding a flag for 4096:4131?


You may have missed one of my messages. I actually asked how much that
bothered you.


>  If the 
> flags for the two 17-commas are used together, we would then have a 
> simple way to notate tones modified by a Pythagorean comma, which 
> might be more useful to theorists than composers, but useful 
> nonetheless.


I figure if we're gonna add yet another flag (is this getting ridiculous?
:-) then it had better give us something else besides just 17'.

There are 3 possible values for an extra flag that would give us the 17'
comma (4096:4131). It could be 17' directly, or 17'-17, or 17'-19.

I had hoped one of these might give us 23 (and all the others that depend
on 23), but none of them do (in 1600-ET).

They might give us a 41-comma symbol.

They might fall near the middle of a big gap in flag-comma sizes.

17' doesn't fall in any big gaps (it's very near 23). 17'-19 falls near the
middle between 17 and 23. 17'-17 falls near the middle between 19 and 17.

Only one of the three gives us a 41-comma symbol, albeit with 3 flags on
the same side! That's 17'-17, which is 288:289, 6.0008 cents.

The (17'-17) comma is the same size as the 19 comma (1 step) in 217-ET,
253-ET and 311-ET. So it doesn't help us notate any higher ET than 217. But
who cares.

Here's how we name the commas:

Name Comma    Name Comma
-----------------------------
5  80:81
7  63:64
11 32:33      11' 704:729
(11-5) 54:55
13 1024:1053  13' 26:27
17 2176:2187  17' 4096:4131
(17'-17) 288:289
19 512:513    19' 19456:19683
23 729:736    23' 16384:16767
29 256:261
31 243:248    31' 31:32
37 999:1024   37' 36:37
41 81:82
43 128:129
47 47:48

Assume we have a flag for each of the following 8 commas.

5
7
(11-5)
17
(17'-17)
19
23
29

Now here's how to make a symbol for each comma, using the flags for the
above commas, and schismas valid in 1600-ET, with a possible allocation to
left and right flags.

Symbol  Left    Right
for     flags   flags
------------------------------
5   =       5
7   =           7
11  =       5 + (11-5)
11' =      29 + 7
13  =       5 + 7
13' =      29 + (11-5)
17  =      17
17' = 17 + (17'-17)               [2 left flags]
19  =           19
19' =           23 + 19           [2 right flags)
23  =           23                [also 17 + 19 + 19]
23' =      17 + (11-5)
29  =      29
31  =           19 + (11-5)       [2 right flags]
31' =       5 + 23 + 23           [also 5 + 7 + (17'-17)]
37  = 29 + 17
37' =  5 +  5 + 19                [also  5 + 17 + 23, also 19 + 7 + 23]
41  = 17 + (17'-17) + (17'-17)    [3 left flags]
43  =(17'-17) + 19 + 19
47  =           19 + 23 + 23      [also 5 + 17 + (17'-17)]

In plan C, as symbolised in my recent bitmaps, there's one flag that hasn't
been used. That's a concave left flag (vL). I propose we use that for
(17'-17).

So my proposal for the flags is

         |  Left    Right
---------+---------------
Convex   |  29        7
Straight |   5     (11-5)
Wavy     |  17       23
Concave  | (17'-17)  19

Want to propose an alternative?


-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)

Message: 4614 - Contents - Hide Contents

Date: Tue, 16 Apr 2002 03:27:08

Subject: Re: the harmonic series segment test

From: Carl Lumma

>> >. 0.36 according to Scala 1.8; diatonic scale in 12-et is 0.77
>> This is failing.
>> 
>> 4. R. stability 0.87 by Scala 1.8; diatonic scale in 12 is 0.95.
>> This passes.
 >

 >How do you get Scala to compute these?

show data

-Ca.

Message: 4615 - Contents - Hide Contents

Date: Tue, 16 Apr 2002 14:49 +0

Subject: Re: the first six criteria

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <4.2.2.20020416032156.00b2c128@xxxxx.xxx>
Carl Lumma wrote:


> Scales that pass all of the first six criteria


Hey, there are a lot of these!  They're all strictly proper, though.  Can 
any of them be altered to have ambiguous intervals?


> 10- Decatonic MOS in MIRACLE
>      [0 7 14 21 28 35 42 49 56 63 72]
>       3. efficiency 0.73
>       4. strictly proper
>       5. yes; 9th is 7:4 in 8 of 10 modes
>       6. no.


The "5th" could be either 7:5 or 10:7 for (6).  That gives 4:7:10 
1/(4:5:7) as triads.  It's (5) I'm worried about because you're 
overlooking 12:7.  I suppose it could make some sense to draw the cutoff 
between 10:7 and 12:7.


                       Graham

Message: 4616 - Contents - Hide Contents

Date: Tue, 16 Apr 2002 17:47:09

Subject: scala stability logic

From: Carl Lumma

Manuel,

Any reason you don't display Rothenberg stability for improper
scales?

-Carl

Message: 4617 - Contents - Hide Contents

Date: Tue, 16 Apr 2002 13:15 +0

Subject: Re: My Approach Generalized Diatonicity

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a9fiuo+lh4v@xxxxxxx.xxx>
Me:

>> Terhardt 
>> relates the traditional root to virtual pitch, so major and minor 
> triads 

>> come out the same.

Paul:

> they don't tend to come out the same in first inversion, though -- 
> the perception of a first-inversion minor triad is very close to the 
> perception of the root-position major triad with two notes (bottom 
> major third) in common.


The root finding algorithm given in 
<Harmony * [with cont.]  (Wayb.)> is clearly 
octave-equivalent.  He actually says "by definition, octave-equivalent 
pitches are not distinguished".

Me:

>> As long as we have a variety of consonances to hand, it shouldn't 
> matter 

>> if they're all the same diatonic interval class or not.

Paul:

> it matters in tonal music. this is why the symmetrical decatonic 
> scale in 22-equal, which is an MOS (we now know), 'wants' to 
> be 'broken' into the non-MOS pentachordal decatonic. 
> omnitetrachordality implies that not only can one find one's place 
> within the octave, but that the correct placement of all the pitches 
> in the scale can be grasped in as simple and perceptually direct a 
> manner as possible.


That's an argument for a tetrachordality, not one diatonic interval class 
having all the consonance.

Given that special casing fourths is phooey, that means Palestrina 
counterpoint recognises 5 diatonic consonances within the octave -- 
unisons, thirds, fourths, fifths and sixths.  Why does pitch placement go 
to the four winds when you increase it to 7?


                Graham

Message: 4619 - Contents - Hide Contents

Date: Tue, 16 Apr 2002 13:15 +0

Subject: Re: My Approach Generalized Diatonicity

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <4.2.2.20020415110352.01eac2e8@xxxxx.xxx>
Carl Lumma wrote:


>>> The entire model assumes a fixed set of notes, but could probably be
>>> successfully extended with statistical definitions.  I don't see how
>>> it would buy you anything when searching for scales, though.
>> 

>> One way of interpreting the 7 from 10 MOS, with propriety grid
>> 
>> 1  2  1  2  1  2  1
>> 3  3  3  3  3  3  2
>> 4  5  4  5  4  4  4
>> 6  6  6  6  5  6  5
>> 7  8  7  7  7  7  7
>> 9  9  8  9  8  9  8
>> 
>> is that the 3 could be either 5:4 or 6:5.
> 

> The interval between 5:4 and 6:5 is very high in harmonic
> entropy.  I don't think any single interval can reasonably
> be said to serve as a 5:4 and 6:5.  If you claim 11:9 is
> consonant, I'd consider it.


If 11:9 is consonant, so is almost everything else, so there's no way (5) 
can be fulfilled.


> Something has got to change, to change the context in which
> the single interval is heard.  Triads where the other interval
> changes would count.  Maybe even octave registration changes
> could count.


Yes, the tuning changes.


>> You could say 7:6 is allowed as a consonance,
> 

> And I do.


Well, not in another message you don't.  Still the problem is that if 7:6 
counts, so should 7:5 and 10:7 and we lose (5) again.


>> but only in the diatonic pitch class with 3.
> 
> ?


Back to that grid

1  2  1  2  1  2  1
3  3  3  3  3  3  2
4  5  4  5  4  4  4
6  6  6  6  5  6  5
7  8  7  7  7  7  7

>9  9  8  9  8  9  8


In the top row, 2 could be 9:8 or 8:7, and in the second row it could be 
8:7 or 7:6.  So that means 2 and 3 are both consonances, and 2/7 is the 
only interval class where 2/10 is this consonance.  It also runs into 
problems when intervals add up, hence:


>> Or that 6:5 can be consonant, but only if it isn't in the root, like
>> fourths are treated in traditional counterpoint.  I don't know how
>> it's going to end up, but other scales might do the same kind of
>> things.
> 
> Phooey!


If major thirds and perfect fifths are both legal consonances, why not put 
them together?


> Oh, you mean to actually re-tune it one way or the other, not just
> have it be re-interpreted in the Erlich sense.  If you can do this
> without breaking propriety, fine.  But then there would be no point
> in expressing the scale without the adjustments in the first place.


Yes.  There are problems in getting thirds and fourths to work together.  
The scale also ends up strictly proper, and I don't want that.  The main 
reason is that it comes out distinctly unexciting:

0 1  3v 4^ 6  7  9v 0
 3  5  5  5  3  5  5
 4  7  6  7  4  7  6

The propriety grids are like this:

 3  5  5  5  3  5  5
 8 10 10  8  8 10  8
13 15 13 13 13 13 13
18 18 18 18 16 18 18
21 23 23 21 21 23 23
26 28 26 26 26 28 26

 4  7  6  7  4  7  6
11 13 13 11 11 13 10
17 20 17 18 17 17 17
24 24 24 24 21 24 23
28 31 30 28 28 30 30
35 37 34 35 34 37 34

so it's strictly proper.  I don't have Scala to hand, so I can't work out 
the efficiency.  But it's similar to, and will probably be heard as, 
12-equal melodic minor.  Rothenberg gives an efficiency of 63% for that.  
Why isn't it on your list?


>> If those consonances can be ambiguous, it does open up new ways of 
>> interpreting the 7 from 10 scale, which is what I'm looking at now.
> 

> You mean Rothenberg ambiguous, or ambiguous in the sense that they
> may have more than one harmonic series representation?

Rothenberg ambiguous.


>> When I originally tried it out, after he mentioned it on the big 

> list, I  >found myself not using notes too far from the tonic.
> 
> Any subsets in particular?


The notes nearest the tonic.  One down, four up, or something.


>> In that case less equal chords, like 7:8:11 might work.
> 
> You bet!


It should be possible to automate the search.  I'll think about it.


>> As long as we have a variety of consonances to hand, it shouldn't 

> matter  >if they're all the same diatonic interval class or not.
> 
> Wrong!  That's what diatonicity is all about -- tying scale objects
> to harmonic objects.  There are a million ways to have lots of
> different consonances, but very few to have them make sense in terms
> of scale intervals.


But isn't that what stability gives us?


                     Graham

Message: 4620 - Contents - Hide Contents

Date: Tue, 16 Apr 2002 14:31:16

Subject: Re: the first six criteria

From: Carl Lumma

>> >cales that pass all of the first six criteria
 >

 >Hey, there are a lot of these!


I mainly searched scales that I thought would pass,
which I've been collecting for years.


 >Can any of them be altered to have ambiguous intervals?


If you want.


>> 10- Decatonic MOS in MIRACLE
>> [0 7 14 21 28 35 42 49 56 63 72]
>> 3. efficiency 0.73
>> 4. strictly proper
>> 5. yes; 9th is 7:4 in 8 of 10 modes
>> 6. no.
 >

 >The "5th" could be either 7:5 or 10:7 for (6).  That gives 4:7:10 
 >1/(4:5:7) as triads.


That's a 6th, I believe, but good catch!  I don't normally consider
7:10 consonant, but with a 4: on the bottom, it is very consonant.
Just goes to show that the by-hand method I was using isn't very
good.


 >It's (5) I'm worried about because you're overlooking 12:7.


Because the rule says that degree can have no other consonances?
Yes, that is a concern.  I'm thinking of dropping that rule, though.


 >I suppose it could make some sense to draw the cutoff 
 >between 10:7 and 12:7.


I think 12:7 can be consonant on a good day, and it can be very
consonant in the septimal minor sixth tetrad, though I believe it
will be able to function as a dissonance next to 7:4 in the above
case.  And I'm still deciding how important this "characteristic
dissonance" like rule is.

-Carl

Message: 4621 - Contents - Hide Contents

Date: Tue, 16 Apr 2002 14:49 +0

Subject: Re: My Approach Generalized Diatonicity

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a9fife+lpo6@xxxxxxx.xxx>
emotionaljourney22 wrote:


> this is where i depart from both mark and carl, maybe. i don't think 
> 11 notes will be enough for the listener to extrapolate a 41-note 
> universe. and (directed to balzano and his followers), for similar 
> reasons, i don't think the scale's important properties should depend 
> in any way upon its tuning as a subset of 41. the scale should be 
> viewable in its own terms, comprising intervals with some allowed 
> ranges of values, etc. . . . the reference to a 41-universe, aside 
> from convenience, should not play any role in evaluating the scale's 
> suitability or unsuitability. (i make a similar argument about the 
> diatonic scale vis-a-vis the 12-tone "universe").


Where is Mark these days?  The 11 from 41 should work fine so long as you 
can distinguish 3 and 4 steps.  You can temper it a bit to improve that.  
It'd actually work in 19-equal if you turn

4 3 4 4 4 3 4 4 4 3 4

into

2 1 2 2 2 1 2 2 2 1 2

Now, I'm wondering how

7 4 4 7 4 4 7 4

or

3 2 2 3 2 2 3 2

would do.  Here's the grid

 7  4  4  7  4  4  7  4
11  8 11 11  8 11 11 11
15 15 15 15 15 15 18 15
22 19 19 22 19 22 22 19
26 23 26 26 26 26 26 26
30 30 30 33 30 30 33 30
37 34 37 37 34 37 37 34

9:7 is still the primary consonance and the secondaries are 6:5 and 8:7.  
There isn't a 9-limit consonance at 18 steps from 41 is there?  Oh, 
0-11-15 doesn't work unless you allow 16:15 and 15:14.

The grid in 19=

 3  2  2  3  2  2  3  2
 5  4  5  5  4  5  5  5
 7  7  7  7  7  7  8  7
10  9  9 10  9 10 10  9
12 11 12 12 12 12 12 12
14 14 14 15 14 14 15 14
17 16 17 17 16 17 17 16


                    Graham

Message: 4622 - Contents - Hide Contents

Date: Tue, 16 Apr 2002 19:56:28

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

>> Judging from your message #4008, you found this one also:
>> 
>> symb  lft-flgs  rt-flgs
>> ------------------------
>> 19' =      23 + 19
>> 
>> but I don't think you mentioned it specifically.
> 

> Only recently. I also pointed out that the 19' symbol therefore 
consists of
> two flags on the same side.
> 

>> The one thing that still bothers me is that there are two useful 
17-
>> commas, 2176:2187 and 4096:4131, and neither one is arrived at by 
a 
>> schisma.  Would you consider adding a flag for 4096:4131?
> 

> You may have missed one of my messages. I actually asked how much 
that
> bothered you.


You were concerned about keeping one comma per prime, and I didn't 
want to pursue the issue any further at the time, because I wan't 
sure how many more of these would be needed.  As it turns out, 17 
looks like the only prime with this situation, and I personally feel 
that the 17-as-flat scale function (hence 4096:4131) is musically 
more useful (Margo Schulter uses the 14:17:21 triad) than the 17-as-
sharp function (2176:2187).  However, we have seen that 2176:2187 is 
more useful in combination with other flags, while a use for 
4096:4131 in combination has yet to be found.  (I'm not trying to 
make a case concerning which of the two is more important, but rather 
to make the point that each one is as important as the other.)

Since we have 4 flag types and 7 ratios for flags, then using the 
remaining flag for 4096:4131 (either directly, or using an 
alternative such as those you gave in the rest of your message) would 
make the 8th one, giving us four pairs of flags.

So far this is in agreement with what you proposed in the rest of 
your message.  I haven't had time to work through all the details of 
that yet (as well as some of the previous things that you sent 
recently about this), but I was expecting to see the 23 flag on the 
left.  So I need to take my time and study this carefully before 
replying.


> So my proposal for the flags is
> 
>          |  Left    Right
> ---------+---------------
> Convex   |  29        7
> Straight |   5     (11-5)
> Wavy     |  17       23
> Concave  | (17'-17)  19
> 
> Want to propose an alternative?


Maybe -- and maybe not.  We shall see.

--George

Message: 4623 - Contents - Hide Contents

Date: Tue, 16 Apr 2002 15:09 +0

Subject: Re: My Approach Generalized Diatonicity

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <memo.579417@xxx.xxxxxxxxx.xx.xx>
I wrote:


> so it's strictly proper.  I don't have Scala to hand, so I can't work 
> out the efficiency.  But it's similar to, and will probably be heard 
> as, 12-equal melodic minor.  Rothenberg gives an efficiency of 63% for 
> that.  Why isn't it on your list?


It's even worse!  It's the usual diatonic, at least in 31-equal.  But 
still, why isn't melodic minor on the list?


                      Graham

Message: 4624 - Contents - Hide Contents

Date: Tue, 16 Apr 2002 15:17:44

Subject: Re: My Approach Generalized Diatonicity

From: Carl Lumma

>> >he interval between 5:4 and 6:5 is very high in harmonic
>> entropy.  I don't think any single interval can reasonably
>> be said to serve as a 5:4 and 6:5.  If you claim 11:9 is
>> consonant, I'd consider it.
 >

 >If 11:9 is consonant, so is almost everything else, so there's no
 >way (5) can be fulfilled.


We agree, then, that 11:9 is not really consonant by itself.


>>> You could say 7:6 is allowed as a consonance,
>> 

>> And I do.
 >

 >Well, not in another message you don't.  Still the problem is that if 7:6 
 >counts, so should 7:5 and 10:7 and we lose (5) again.


I generally think of 7:6, 7:5 as barely strong enough, and 10:7 too weak,
to be considered consonant dyads.

Which scale are you talking about loosing (5) in?


>>> but only in the diatonic pitch class with 3.
>> 
>> ?
 >

 >Back to that grid
 >
 > 1  2  1  2  1  2  1
 > 3  3  3  3  3  3  2
 > 4  5  4  5  4  4  4
 > 6  6  6  6  5  6  5
 > 7  8  7  7  7  7  7
 > 9  9  8  9  8  9  8
 >
 >In the top row, 2 could be 9:8 or 8:7, and in the second row it could be 
 >8:7 or 7:6.


Granted for the sake of argument...


 >So that means 2 and 3 are both consonances, and 2/7 is the 
 >only interval class where 2/10 is this consonance.

??

 >It also runs into problems when intervals add up, hence:
 >

>>> Or that 6:5 can be consonant, but only if it isn't in the root, like
>>> fourths are treated in traditional counterpoint.  I don't know how
>>> it's going to end up, but other scales might do the same kind of
>>> things.
>> 
>> Phooey!
 >

 >If major thirds and perfect fifths are both legal consonances, why not
 >put them together?


You've lost me completely.


>> Oh, you mean to actually re-tune it one way or the other, not just
>> have it be re-interpreted in the Erlich sense.  If you can do this
>> without breaking propriety, fine.  But then there would be no point
>> in expressing the scale without the adjustments in the first place.
 >

 >Yes.  There are problems in getting thirds and fourths to work together.  
 >The scale also ends up strictly proper, and I don't want that.  The
 >main reason is that it comes out distinctly unexciting:
 >
 >0  1  3v 4^ 6  7  9v 0
 >  3  5  5  5  3  5  5
 >  4  7  6  7  4  7  6
 >
 >The propriety grids are like this:
 >
 > 3  5  5  5  3  5  5
 > 8 10 10  8  8 10  8
 >13 15 13 13 13 13 13
 >18 18 18 18 16 18 18
 >21 23 23 21 21 23 23
 >26 28 26 26 26 28 26


Isn't this just the diatonic scale?


 > 4  7  6  7  4  7  6
 >11 13 13 11 11 13 10
 >17 20 17 18 17 17 17
 >24 24 24 24 21 24 23
 >28 31 30 28 28 30 30
 >35 37 34 35 34 37 34


Doesn't look very consonant.


 >so it's strictly proper.  I don't have Scala to hand, so I can't work out 
 >the efficiency.  But it's similar to, and will probably be heard as, 
 >12-equal melodic minor.  Rothenberg gives an efficiency of 63% for that.  
 >Why isn't it on your list?


The Harmonic and Hungarian minors should be in there too.  I'll add
them.


>>> If those consonances can be ambiguous, it does open up new ways of 
>>> interpreting the 7 from 10 scale, which is what I'm looking at now.
>> 

>> You mean Rothenberg ambiguous, or ambiguous in the sense that they
>> may have more than one harmonic series representation?
 >
 >Rothenberg ambiguous.


I'll be anxious to see the results.


>>> As long as we have a variety of consonances to hand, it shouldn't 

>> matter  >if they're all the same diatonic interval class or not.
>> 
>> Wrong!  That's what diatonicity is all about -- tying scale objects
>> to harmonic objects.  There are a million ways to have lots of
>> different consonances, but very few to have them make sense in terms
>> of scale intervals.
 >

 >But isn't that what stability gives us?


No, stability ties acoustic objects with scales objects.  Diatonicity
restricts things further to acoustic objects which are harmonious.

-Carl

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