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Message: 3050 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 10:56:51

Subject: Re: Enneadecal?

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Gene, you didn't reply:
> 
> 'Gene, your "Enneadecal" comma should have a power of 2 equal to 14, 
> not 15 as you said, right?'


Right. Commas are small...

Message: 3051 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:07:20

Subject: Re: A 72-et decatonic

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

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

>> There were 66 scales of this type, but one was something of a 

> standout, so I'll give just it:
>> 

>> [0, 5, 14, 19, 28, 33, 42, 49, 58, 63]
>> [5, 9, 5, 9, 5, 9, 7, 9, 5, 9]
>> edges 11 24 34   connectivity 0 4 6
> 

> Are you now taking into account _all_ the consonances of 72-tET?


I'm doing the same thing as before--looking at the 5, 7, and 11-limits. I don't know what you mean by "all the consonances",
but if you define this by means of a particular list of intervals
which you think amount to that, this could be done.

Message: 3052 - Contents - Hide Contents

Date: Mon, 7 Jan 2002 11:56 +00

Subject: Re: Optimal 5-Limit Generators For Dave

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a1atde+mhm5@xxxxxxx.xxx>
Paul wrote:


> No, I don't think this is torsion at all! It's a different 
> phenomenon altogether, for which I gave the name "contortion".


It gives the same wedge product as unison vectors with torsion.

Paul:

>>> That's not a just interval.

Me:
>> So?

Paul:

> You said "just interval".


I also said I wasn't considering systems with this contorsion.


Paul:

> It should be quite straightforward to prove. How could you tell 
> whether 50:49 produces torsion or not in an octave-invariant 
> formulation?


Do you care about it being [dis]proven, then?  I expect your algorithm for 
generating periodicity blocks will solve everything.  But I haven't looked 
it up because people keep saying they aren't interested, while asking more 
and more questions.  It won't change anything musically.


Paul:

> I thought Gene showed that the common-factor rule only works in the 
> octave-specific case.


I don't remember him considering the adjoint, rather than the wedge 
product.  But we may not need it anyway.

Me:

>> Pairs of ETs with 
>> torsion don't work with wedge products either.  It may be that the 

> sign of the 

>> mapping can be used to disambiguate them.  Otherwise, give the 

> range of generators 

>> as part of the definition.

Paul:

> You've lost me. Gene, any comments?


Meaning contorsion here.  The octave-specific wedge product can remove it, 
but not use it.  An octave-equivalent wedge product (the octave-equivalent 
mapping) will treat such systems, wrongly, as requiring a division of the 
octave.  But starting from ETs it does make more sense to use 
octave-specific vectors in the first place.  Perhaps we should only ask if 
unison vectors can work in an octave-equivalent system, in which case this 
problem doesn't apply.

Me:

>> Wouldn't it be nice to say whether or not Fokker's methods would 

> have worked if he 

>> had run into torsion?

Paul:

> I'm pretty sure the answer is no. Gene?


The main thing we've added to Fokker (after Wilson) is the mapping, 
instead of merely counting the number of notes in the periodicity block.  
The Monz-shruti example gives a periodicity block with more notes than you 
need for the temperament, but the mappings still come out.  There are more 
insidious examples of torsion where the mappings don't work either.  The 
problem being that octave-equivalent matrices don't differentiate commatic 
torsion from systems requiring a period that isn't the octave.


                       Graham

Message: 3053 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:08:46

Subject: Re: Some 12-tone, 2-step 46-et scales

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


> If you do that, you take _all_ the commas of 72-tET into account.


I'm not getting you. I'm connecting things via intervals; the commas do not directly enter the picture.

Message: 3054 - Contents - Hide Contents

Date: Mon, 7 Jan 2002 13:52:43

Subject: Re: Distinct p-limit intervals and ets

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>Monz, the links to the tables are outdated. Manuel, could you provide
>the updated links?

Consistency limits of equal temperaments * [with cont.]  (Wayb.) and
Equal temperament step size ranges for consist... * [with cont.]  (Wayb.)

Message: 3055 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:09:46

Subject: Re: Some 12-tone, 2-step 46-et scales

From: paulerlich

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

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

>> I was facinated to discover that the 7,5 system did a little 

better than the completely symmetrical 6,6 system.

> 
> Here are the graphs. Looking at these, 12 might be a good place to 
center.
> 

>> [0, 4, 8, 12, 16, 20, 23, 27, 31, 35, 39, 43]
>> [4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3]
>> edges   24   24   40   connectivity   3   3   6
> 
> Yahoo groups: /tuning-math/files/Gene/graph/g5... * [with cont.] 
> Yahoo groups: /tuning-math/files/Gene/graph/g7... * [with cont.] 
> Yahoo groups: /tuning-math/files/Gene/graph/g1... * [with cont.] 
> 

>> [0, 4, 8, 12, 16, 20, 24, 27, 31, 35, 39, 43]
>> [4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 3]
>> edges   24   25   41   connectivity   3   3   6
> 
> Yahoo groups: /tuning-math/files/Gene/graph/g5... * [with cont.] 
> Yahoo groups: /tuning-math/files/Gene/graph/g7... * [with cont.] 
> Yahoo groups: /tuning-math/files/Gene/graph/g1... * [with cont.] 


The note 31 would be the usual "tonic" of the Modern Indian Gamut 
when equated with the 7,5 system.

Is there any way you could color the connecting lines, say, red for 
ratios of 3, orange for ratios of 5, yellow for ratios of 7, green 
for ratios of 9, and blue for ratios of 11?

Message: 3056 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 19:08:59

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>> >s in, part or parts in the music sharing the same rhythm.
> 

>So what does the sentence,
> 
>"I've never heard a voice in the music that was triads, Paul."
>
>mean. You haven't heard parallel triads? Me either!


I've never heard a voice that played triads, one after the
other.  -C.

Message: 3057 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:11:27

Subject: Re: A 72-et decatonic

From: paulerlich

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

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

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

>>> There were 66 scales of this type, but one was something of a 

>> standout, so I'll give just it:
>>> 

>>> [0, 5, 14, 19, 28, 33, 42, 49, 58, 63]
>>> [5, 9, 5, 9, 5, 9, 7, 9, 5, 9]
>>> edges 11 24 34   connectivity 0 4 6
>> 

>> Are you now taking into account _all_ the consonances of 72-tET?
> 

> I'm doing the same thing as before--looking at the 5, 7, and 11-
>limits. I don't know what you mean by "all the consonances", but if 
>you define this by means of a particular list of intervals which you 
>think amount to that, this could be done.


If you were already doing that all along, what did it mean when you 
said you were only taking one particular comma into account? I though 
it meant you were treating some of the consonant 72-tET intervals as 
dissonances, since they would be broken if not _all_ the members of a 
complete defining basis of commas of 72-tET were being tempered out.

Message: 3058 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 20:04:07

Subject: Re: Distinct p-limit intervals and ets

From: genewardsmith

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:


> Consistency limits of equal temperaments * [with cont.]  (Wayb.) and
> Equal temperament step size ranges for consist... * [with cont.]  (Wayb.)


These
don't contain the same information as I was looking at; I only
considered the standard et val which rounds to the nearest integer for
each prime, and then looked at the first eight unique examples.
I already went farther than Manual's tables, and was pondering such
questions as whether 311 would turn out unique in the 41-limit.

We can define a funtion unq(n) from odd numbers>1, which tells us the
first unique standard et for odd limit n. So, unq(3)=3, unq(5)=9, 
unq(7)=27, unq(9)=?, unq(11)= 58 ... calculating unq to some point
(49?) might be an interesting project sometime.

Message: 3059 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:13:20

Subject: Re: Some 12-tone, 2-step 46-et scales

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


> Is there any way you could color the connecting lines, say, red for 
> ratios of 3, orange for ratios of 5, yellow for ratios of 7, green 
> for ratios of 9, and blue for ratios of 11?


I've
been wishing I could do that very thing. The output is the Maple graph
drawing program output; if I could figure out a way of getting it to
change color, and also of drawing more than one graph at a time, it
could be done, but it doesn't seem to be implimented.

Message: 3060 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 00:00:50

Subject: Re: please simplify equation

From: genewardsmith

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


>   2^[ (8r+1) / (13r+3) ]
> 
> And Paul gave me these equivalent simplifications of it:
> 
> = 2^[ (2r-1) / (3r-1) ]
> 
> = 2^[ (3-r) / (4-r) ] 
> 
> 
> I plotted the numbers of all three of the above formulas
> into a graph, and can see how they're all related linearly.
> Can you explain algebraically what's going on?  Please
> be as detailed as possible.  Thanks.


Not really. My (3r+1)/(5r+1) is (r+9)/19, your (8r+1)/(13r+3) is
(r+18)/31,
and Paul's (2r-1)/(3r-1) = (3-r)/(4-r) = (8-r)/11, so these are not
the same. If you tell me what recurrence you are seeking the limit of,
I'll tell you the answer.

Message: 3061 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:18:03

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>> >'m trying to show that the things in Western music that led
>> to temperament are absent in Indonesian music.
> 

>So what? They may have had their own reasons,


Sure.  What might they be?


>and inharmonicity makes the situations rather different.
>Gamelan intervals are "pastelized", as Margo Schulter says.

?
 

>>> So C major and E major ???
>> 

>> C and F major.
> 

> How did the note A get in there?


It didn't; these aren't triads after all.  Is
there an Ab?  No.  But it sounds like if there
was a note, it'd be major.


>>> 2 minor.
>> 
>> E minor.
>> 

>>> I can't tell 4.
>> 
>> G something.
> 

>You mean you can't hear which notes make up the chord?


I've never heard a voice in the music that was triads, Paul.
Have you?  I've heard triads formed between voices, in
between all other kinds of chords made up of degrees of the
scale.  Nonetheless, I do admit that it sounds like triadic
motion, to me.  It sounds like a progression involving the
chords Cmaj, Emin, Fmaj, and G... something.  Obviously, the
note, when it's there, is B natural.

-Carl

Message: 3062 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 00:23:56

Subject: More 72-et hexatonics

From: genewardsmith

These I found starting from 2401/2400~1, which is a very small 7-limit
comma. Perhaps not surprisingly, the result is actually RI in the
7-limit; but the 11-limit results are quite good.

[0, 9, 23, 35, 49, 58]
[9, 14, 12, 14, 9, 14]
edges   4   9   14   connectivity   0   2   4

[0, 14, 23, 35, 49, 58]
[14, 9, 12, 14, 9, 14]
edges   3   9   14   connectivity   0   2   4

[0, 14, 23, 35, 44, 58]
[14, 9, 12, 9, 14, 14]
edges   3   8   13   connectivity   0   2   4

[0, 14, 23, 35, 49, 63]
[14, 9, 12, 14, 14, 9]
edges   3   8   13   connectivity   0   1   3

[0, 14, 28, 37, 46, 58]
[14, 14, 9, 9, 12, 14]
edges   2   6   11   connectivity   0   0   3

[0, 9, 23, 35, 49, 63]
[9, 14, 12, 14, 14, 9]
edges   2   6   11   connectivity   0   1   2

Six things taken two at a time is 15, so the two scales with 14 edges
are missing only a single consonant interval to be maximally
connected, and even this counts as consonant if we are willing to go
to 15/11. The first scale listed in the 7-limit can be tuned as

1--35/32--5/4--7/5--8/5--7/4

It is based on three steps which avoid the use of 3, so these are in
the 2^a 5^b 7^c system, with scale steps

(35/32)^2 (8/7)^3 (28/25) = 2

Because of the high degee of connectedness in the 11-limit, all kinds
of modal transposition and other games could be played with this
scale, somewhat along the lines of a hexany.

Message: 3063 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:22:28

Subject: Re: A 72-et decatonic

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


> If you were already doing that all along, what did it mean when you 
> said you were only taking one particular comma into account? 


In
the 7-limit, four vals which make a unimodular matrix, and all of
which are positive, can be considered to define the step sizes and
multiplicities of an RI scale. If I take three vals instead, all of
which have a certain comma in the kernel, and such that the 72-et val
or whatever et I am looking at is an integer combination with positive
coefficients of these vals, I get intervals which I can use to
construct 72-et scales. I could also use one or two 11-limit vals, two
7-limit vals (a linear temperament), and so forth.

In general, I might simply take any partition of 72 and its dual, and
look at all permutations. In practice this is far too much to deal,
and I am trying to look at things which would tend to take advantage
of the 72-et commas.

Message: 3064 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 00:31:43

Subject: Re: tetrachordality

From: clumma

>> >hat's the def. in your paper.  But:
>> 
>> () I never understood how it reflects symmetry at the 3:2.
> 

>4:3 more clearly than 3:2. However, you could look at 3:2 spans
>if you wished, and still see a large gulf between the pentachordal
>and symmetrical decatonic scales.


I accept symmetry at 4:3 being tied to symmetry 3:2 in an octave-
equivalent universe.  Just trying to see why we're getting
different results.


>> () "homotetrachordal" is a new term on me.  Are there precise
>> defs. of homo- vs. omni- around?
> 

> Were those not precise enough for you?


Never saw them!


>> How did you choose these prefixes?
> 

> Homo = same -- two 4:3 spans that are the same
> Omni = all -- all octave species are homotetrachordal.


Aha!  Now I've seen them!


>> () We agreed a bit ago that 'the number of notes that change
>> when a scale is transposed by 3:2 index its omnitetrachordality',
>> right?
> 

>We did? I don't see transposition as coming into this -- rather,
>it's a property of the _untransposed_ scale, heard in its full,
>unmodulating glory.


You said something like "aren't we done?  we just count the number
of notes that change...".

I always thought the basis for tetrachordality was that for any
note heard, it's 3:2 transposition was floating in the listener's
ear.  Thus, if _that_ note is then played, it sounds natural (or,
you could say, it makes it easier to sing melodies).  Also, there
was the anecdotal stories about folks at parties singing tunes
a 3:2 off.  So that's an absolute pitch thing.

The interval pattern stuff (the L-L-s of conventional theory) is
a relative pitch thing... ?

-Carl

Message: 3065 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:23:34

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>Carl, it seemed you yourself just gave evidence for the 135:128
>vanishing, didn't you?


They treat it like it's vanished, but I don't think it's
vanished.  This 4th sounds different to me.  I don't think
they've tempered it, I think they just use it, despite it
being out.  Just like in Wilson's keyboard layouts.  How
does it sound to you, and on what recordings does it sound
that way?

Sometime this week, I may be able to make some mp3s, but
don't hold your breath.  It will be busy at work, since
we're starting up after basically a month off, and we're
broke, since nobody's buying serial adapters.

-Carl

Message: 3066 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 00:46:06

Subject: Re: Some 9-tone 72-et scales

From: clumma

>> >sn't it proper
> 

> No: 4 + 4 + 4 > 1 + 4 + 4 + 1.


The point non-strict propriety for the diatonic scale is 12-tET.
It will be improper in any tuning with pos. fifths (22-tET), and
strictly proper in any tuning with negative fifths (meantone).
None of this matters too much, because you still have all the
other properties of the diatonic scale, namely:

() Pitch set under Miller limit of 7-9.

Pitch tracking possible.  Other properties may be applied
to entire scale.

() Low mean variety.
() Tetrachordal.

Singable.

() One interval class gives the same consonance mosts modes (5th).

As easy to locate yourself in a mode as it is in the entire scale
(efficiency).

() One int. class gives different consonances every mode (3rd).

Possible to harmonize with consecutive scale degrees without
timbre-fusing.  Parallel fifths don't work because it's the same
interval every time, and you don't hear two parts.  To put it
another way, you can write harmony parts that themselves are
shaped like the melody.


And since you've already learned the proper map for the diatonic
scale, it's easy to put the 22-tET diatonic in the same map.
Remember, propriety refers to a mental object, _not_ a scale.
Applying it to a scale assumes the listener has a matching map.
When I hear a new scale, I almost always hear subsets of a diatonic
scale.  I've only recently learned to hear the wholetone and
pentatonic scales, and am still working on the diminished scale.

If you read R.'s papers, he _never_ applies the concept of
propriety like you guys tried to here.

-Carl

Message: 3067 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:35:32

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

>>> I'm trying to show that the things in Western music that led
>>> to temperament are absent in Indonesian music.
>> 

>> So what? They may have had their own reasons,
> 

> Sure.  What might they be?


Wanting to play 4/3 together with 15/8 and make them consonant with 
one another, as you observed.


> 

>> and inharmonicity makes the situations rather different.
>> Gamelan intervals are "pastelized", as Margo Schulter says.
> 
> ?


Search for "pastelize".

>  

>>>> So C major and E major ???
>>> 

>>> C and F major.
>> 

>> How did the note A get in there?
> 

> It didn't; these aren't triads after all.  Is
> there an Ab?  No.  But it sounds like if there
> was a note, it'd be major.


Maybe the music moves to one of the other two Pelog pitches on the 7-
tone instruments.


>>>> 2 minor.
>>> 
>>> E minor.
>>> 

>>>> I can't tell 4.
>>> 
>>> G something.
>> 

>> You mean you can't hear which notes make up the chord?
> 

> I've never heard a voice

Voice?

> in the music that was triads, Paul.
> Have you?


No, but you used roman numerals, so I thought you did.


> I've heard triads formed between voices,


Ok, what does the voice thing mean in your first statement above?

Message: 3068 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 01:25:02

Subject: Re: Optimal 5-Limit Generators For Dave

From: paulerlich

--- In tuning-math@y..., graham@m... wrote:


> Me:

>>> But if you mean the case where all consonances are specified 
in 
>>> terms of fifths, but the generator is a half-fifth, I thought 
I 
>> defined 

>>> those out of existence above.
> 
> Paul:

>> Defined those out of existence? I thought you were saying this 
was 
>> the Vicentino enharmonic case.
> 

> Yes, and it can't be unambiguously expressed as an 

octave-equivalent mapping.  It 

> has torsion.


No, I don't think this is torsion at all! It's a different 
phenomenon altogether, for which I gave the name "contortion".


>> That's not a just interval.
> 
> So?


You said "just interval".

> 
> Paul:

>> I think Gene has convinced be that they won't work. The only way 
you 
>> can possibly distinguish cases of torsion correctly is with the 
>> octave-specific mapping.
> 

> I haven't seen that proven yet.


It should be quite straightforward to prove. How could you tell 
whether 50:49 produces torsion or not in an octave-invariant 
formulation?


> Let's get an algorithm first, and see if it 
> doesn't work.  Where do you think torsion is a problem?  An 
octave-equivalent 
> mapping can do everything a wedge product can.  You can add a 

parameter if you want 

> to distinguish torsion from equal divisions of the octave.  In 

going from unison 

> vectors to a mapping, torsion might show up as a common factor in 

the adjoint where 

> it's a problem.  I haven't even got round to checking yet.


I thought Gene showed that the common-factor rule only works in the 
octave-specific case.


> Pairs of ETs with 
> torsion don't work with wedge products either.  It may be that the 

sign of the 

> mapping can be used to disambiguate them.  Otherwise, give the 

range of generators 

> as part of the definition.


You've lost me. Gene, any comments?

> 

>> Fokker didn't run into any cases of torsion, but we have! The 
paper 
>> can cover Fokker's methods but doesn't need to be restricted to 
them.
> 

> Wouldn't it be nice to say whether or not Fokker's methods would 

have worked if he 

> had run into torsion?


I'm pretty sure the answer is no. Gene?

Message: 3069 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:38:29

Subject: Re: A 72-et decatonic

From: paulerlich

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

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 

>> If you were already doing that all along, what did it mean when 
you 
>> said you were only taking one particular comma into account? 
> 

> In the 7-limit, four vals which make a unimodular matrix, and all 

of which are positive, can be considered to define the step sizes and 
multiplicities of an RI scale. If I take three vals instead, all of 
which have a certain comma in the kernel, and such that the 72-et val 
or whatever et I am looking at is an integer combination with 
positive coefficients of these vals, I get intervals which I can use 
to construct 72-et scales. I could also use one or two 11-limit vals, 
two 7-limit vals (a linear temperament), and so forth.

> 
> In general, I might simply take any partition of 72 and its dual, 

and look at all permutations. In practice this is far too much to 
deal, and I am trying to look at things which would tend to take 
advantage of the 72-et commas.

So were you taking all 72-tET consonances into account all along, and 
simply using certain generators?

I'm unclear . . . why don't we go over this process in 12-tone, 
starting from the beginning . . . or better yet, focus on our linear 
temperament paper first. I'd really like to work the "heuristic" into 
it.

Message: 3070 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 01:26:17

Subject: Paper (was Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVEE

From: paulerlich

--- In tuning-math@y..., graham@m... wrote:

> Paul wrote:
> 

>> The proof that MOSs are linear might be said to be published. 
The 
>> periodicity block concept was of course published by Fokker, 
though 
>> the explanation of periodicity blocks might better take off from 
this 
>> starting point, which you are all welcome to suggest changes to:
>> 
>> A gentle introduction to Fokker periodicity bl... * [with cont.]  (Wayb.)
>> 
>> As for the rest, I'm fairly certain it's entirely new work.
> 

> C Karp's "Analyzing Musical Tuning Systems" from Acustica Vo.54 
(1984) 
> should be considered.  He uses octave-specific, 5-limit matrices, 
> including some inverses.  He does say, p.212, "... the temperament 
vector 
> of any interval (a, b, c)_t, is associated with the c/b comma 
division 
> temperament" and works through examples for fractional meantones.
> 
> Brian McLaren sent me a copy, in the days when he deigned to 
recognize 
> mathematical theory.  It acknowledges one "Bob Marvin, who devised 
the 
> matrix representation of tuning systems used here, and introduced 
it to 
> the author."
> 
> 
>                            Graham


Would you send me a copy?

Paul Erlich
57 Grove St.
Somerville, MA 02144

Message: 3071 - Contents - Hide Contents

Date: Mon, 07 Jan 2002 07:40:49

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

>> Carl, it seemed you yourself just gave evidence for the 135:128
>> vanishing, didn't you?
> 

> They treat it like it's vanished, but I don't think it's
> vanished.  This 4th sounds different to me.


I think it's very difficult for ears with Western=trained categorical 
perception not to hear it as different. We're talking about 4ths 
which average about 523 cents, after all! Texturally, though, it 
comes out sounding a lot like the other 4ths, to me.


> I don't think
> they've tempered it, I think they just use it, despite it
> being out.  Just like in Wilson's keyboard layouts.  How
> does it sound to you, and on what recordings does it sound
> that way?


I used to spend lots of time in libraries listening to this stuff. 
Maybe I should start again.

Message: 3072 - Contents - Hide Contents

Date: Tue, 8 Jan 2002 09:00:45

Subject: Dictionary query

From: monz

My Dictionary entries
Definitions of tuning terms: positive system, ... * [with cont.]  (Wayb.)
Definitions of tuning terms: negative system, ... * [with cont.]  (Wayb.)

define positive and negative tuning systems as those
which have "5ths" larger or smaller, respectively,
than the 700-cent 12-EDO "5th".


Isn't that wrong?  Doesn't the 3:2 ratio define the
boundary between positive and negative?  Is there
more than one accepted usage?  Help!



-monz


 




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Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.]  (Wayb.)

Message: 3073 - Contents - Hide Contents

Date: Tue, 8 Jan 2002 18:15:47

Subject: Re: Dictionary query

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

It's correct. The interval of 12 fifths minus 7 octaves
(the Pythagorean comma) defines whether a system is positive
or negative. So with a fifth of 3/2 it's positive because
P is greater than 1/1.

Manuel

Message: 3074 - Contents - Hide Contents

Date: Tue, 8 Jan 2002 17:41 +00

Subject: Re: Dictionary query

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <003f01c19866$03a02e80$af48620c@xxx.xxx.xxx>
monz wrote:


> define positive and negative tuning systems as those
> which have "5ths" larger or smaller, respectively,
> than the 700-cent 12-EDO "5th".

That's correct.

> Isn't that wrong?  Doesn't the 3:2 ratio define the
> boundary between positive and negative?  Is there
> more than one accepted usage?  Help!


You remember when we were at the Huygens Fokker Institute in Amsterdam, 
and I pulled a Bosanquet book off the shelves?  That's where the 
"positive"/"negative" terminology is defined, and it is relative to 
12-equal, not Pythagorean.  If you'd been paying attention, you could 
have checked it.  One of Erv Wilson's early Xenharmonikon articles 
reiterates this, and another (I think "On Linear Notations ...") extends 
it for ETs other than 12.


                    Graham

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