Tuning-Math Digests messages 5750 - 5774

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Message: 5750

Date: Sat, 14 Dec 2002 05:28:39

Subject: Re: Relative complexity and scale construction

From: Carl Lumma

>>Graham complexity tells me the minimum number of notes of the
>>temperament I need to play all the identities in question.
>>Does relative complexity?
> 
>Graham complexity along both generators, and thier product,
>might be what we need. We could attempt to minimize the product.

Sounds right.  I wonder how this follows relative and/or
geometric complexity...

I should point out that I *don't know* the difference between
this notes measure, and geometric complexity measures -- I'm
asking.  What is it exactly that we're trying to measure about
temperaments, from a music-theoretical POV?

-Carl


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Message: 5751

Date: Sat, 14 Dec 2002 05:29:12

Subject: Re: Relative complexity and scale construction

From: Carl Lumma

>>>>Perhaps you don't have enough uvs to close a block, but you're
>>>>certainly on you're way.  No?
>>>
>>>If you think 5-limit JI is on the way to being a block.
>>
>>Maybe Paul can shed some light on this, when he's feeling
>>better.
>>
>>-Carl
>
>i can't.
>
>now answer the question, carl!

What question?

-Carl


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Message: 5756

Date: Sun, 15 Dec 2002 14:38:05

Subject: Re: Relative complexity and scale construction

From: Graham Breed

Me:
>> So you can then add planar temperaments to that graph of size against 
>>accuracy.  They'll appear for high accuracies with a large but not 
>>absurdly large number of notes.  You can also say the number of notes 
>>beyond which a planar temperament is equivalent to a linear temperament 
>>that's consistent with it.

Gene:
> Why
a large number of notes? The Pauline tempered Duodene only has 12, and
I was wrong about then all being covered by tetrads. You can cut the
corners off, and get the following scale:

Oh, I was assuming you'd need a lot of notes to get an accurate 
temperament that's still simpler than JI.

> ! pship.scl
> !
> Pauline (225/224) tempered 10 note scale

Why is this still being called the Pauline temperament?  I showed that

it isn't the temperament Pauline gave.

> This has 4 major and 4 minor triads, 2 major and 2 minor tetrads,
> 2 supermajor and 2 subminor triads, one each of what I call
supermajor and subminor tetrads (1-9/7-3/2-9/5 and 1-7/6-3/2-5/3), and
> a 1-7/6-7/5-5/3 dimininished 7th chord. I wouldn't call this a lot
of notes, but the tempering is very much in evidence.

It isn't that many notes, but it isn't that accurate either.  It's 
simple as planar temperaments go.  I make it equivalent to h12&h60 
beyond 33 notes.  Or 34 notes if you lop the corners off.  36 in 
practice as h12&h60 goes up in steps of 12 notes.


                      Graham


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Message: 5757

Date: Sun, 15 Dec 2002 18:56:26

Subject: Re: Relative complexity and scale construction

From: Graham Breed

I wrote:
> It isn't that many notes, but it isn't that accurate either.  It's 
> simple as planar temperaments go.  I make it equivalent to h12&h60 
> beyond 33 notes.  Or 34 notes if you lop the corners off.  36 in 
> practice as h12&h60 goes up in steps of 12 notes.

Well, that's wrong because my simplification of planar temperaments is 
wrong.  It looks like you get more chords per note if you have equal 
numbers of steps in each generator direction, instead of holding one at 
its smallest value and expanding the scale in only one direction.

Here's a 10 note scale with 2 utonal and 2 otonal 7-limit tetrads when 
you temper out 225:224

    D# A# E#
E  B  F# C#
C  G  D


Expanding in one direction, you get a 13 note scale with 3 of each tetrad

    D# A# E# B#
E  B  F# C# G#
C  G  D  A

And the next in the pattern is a 16 note scale with 4 of each tetrad

    G# D# A# E# B#
A  E  B  F# C# G#
F  C  G  D  A

But there's also this 14 note scale that has 4 of each tetrad

    D# A# E#
E  B  F# C#
C  G  D  A
Ab Eb Bb

So it can be done with 14 notes when the theory predicts 16 (and the 
even cruder theory 18)

18 notes can give 6 of each tetrad

    G# D# A# E#
A  E  B  F# C#
F  C  G  D  A
Db Ab Eb Bb

This is the point where we get extra tetrads with meantone.  Then 
there's a 23 note scale with 9 of each tetrad

    B# Fx Cx Gx
C# G# D# A# E#
A  E  B  F# C#
F  C  G  D  A
Db Ab Eb Bb

By my original formula, it would take 30 notes to get this many tetrads.


So, the complexity of a planar temperament is (i, j) and the scale is 
(I, J).  The number of notes is then I*J.  The number of complete chords 
of each type is (I-i)*(J-j).  This still slightly overestimates the 
number of notes, because some notes in the rectangle won't be used in 
any chords, but that's only a minor correction.

I'm going to set I-i = J-j = n.  Then the number of chords is n**2 (n 
squared) and the number of notes is ij + n(i+j) + n**2.

I've worked out that the number of notes where the number of chords in 
an equal and planar temperament are the same is

((c-ij)/(i+j))**2 + c

where c is the complexity of the linear temperament and (i, j) is the 
complexity of the planar temperament.  Using that formula, the 225:224 
planar temperament doesn't become equivalent to the h12&h60 linear 
temperament until they both have 49 notes.  So the planar temperament 
holds it's own for larger scales than I said before.

Ennealimmal (which is much more accurate) becomes simpler around 60 notes.

It also takes more notes before the planar temperament becomes simpler 
than JI.  This scale has 12 notes and 3 of each tetrad, a feat that 
requires 13 notes for a 225:224 tempered scale.  You'll really need to 
view source and turn off word wrapping for this to come out right.

         C#
        / \
B#------Fx------Cx
  \   / / \ \   /
   \ A-/---\-E /
    \ /     \ /
   / D#------A#\
  /   \ \ / /   \
F-----\-C-/-----G
        \ /
         F#

You can also get 4 of each tetrad from 14 notes by filling in the two 
5-limit triads in the middle (is that a stellated hexany?)  So the 
planar temperament rules where you have more than 14 up to around 49 or 
60 notes -- a large, but not absurdly large number of notes to me.

I may still be wrong on the details, but I think I'm getting closer


                    Graham


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Message: 5761

Date: Sun, 15 Dec 2002 06:54:26

Subject: Re: Relative complexity and scale construction

From: Carl Lumma

> > >
> > >now answer the question, carl!
> > 
> > What question?
> 
> do you think 5-limit JI is on the way to being a block?

No.

The lattices Gene are posting contain pitches
outside of 5-limit JI.

-C.


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Message: 5763

Date: Mon, 16 Dec 2002 12:26:43

Subject: Re: Relative complexity and scale construction

From: Graham Breed

Gene Ward Smith  wrote:

> The 225/224 planar is pretty accurate; you can use 72-et or for more accuracy even 228-et. Why do you say it isn't accurate?

I was comparing it with other 7-limit planar temperaments, not equal 
temperaments.  It's only a marginal improvement on Miracle, whereas some 
planar temperaments can do a lot better (such as those 
Schismic-with-explicit-schisma notations).

But this isn't important.  You can call it "pretty accurate" if you like 
and say that planar temperaments occupy the "pretty accurate" zone.

> I don't have a name listed for this temperament yet, but Duodecimal ought to do.

Why does it need a name?  I didn't think anybody was writing in it. 
I've a feeling it came up on Usenet earlier in the year.

> It is equivalent in the sense that if you use 72-et or
> 228-et
as above, they are Duodecimal compatible; you can describe it as
225/224 plus the Pythagorean comma, though its actual TM reduced basis
is <225/224, 250047/250000>. If you use the mapping
> [[12,19,28,34],[0,0,-1,-2]] for it, you get generators of the
Pythagorean minor second (256/243) and the 81/80~126/25 comma; this
not neccessarily the best setup for 225/224 alone, it seems to me.

It's equivalent in the sense that it tempers out 225:224, and gets 
within 98% of the accuracy of 225:224.

> You can also add commas to get, for instance, Catakleismic or
Miracle, so this isn't uniquely attached to 225/224, though the
connection is close.

225:224 planar's worst error is 79.4% of Miracle's.  You can call that

equivalent if you like.  In which case the equivalence occurs beyond
18 
notes.  Catakleismic is more complex and less accurate, so what's the 
point?  h31&h63 gets to within nearly 85%, and has a complexity of 23.

But as h12&h60's so much more accurate, and only slightly more complex

(24) I went with that.


                      Graham


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Message: 5767

Date: Mon, 16 Dec 2002 23:17:27

Subject: Re: A common notation for JI and ET's

From: M. Schulter

Hello, everyone, and for my first post on tuning-math, I'd like to
propose a new sagittal symbol that might illustrate an approach to the
notation of certain tempered systems not based on an equal division of
an interval such as the octave and involving more than one chain of
fifths.

Before presenting my proposal, I should caution that I am still in my
early beginner's stage of learning sagittal notation, and only within
the last day or so realized that I hadn't been quite clear on the
effect of comma signs in the region between an 11-diesis and a sharp
or flat (apotome-less-comma rather than diesis-plus-comma). Happily,
reading again the relevant portions of the draft by George and Dave
for _Xenharmonikon_ 18 brought this nuance to my attention, so that I
can at least construct my suggested new symbol correctly.

Also, I would caution that especially given my inexperience, this
article is not unlikely to include its share of obvious beginner's
errors, and that I warmly welcome any corrections.

As I discuss in some longer articles which I might want to proofread
and possibly revise accordingly, one approach to the kind of tempered
system I here discuss is a "quasi-JI" transcription which shows
precise or approximate deviations from Pythagorean intonation,
including significant commas which are dispersed in the system. 
Consider, for example, this diatonic scale and a possible "quasi-JI"
representation, with octave numbers (C4=middle C) appearing before the
note names and sagittal signs:

   4C       4D|(      4E)|(    4F!(     4G      4A|(    4B)|(   5C
   1/1     ~44/39    ~14/11   ~4/3     ~3/2    ~22/13  ~21/11   2/1
    0      208.19    416.38   495.90  704.10   912.29  1120.48 1200

Mapping a regular temperament to a JI notation, like mapping a
three-dimensional globe on a 2-D surface, inevitably involves some
distortion. Thus the notation for this scale based on a regular
temperament with fifths around 704.096 cents (the Wilson/Pepper "Noble
Fifth") accurately represents the sizes of many intervals, including
the whole-tones near 39:44 -- 8:9 + 351:352 or |( -- and the diatonic
semitones near 21:22 -- 243:256 less 891:896 or )|(.

From a vertical perspective, the notation here also accurately
suggests that the major thirds are very close to 11:14, or 64:81 plus
891:896 -- e.g. 4C-4E)|( or 4F!(-4A|(.

However, certain small vertical and melodic anomalies occur because
the notation does not actually show the fractional commas by which
each fifth or fourth is tempered -- not quite half of a 351:352, or
1/4 of an 891:896. Thus 4C-4G and 4E)|(-4B)|( are arbitrarily shown as
pure, while 4C-4F!( and 4E)|(-4A are shown as tempered by about a full
351:352 (~4.925 cents), although in reality all of these intervals are
tempered by an identical of amount of ~2.141 cents.

Apart from the issue of such unavoidable distortions, the main
drawback of this "quasi-JI" style of notation is that it is in tension
with the usual sagittal principle of notating routine intervals along
a single chain of fifths with routine symbols. Here 4C-4E)|( indeed
suggests that the size of this regular major third is very close to a
pure 11:14 (~417.51 cents), but the extra symbols might distract from
the basic fact that it is the _usual_ major third to be found simply
by pressing the keys 4C-4E.

Of course, for many linear temperaments based on a single chain of
fifths, one might simply pick a nearby equal temperament whose
standard symbol set fits. However, I here consider the system called
Peppermint 24, with two 12-note chains of fifths (~704.096 cents) at a
distance of ~58.680 cents, the "quasi-diesis" which when added to the
regular major second yields a pure 6:7 minor third (~266.871 cents).

My solution for communicating some intonational information about the
system while following the usual conventions of sagittal notation
within a chain of fifths is to use a symbol rather accurately defining
the size of the quasi-diesis as around 88:91 (11-diesis at 32:33 plus
11:13 comma at 351:352, ~58.036 cents) or 117:121 (32:33 plus 363:364
at about 4.763 cents, ~58.198 cents). The ratios of 88:91 and 117:121,
like 351:352 and 363:364, differ by the harmonisma at 10647:10648
(~0.163 cents).

In JI, 88:91 defines the difference for example between 39:44 and 6:7,
7:11 and 8:13, or 13:22 and 4:7; the very slightly larger 117:121
defines the difference between 11:13 and 9:11, or 22:39 and 6:11.

In a tempered system such as Peppermint 24, these 88:91 or 117:121
relationships are closely approximated, with the first interval of
each of the above pairs as a regular interval along the chain of
fifths, and the second as an interval realized by the addition of a
58.68-cent quasi-diesis.

To show the approximate size of the quasi-diesis, and thus to imply
this type of intonational structure, I propose the following sign
showing a modification of 32:33 or /|\ plus 351:352 or |(.

                             /|\(

I am tempted to call this an "intonational signature," since the
sagittal symbol looks a bit like the usual 11-diesis arrow plus the
sign "(" or "C" associated metrically with "common time" (4/4). Here
the sign could be said to stand for "common temperament," a term
applied to 24-note systems with two chains of fifths at around 704
cents or so spaced so as to optimize certain ratios, and especially
those of 2-3-7-9 (e.g. 6:7, 7:9, 4:7).

In such a scheme, the approximate 88:91 diesis or /|\( serves at once
as the 11-diesis (e.g. 11:12 vs. 8:9) or /|\ in JI, and the septimal
or Archytas semitone at 27:28 (~62.96 cents), or )/|\( as it might be
written in JI (32:33 plus 891:896). The latter interval marks the
difference, for example, between 8:9 and 6:7, or 7:9 and 3:4.

Following this approach, we might notate some characteristic
progressions as follows, with JI approximations given below the
examples:

                                       5E\!!/  5F      5E\!!/  5F
                                       4B\!!/  5C      4B\!!/  5C  
    4G  4F     4E   4F      4G         4F/|\(  4F  or  4G!!!)  4F
    4C  4D\!/( 4E   4D\!/(  4C                            
                                        16/9   2/1
    3/2  4/3  14/11  4/3    3/2          4/3   3/2
    1/1 12/11 14/11  12/11  1/1         28/27  1/1


4A/|\(  4G/|\(        4B!!!) 4A!!!)        4A/|\( 4A      4B!!!) 4A
4G      4G/|\(        4G     4A!!!)        4G     4A      4G     4A
4D/|\(  4C/|\(        4E!!!) 4D!!!)        4E/|\( 4D      4F!)   4D
4C      4C/|\(   or   4C     4D!!!)        4C     4D  or  4C     4D

7/4     14/9                               7/4   27/16
3/2     14/9                               3/2   27/16
7/6     28/27                             21/16   9/8
1/1     28/27                              1/1    9/8

From the alternative notations for some of these examples, it can be
seen that in this kind of tempered system, the "limma complement" of
the 88:91 or /|\( is the representation of the 63:64 Archytas comma or
7-comma, !) -- thus D/|\( is equivalent to E!!!) at a 6:7 above C.

One "refinement" -- if that is the right word -- to go along with this
innovation is an "apotome complement" of sorts for /|\(. On this
point, I will propose a poetic liberty with the sagittal system.

Just as /|\( closely approximates the _absolute_ size of the
quasi-diesis in a tempered system like Peppermint 24, so its proposed
apotome complement also represents an absolute size close to that
obtaining in such a system, where the apotome /||\ or \!!/ has a size
of around 13:14 or ~128.30 cents (in Peppermint 24, actually ~128.67
cents).

Thus this "system-specific" apotome complement is equal to about
169:176 (~70.262 cents), the difference between 13:14 and 88:91, and
also the sum of 26:27 (the 13' diesis, ~65.337 cents) and 351:352. 
I suggest this symbol for the 169:176, or also 121:126 (~70.100 cents)
as the difference of 13:14 and 117:121, or sum of 26:27 and 363:364;

          (|\(

This is the usual 13' diesis sign plus a 351:352. While its use as an
apotome complement, at least in relation to 88:91 or 117:121, is
highly "system specific," the 169:176-like size could also represent
the Pythagorean "tricomma" of ~70.380 cents or (531441:524288)^3, a
ratio of 150094635296999121:144115188075855872. The schisma between
this tricomma and 169:176 is 25365993365192851449:25364273101350633472
or ~0.117 cents, well within usual sagittal tolerances.

A nice touch here, at least for those of us with a taste for this kind
of thing, is that both the 88:91 sign /|\( and its system-specific
apotome complement at around 169:176 of (|\( share a 351:352 comma as
a distinguishing mark, also the intonational signature for a "common
temperament" of this sort. In Peppermint 24, the quasi-diesis and its
apotome complement have sizes of around 58.680 cents and 69.990 cents,
quite close to the advertised rational representations.

Using (|\(, we can conveniently write a progression like the following
using single-symbol notation, with a double-symbol version also shown:

            5E\!!/  5E(!\(        5Eb     5Eb/|\(
            5C/|\(  4B(!\(        5C/|\(  5Bb/|\(
            4B\!!/  4B(!\(        4Bb     4Bb/|\(
            4F/|\(  4E(!\(   or   4F/|\(  4Eb/|\(

            27/14    2/1
            27/16    3/2
            81/56    3/2
             9/8     1/1            

I must admit that the second style of notation looks much more
intuitive to me: it quickly reveals the location of regular whole-tone
and near-27:28 steps -- e.g. 4F/|\(-4Eb/|\( and 4Bb-4Bb/|\(. This
consideration might motivate the use of some double-symbol notation in
pieces where the single-symbol style is generally preferred.

If we wish to stick with a single-symbol style, another solution is
also possible:

         5E\!!/   5F!!!)
         5D!!!)   5C!!!)
         4B\!!/   5C!!!)
         4G!!!)   4F!!!)

Here 4F!!!) -- or 4Fb!) in double-symbol style -- is the system's
Archytas comma lower than the diminished fourth 4C-4F\!!/ or 4C-4Fb at
~367.235 cents, yielding an interval of about 346.393 cents, also
written conveniently in double-symbol notation as 4C-4Eb/|\(. Thus
while a regular diminished fourth approximates 14:17 (~365.825 cents),
the diminished fourth less Archytas comma (or minor third plus
quasi-diesis) approximates 9:11 (~347.408 cents). This notational
solution involves more "remote" accidentals like F!!!), but shares
with the double-symbol example the advantage of readily communicating
usual whole-tone steps and approximate 27:28 steps.

Getting back to my main proposal for the symbol /|\( to show the
"quasi-diesis" of around 88:91 in a tempered system with two chains of
fifths at about 704 cents, this approach seeks to analyze some
important intonational and harmonic relationships in the system and
then give them a "shorthand" signature.

For example, /|\( signals that the near-88:91 is serving as a
representation of both 32:33 and 27:28, and that regular major and
minor thirds are rather close to 11:14 and 11:13 or 28:33, regular
whole-tones to 39:44, and diatonic semitones to 21:22. Also implied
are "39:44 + 88:91 = 6:7" and "7:11 + 88:91 = 8:13" -- and so forth.

As compared to a "quasi-JI" style of transcription showing approximate
deviations from Pythagorean intonation, this solution follows the
usual sagittal approach of regarding the reference intervals as those
following the chain of fifths -- thus 4C-4E for a regular major third,
rather than 4C-4E)|( to show that this ~11:14 interval is ~891:896
wider than a Pythagorean 64:81 (~407.820 cents).

As compared to the approach of attempting to map a non-equal system to
some very large equal temperament, the "intonational signature"
approach reflects the structure of two regular chains of fifths placed
an arbitrary distance to produce certain ratios of 2-3-7-9-11-13
complementing the ratios represented within each chain (e.g. 22:26:33
and 14:17:21).

The resulting sagittal notation looks rather like a style of keyboard
notation showing simply which note is played on which keyboard, with a
symbol like the asterisk (*) used to show a note on the upper keyboad
raised by a quasi-diesis, e.g. F*4-Bb4 for a near-7:9 third, written
as 4F/|\(-4B\!!/ or 4F/|\(-4Bb in this sagittal approach. One could
also write 4G!!!)-4B\!!/ to show the represented 7:9 relationship.

In view of the accustomed subject of this thread, "A Common Notation
for JI and ET's," I might seem in a curious position devoting my first
post to a "not-so-common" sagittal approach to a nonequal tempered
system based on two chains of fifths at an arbitrary distance. Such is
what can happen when a good system falls into certain not so
experienced hands.

Most appreciatively,

Margo Schulter
mschulter@xxxxx.xxx


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Message: 5773

Date: Fri, 20 Dec 2002 12:05:10

Subject: Re: Tempering commas

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

>This worked! I then tried to get it to work via the command line, 
>and it didn't, until I entered "2" by itself. 
>It then asked me for the rest of the data.

It's not quite clear to me what exactly you typed, or how
you misinterpreted the help file.
It's always important to look at the first line of a
help subject, to see what the command parameters are.
Any other subsequent data will be prompted for.

Manuel


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