Re: A readily divisible journey into microtonalism

Dan Stearns:

>Of course in this
>sense, it's then not too much of a leap to say that a generalized
>diatonic scale such as Paul Erlich's 22e decatonic has a brute
>representation in 12e (as 0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12), and yet
>this would no doubt be some sort of near heresy for Paul, as it flies
>in the face of most everything that he's very carefully considered his
>scale to accomplish...

Only some things would be problematic:

1. Two of the three types of "dissonant" tetrads would come out as different
inversions of the minor seventh chord, a pretty consonant chord in 12-equal.

2. The two types of "consonant" tetrads would come out as a dominant seventh
and a minor added sixth, not too consonant in 12-equal but good enough
approximations to the consonant 7-limit tetrads for books like _Mathematical
Representations of Musical Scales_ and _On the Relations of Tone_ to ascribe
this meaning to them in the works of Wagner and his contemporaries.

3. The number of exceptions to strict propriety would be numerous. While the
diatonic scale in 12-equal has one exception (aug. 4th = dim. 5th), and both
decatonic scales in 22-equal as well as the "diatonic pentatonic" in whatever
have none, the decatonic scale in 12-equal has M"2"=m"3", M"3"=m"4", etc.
Propriety isn't one of the things in my paper but it can't hurt.

Other than that, I'd say the decatonic scale in 12-equal is great and I hear
quite a bit of it in country music (used triadically, not tetradically).

[Paul Erlich:]
> 3. The number of exceptions to strict propriety would be numerous.
While the diatonic scale in 12-equal has one exception (aug. 4th =
dim. 5th), and both decatonic scales in 22-equal as well as the
"diatonic pentatonic" in whatever have none,

Yes, I guess you could say that 12 (and its successive multiples) is a
sort of the propriety borderline when mapping a 2L & 8s step
structure, as everything > would be strictly proper (though less and
less meaningfully so, the higher and further you go), and everything <
would be improper... there seems to be a lot of pretty nice 2L & 8s
strictly proper decatonics that fall between the 12 and the 22, where
the equal division of the octave isn't either too large, or, so close
to the 12 that a difference between the largest interval of one class
and the smallest interval of the next class approaches triviality.
Once you start to exceed 22*n, where n is the n of L=s+n, you can work
the 2nds & 7ths closer to 16/15 & 8/7, and 15/8 & 7/4, but you'd also
be working everything else closer and closer to 10e, hence, nearer to
neutral 3rds & 6ths, and 5e 5ths & 4ths... Here are some of those that
I was referring to where the 2L & 8s is <22*n & >12*n (and the
additional chromatic step is therefore >0 & <L-s).

L=5 s=3
0 3 6 11 14 17 20 25 28 31 34

L=7 s=4
0 4 8 15 19 23 27 34 38 42 46

L=8 s=5
0 5 10 18 23 28 33 41 46 51 56

L=11 s=7
0 7 14 25 32 39 46 57 64 71 78

L=12 s=7
0 7 14 26 33 40 47 59 66 73 80

L=13 s=8
0 8 16 29 37 45 53 66 74 82 90

L=14 s=9
0 9 18 32 41 50 59 73 82 91 100

L=17 s=11
0 11 22 39 50 61 72 89 100 111 122

etc.

Dan

PS - This list doesn't include 2L 8s decatonics like 58, 70 & 82e,

L=9 s=5 0 5 10 19 24 29 34 43 48 53 58
L=11 s=6 0 6 12 23 29 35 41 52 58 64 70
L=13 s=7 0 7 14 27 34 41 48 61 68 75 82,

where the difference between the largest interval of one class and the
smallest interval of the next aren't necessarily so bad, but are
growing successively smaller than a syntonic comma - in other words I
used the 81/80 as a cut off point (and I think that all of the
approximate ratio interpretations of the interval classes in your 22e
decatonic are also separated by at least an 81/80?).

Dan wrote,

>(and I think that all of the
>approximate ratio interpretations of the interval classes in your 22e
>decatonic are also separated by at least an 81/80?).

There are an infinite number of appropriate ratio interpretations for each
interval class, even if you're totally inside the 5-limit or 7-limit. 81/80
comes out compositionally as 1 degree in 22-equal, which is the difference
between steps of 2 and 3 degrees. However, there could easily be a complex
ratio smaller than 81/80 which would come out as more than 1 degree in
22-equal -- or in any ET. Once you've decided how you're approximating the
prime intervals in an ET, one can find ratios which magnify the ET's errors
to any degree desired.

[Paul H. Erlich:]
>However, there could easily be a complex ratio smaller than 81/80
which would come out as more than 1 degree in 22-equal -- or in any
ET. Once you've decided how you're approximating the prime intervals
in an ET, one can find ratios which magnify the ET's errors to any
degree desired.

Yes, and I'm pretty sure that I understand what you're saying here,
but there aren't any interpretations of a ratio smaller than ~22�
which would have to be =/> ~109� to make your 22e decatonic fly,
right?

Dan

>Yes, and I'm pretty sure that I understand what you're saying here,
>but there aren't any interpretations of a ratio smaller than ~22�
>which would have to be =/> ~109� to make your 22e decatonic fly,
>right?

I don't mean to be obstinate, but it depends on what you mean by "fly". The
decatonic scale, like the ordinary diatonic scale, depends on "puns"
relative to JI. A diatonic piece may descend by dozens of 81:80 commas if
performed in strict JI, giving a ratio between one scale degree at the
beginning of the piece and another scale degree at the end of the piece that
is hundreds of cents off the corresponding ET or meantone interval -- and
this ratio could easily be between 0 and 22�. In the decatonic case, there
are two commas (50:49 and 64:63) instead of one, and they are both larger
than 81:80, so this sort of thing can happen even more easily.

[Paul Erlich:]
>it depends on what you mean by "fly".

Basically the gist, or the last bit ("Therefore an entirely new scale
is required, in which the tetrads can act as the basic consonances.")
of the introduction to "Tuning, Tonality, and Twenty-Two-Tone
Temperament."

>In the decatonic case, there are two commas (50:49 and 64:63) instead
of one, and they are both larger than 81:80,

I guess what I was specifically referencing would be a large
approximation of a 3/22 as a 10/9, and a small approximation of a 4/22
as a 9/8 (and ditto the 18 & 19/22, and the 16/9 & 9/5), and that
this -- 81/80 -- was probably the smallest approximate ratio that
separated different interval classes of the 0, 2, 4, 7, 9, 11, 13, 16,
18, 20, 22 (standard Pentachordal Major).

Dan

I wrote,

>>it depends on what you mean by "fly".

Dan replied,

>Basically the gist, or the last bit ("Therefore an entirely new scale
>is required, in which the tetrads can act as the basic consonances.")
>of the introduction to "Tuning, Tonality, and Twenty-Two-Tone
>Temperament."

>I guess what I was specifically referencing would be a large
>approximation of a 3/22 as a 10/9, and a small approximation of a 4/22
>as a 9/8 (and ditto the 18 & 19/22, and the 16/9 & 9/5), and that
>this -- 81/80 -- was probably the smallest approximate ratio that
>separated different interval classes of the 0, 2, 4, 7, 9, 11, 13, 16,
>18, 20, 22 (standard Pentachordal Major).

If you include just enough notes to get all six consonant tetrads to come
out in JI, the pentachordal decatonic scale has twelve notes in JI (kind of
like how the major scale requires eight notes, including both a 10/9 and a
9/8, to have six consonant triads in JI). Figures 5 shows a lattice diagram
for this JI representation. With ratios, the standard pentachordal major
scale is:

25/14
/ \
/ \
5/3-------5/4------15/8
\`. /,'/:\` \ ,'/:\
\10/7-/-:-\15/14/ : \
\ : / 7/4------21/16\
\:/,' \`.\:/,'/ `.\
1/1-------3/2-------9/8
\ : /
\: /
21/20

The 22-equal mappings are

0: 1/1
2: 21/20, 15/14
4: 9/8
7: 5/4
9: 21/16
11: 10/7
13: 3/2
16: 5/3
18: 7/4, 25/14
20: 15/8
22: 2/1

The list of intervals in this JI scale corresponding to each 22-tET interval
in the 22-equal mapping is:

0/22 oct.: 1:1 (0�), 50/49 (35�)
2/22 oct.: 21:20 (84�), 16:15 (112�), 15:14 (119�), 160:147 (147�)
3/22 oct.: 10:9 (182�)
4/22 oct.: 28:25 (196�), 9:8 (204�), 8:7 (231�)
5/22 oct.: 7:6 (267�), 25:21 (302�)
6/22 oct.: 147:125 (281�), 6:5 (316�), 60:49 (350.6�)
7/22 oct.: 49:40 (351.3�), 5:4 (386�), 80:63 (414�)
8/22 oct.: 63:50 (400�), 9:7 (435�)
9/22 oct.: 21:16 (471�), 4:3 (498�), 200:147 (533�)
10/22 oct.: 27:20 (520�)
11/22 oct.: 7:5 (583�), 10:7 (617�)

So, not only are there are ratios much smaller than 81�80 separating
interval classes, for example the 2401�2400 (0.7�) between 60:49 and 49:40,
but there are overlaps between interval classes, one of which (between 25:21
and 147:125) is 3087�3125, or -21.2 cents, nearly a syntonic comma IN THE
WRONG DIRECTION!

[Paul Erlich:]
>If you include just enough notes to get all six consonant tetrads to
come out

Yes, my mistake... (thanks for further explaining)

Dan

PS - I see now that I was thinking in terms of your ("In the following
table, we introduce names for intervals in, and notation for, the
decatonic scale") Table 1.

>>If you include just enough notes to get all six consonant tetrads to
>>come out

>PS - I see now that I was thinking in terms of your ("In the following
>table, we introduce names for intervals in, and notation for, the
>decatonic scale") Table 1.

OK. The ratios in that table are simply the 17-limit consonances (no 13s
allowed) in a consistent mapping to 22-equal. 9:8 and 10:9 are 17-limit
consonances, while 60:49 and 49:40 are not. However, if we take the 7-limit
as the standard of consonance, which is what the whole tetradic business
implies, then 9:8 is 3:2*3:2, 10:9 is 5:3*4:3, 60:49 is 10:7*12:7, and 49:40
is 7:5*7:4. In other words, all these intervals fall into the same category
with respect to tetradic harmony, since each is a dissonant composite of two
consonant intervals.

Perhaps the non-boldface ratios in Table 1 are more confusing than helpful
since the table is placed in the midst of a discussion of tetradic harmony.

is there a resource that I can access, that will supply some information to
get a handle on some of the discussions going on here?

Micah

> -----Original Message-----
> From: Paul H. Erlich [SMTP:PErlich@Acadian-Asset.com]
> Sent: Thursday, October 14, 1999 11:08 AM
> To: 'tuning@onelist.com'
> Subject: RE: [tuning] Re: A readily divisible journey into
> microtonalism
>
> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> >>If you include just enough notes to get all six consonant tetrads to
> >>come out
>
> >PS - I see now that I was thinking in terms of your ("In the following
> >table, we introduce names for intervals in, and notation for, the
> >decatonic scale") Table 1.
>
> OK. The ratios in that table are simply the 17-limit consonances (no 13s
> allowed) in a consistent mapping to 22-equal. 9:8 and 10:9 are 17-limit
> consonances, while 60:49 and 49:40 are not. However, if we take the
> 7-limit
> as the standard of consonance, which is what the whole tetradic business
> implies, then 9:8 is 3:2*3:2, 10:9 is 5:3*4:3, 60:49 is 10:7*12:7, and
> 49:40
> is 7:5*7:4. In other words, all these intervals fall into the same
> category
> with respect to tetradic harmony, since each is a dissonant composite of
> two
> consonant intervals.
>
> Perhaps the non-boldface ratios in Table 1 are more confusing than helpful
> since the table is placed in the midst of a discussion of tetradic
> harmony.
>
> > You do not need web access to participate. You may subscribe through
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Micah wrote,

>is there a resource that I can access, that will supply some information to
>get a handle on some of the discussions going on here?

The paper we are discussing is my _Tuning, Tonality, and Twenty-Two-Tone
Temperament_, published in Xenharmonikon 17 or available in .pdf format at
http://www-math.cudenver.edu/~jstarret/22ALL.pdf (ignore page 20; it's
irrecoverably erroneous). Read the paper and feel free to ask me to explain
anything that is unclear. Then what Dan and I are talking about may become
clearer to you.

-Paul

Thank you.

Micah

> -----Original Message-----
> From: Paul H. Erlich [SMTP:PErlich@Acadian-Asset.com]
> Sent: Thursday, October 14, 1999 11:48 AM
> To: 'tuning@onelist.com'
> Subject: RE: [tuning] Re: A readily divisible journey into
> microtonalism
>
> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Micah wrote,
>
> >is there a resource that I can access, that will supply some information
> to
> >get a handle on some of the discussions going on here?
>
> The paper we are discussing is my _Tuning, Tonality, and Twenty-Two-Tone
> Temperament_, published in Xenharmonikon 17 or available in .pdf format at
> http://www-math.cudenver.edu/~jstarret/22ALL.pdf (ignore page 20; it's
> irrecoverably erroneous). Read the paper and feel free to ask me to
> explain
> anything that is unclear. Then what Dan and I are talking about may become
> clearer to you.
>
> -Paul
>
> > You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@onelist.com - subscribe to the tuning list.
> tuning-unsubscribe@onelist.com - unsubscribe from the tuning list.
> tuning-digest@onelist.com - switch your subscription to digest mode.
> tuning-normal@onelist.com - switch your subscription to normal mode.