Yahoo Tuning Groups Ultimate Backup tuning Some thoughts on indexing L&s mappings in ASCII

When not restricted to the standard ASCII email format I'll often use
superscripts to help denote the huge variety of commatic "errors" and
differences that one encounters when trying to "explain" scales of
more than two step sizes in terms of two step size L&s mappings. These
are some rough ideas (hopefully with an eye towards some sense of
visual and contextual clarity) on using the ASCII format to index
commatic errors and differences.

Using a 5L 2s mapping:

(5) 3 1 6 4 2 7
12 (10) 8 13 11 9 14
19 17 (15)(20) 18 16 21
26 24 22 27 (25) 23 28
33 31 29 34 32 (30)(35)
40 38 36 41 39 37 42
...

and 31e at L=5&s=3 as an example, LLsLLLs can be seen as both a QCM
(like) major scale on a circle of 18/31 fifths (taken F to B), and as
a syntonic major taken (log(N)-log(D))*(e/log(2)) where 10/9=9/8, and
the 81/80 disappears...

05----23----10----28
\ /\ /\ /\
\ / \ / \ / \
\/ \/ \/ \
13----00----18----05

But supposing you take a Pythagorean major the same
(log(N)-log(D))*(e/log(2)) way, then you'd have 0 5 11 13 18 23 29 31,
or say

13----00----18----05----23~~~~11----29

where it's easier to note the break, or "inconsistency" that occurs
between the 81st and the 27th harmonic when this 31e Pythagorean major
is transferred onto a (one dimensional) sequence of 3/2s.

This scale could then be indexed as a 5L 2s (where L=5&s=3) taken

LL's,LLL's,

The apostrophes would then indicate the number of tuning units (in
other words 1/e) a step is raised by, and the commas would indicate
the number of tuning units a step is lowered by. And I suppose it
would also probably be wise to write this inside of brackets, say
[LL's,LLL's,] for example... that should minimize any confusion with
standard (grammatical) commas and apostrophes and whatnot.

In 53e the 5L 2s is taken L=9&s=4. And a LLsLLLs major scale taken
from a circle of 31/53 fifths here is the same as the Pythagorean
major taken (log(N)-log(D))*(e/log(2)), or:

22----00----31----09----40----18----49

But 53e recognizes the syntonic comma and it differentiates between
both the 10/9 and the 9/8, and the 256/243 and the 16/15. So if you
wanted to index this as a "strict" 5L 2s where L=9&s=4, then would
have to spell it:

LL,s'LL,Ls'

While this seems to me a pretty clean and understandable way to index
these types of scales -- where a variety of step sizes are referenced
to a two (L&s) step size cardinality -- the comparisons can get rather
convoluted...

I've said before that I considered a 0 3 7 8 12 15 18 20 scale in 20e
(while it has four different step sizes) to be some variety of a 5L
2s, say a

15/8
/ \
/ \
/21/16\
/ \
1/1-------3/2-------9/8-------27/16------81/64

where 20e would have to be seen as the following intervals...

I II III IV V VI VII
0 1/1 . . . . . .
1 28/27
2 16/15
3 9/8 [9/8] 10/9
4 9/8 8/7
5 7/6 32/27 [32/27] 6/5
6 5/4
7 81/64 9/7
8 21/16 4/3 [4/3] [4/3]
9 4/3 27/20
10 10/7 7/5
11 40/27 3/2
12 3/2 [3/2] 32/21 [3/2]
13 128/81 14/9
14 8/5
15 27/16 5/3 12/7 [27/16]
16 16/9 7/4
17 16/9 [16/9] 9/5
18 15/8
19 27/14
20 2/1 . . . . . .

or put another way,

11------03------15~~~~~~06------18
/\ /\ /\ 10 /\ /\
/ \ / \ / \ /\ / \ / \
/ 01-\--/-13-\--/05~~\~~/~~16\--/-08 \
/ 16/ 08/ \ / \/ \ / 11/ 03
13--05-----17------09----X--20--X--12------04-------15--07
16\ 08\ / \ /\ / \ 11\ 03\ /
\ 12-/--\04~~/~~\~~15/--\-07-/--\-19 /
\ / \ / \/ \ / \ / \ /
\/ \/ 10 \/ \/ \/
02------14~~~~~~05------17------09

You could also describe this scale as a major scale taken F to B on a
circle of fifths where the fifth size is (log(3)-log(2))*(20/log(2))
and each multiple or addition of this ~11.7/20 fifth is then taken as
the rounded fraction of e (and here of course that means a rounded
fraction of 20, as e=20). This would give an internally consistent
mapping of 20e as:

Ax
/|\
/ | \
/ | \
Fx--|---Cx
/|\ | /|\
/ | \ | / | \
/ | \|/ | \
D#--|---A#--|---E#
/|\ | /|\ | /|\
/ | \ | / | \ | / | \
/ | \|/ | \|/ | \
B---|---F#--|---C#--|---G#
\ | /|\ | /|\ | /|
\ | / | \ | / | \ | / |
\|/ | \|/ | \|/ |
D---|---A---|---E |
\ | /|\ | /|\ |
\ | / | \ | / | \ |
\|/ | \|/ | \|
F---|---C---|---G
|\ | /|\ | /|\
| \ | / | \ | / | \
| \|/ | \|/ | \
| Ab--|---Eb--|---Bb
| /|\ | /|\ | /|\
| / | \ | / | \ | / | \
|/ | \|/ | \|/ | \
Fb--|---Cb--|---Gb--|--Db
\ | /|\ | /|\ | /
\ | / | \ | / | \ | /
\|/ | \|/ | \|/
Abb-|---Ebb-|---Bbb
\ | /|\ | /
\ | / | \ | /
\|/ | \|/
Cbb-|---Gbb
\ | /
\ | /
\|/
Ebbb

where the same 0 3 7 8 12 15 18 20 scale is seen as

B
\
D---A---E
\ / \ / \
F---C---G

But if you want to consider this scale as a strict index of 5L 2s,
then L=5&s=0, as 20e sits on the (5e) diagonal border of a 5L 2s
periodic block...

(5) 3 1 6 4 2 7
12 (10) 8 13 11 9 14
19 17 (15)(20) etc.

So in other words you can look at the 5L 2s as a linear mapping of an
interval >4/7 & <3/5:

3/5 4/7
7/12
10/17 11/19
13/22 17/29 18/31 15/26
16/27 23/39 27/46 24/41 25/43 29/50 26/45 19/33

(etc.)

where the periodic block is moving diagonally towards 5e (in other
words "L") and vertically towards 7e (in other words "L+s"), and any
EDO where s=(-n) will not have a fifth size inside of this 1/70 space,
and therefore will be incapable of rendering the 5L 2s scale...

So as 20e sits right on the diagonal or "L" border where s=0, indexing
the four step size 0 3 7 8 12 15 18 20 scale as a 5L 2s (where again,
L=5&s=0) would require a spelling of:

L,,L,s'L,L,,L,,s''

And while this still doesn't seem overly cumbersome to me, the whole
process certainly can become somewhat convoluted...

Dan

🔗D.Stearns <stearns@xxxxxxx.xxxx>

Using a 5L 2s mapping:

(5) 3 1 6 4 2 7
12 (10) 8 13 11 9 14
19 17 (15)(20) 18 16 21
26 24 22 27 (25) 23 28
33 31 29 34 32 (30)(35)
40 38 36 41 39 37 42
...

05----23----10----28
\ /\ /\ /\
\ / \ / \ / \
\/ \/ \/ \
13----00----18----05

But supposing you take a Pythagorean major the same
(log(N)-log(D))*(e/log(2)) way, then you'd have 0 5 11 13 18 23 29 31,
or say

13----00----18----05----23~~~~11----29

where it's easier to note the break, or "inconsistency" that occurs
between the 81st and the 27th harmonic when this 31e Pythagorean major
is transferred onto a (one dimensional) sequence of 3/2s.

This scale could then be indexed as a 5L 2s (where L=5&s=3) taken

LL's,LLL's,

In 53e the 5L 2s is taken L=9&s=4. And a LLsLLLs major scale taken
from a circle of 31/53 fifths here is the same as the Pythagorean
major taken (log(N)-log(D))*(e/log(2)), or:

22----00----31----09----40----18----49

LL,s'LL,Ls'

I've said before that I considered a 0 3 7 8 12 15 18 20 scale in 20e
(while it has four different step sizes) to be some variety of a 5L
2s, say a

15/8
/ \
/ \
/21/16\
/ \
1/1-------3/2-------9/8-------27/16------81/64

where 20e would have to be seen as the following intervals...

or put another way,

where the same 0 3 7 8 12 15 18 20 scale is seen as

B
\
D---A---E
\ / \ / \
F---C---G

But if you want to consider this scale as a strict index of 5L 2s,
then L=5&s=0, as 20e sits on the (5e) diagonal border of a 5L 2s
periodic block...

(5) 3 1 6 4 2 7
12 (10) 8 13 11 9 14
19 17 (15)(20) etc.

So in other words you can look at the 5L 2s as a linear mapping of an
interval >4/7 & <3/5:

3/5 4/7
7/12
10/17 11/19
13/22 17/29 18/31 15/26
16/27 23/39 27/46 24/41 25/43 29/50 26/45 19/33

(etc.)

So as 20e sits right on the diagonal or "L" border where s=0, indexing
the four step size 0 3 7 8 12 15 18 20 scale as a 5L 2s (where again,
L=5&s=0) would require a spelling of:

L,,L,s'L,L,,L,,s''

And while this still doesn't seem overly cumbersome to me, the whole
process certainly can become somewhat convoluted...

Dan

🔗D.Stearns <stearns@xxxxxxx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>

Hey folks,

I had been working on this posting a couple of weeks ago
and haven't added anything to it lately, so I figured I
should just send it as it is before I leave for vacation
in the morning, rather than letting it sit.

Unless I have time to respond to TDs 465 and 466, this
will probably be my last post for awhile. Ciao!

----------------------------------------------------

Like Erv Wilson's theoretical articles, Dan Stearns's posts
often contain a wealth of information in a very concise
presentation, which leaves a lot of the mathematical calculation
implied rather than made explicit.

As with my attempts to understand Wilson's work, I often find
it difficult to follow Dan until I actually do the calculations.

In the expectation that others are equally mystified and may
benefit from the work I've done on this, here's an expansion
of his recent posting on 'L&s mapping', with some commentary
by me also.

(Dan is uncomfortable with my analogy between his work and
Wilson's, so let it be said here that this is strictly
*my own* feeling...)

--------------------------------------

> [Dan Stearns, TD 431.22 & 25; original post sent twice by mistake]
>

> Using a 5L 2s mapping:
>
> (5) 3 1 6 4 2 7
> 12 (10) 8 13 11 9 14
> 19 17 (15)(20) 18 16 21
> 26 24 22 27 (25) 23 28
> 33 31 29 34 32 (30)(35)
> 40 38 36 41 39 37 42
> ...

Dan,

I think I'd like to present this explanation of your
Periodic Table at the beginning of my analysis. Tell
me what you think.

---------------

> (5) 3 1 6 4 2 7
> 12 (10) 8 13 11 9 14
> 19 17 (15)(20) 18 16 21
> 26 24 22 27 (25) 23 28
> 33 31 29 34 32 (30)(35)
> 40 38 36 41 39 37 42
> ...

This 'Periodic Table', as Dan calls it, is derived
from following a sequence of

-s (mod L+s)

horizontally, which also results in a sequence of (L+s)
vertically, and creates this pattern:

L=s+1 L=s+2 L=s+3 L=s+4 L=s+5 L=s+6 L=s+7

L s L s L s L s L s L s L s

(1 0) 1 -1 1 -2 1 -3 1 -4 1 -5 1 -6
2 1 (2 0) 2 -1 2 -2 2 -3 2 -4 2 -5
3 2 3 1 (3 0) 3 -1 3 -2 3 -3 3 -4
4 3 4 2 4 1 (4 0) 4 -1 4 -2 4 -3
5 4 5 3 5 2 5 1 (5 0) 5 -1 5 -2
6 5 6 4 6 3 6 2 6 1 (6 0) 6 -1
7 6 7 5 7 4 7 3 7 2 7 1 (7 0)
8 7 8 6 8 5 8 4 8 3 8 2 8 1
...

[DAN: is that correct?]

Applying the L=5,s=2 mapping to this gives the following:

(5) 3 1 6 4 2 7
12 (10) 8 6 4 2 7
19 17 (15) 13 11 9 7
26 24 22 (20) 18 16 14
33 31 29 27 (25) 23 21
40 38 36 34 32 (30) 28
47 45 43 41 39 37 (35)
...

Eliminating redundant portions of a row and shifting
the following rows upwards gives the concise table
Dan presented:

> (5) 3 1 6 4 2 7
> 12 (10) 8 13 11 9 14
> 19 17 (15)(20) 18 16 21
> 26 24 22 27 (25) 23 28
> 33 31 29 34 32 (30)(35)
> 40 38 36 41 39 37 42
> ...

> ... if s>0, then the mapping (which is however many large
> and small steps the scale has, and here that's five large
> and two small, i.e., 5L&2s) will give an EDO than can render
> the [LLsLLLs] scale.
>
> Where s<0, those EDOs cannot accomplish the mapping.

I puzzled over this for quite a while, until I realized that
(Doh!) *obviously*, if s<0 then there can be no 5L,2s mapping
because *there is no s* !

So this being the case, all the EDOs which lie below and
to the left of the 'border' EDOs, which in this case are
all multiples of 5, will be able to map 5L,2s. All those
on the other side will not, and those right on the border
will map 5L,2s but only with a great deal of ambiguity.

*******************************

no longer needed:

(5) 3 1 6 4 2 7
-2 -2 +5 -2 -2 +5 (+5)

12 (10) 8 13 11 9 14
+5 -2 -2 +5 -2 -2 +5 (+5)

19 17 (15)(20) 18 16 21
+5 -2 -2 +5 -2 -2 +5 (+5)

26 24 22 27 (25) 23 28
+5 -2 -2 +5 -2 -2 +5 (+5)

33 31 29 34 32 (30)(35)
+5 -2 -2 +5 -2 -2 +5 (+5)

40 38 36 41 39 37 42
+5 -2 -2 +5 -2 -2 +5

...

**************************************

> and 31e at L=5&s=3 as an example,

That is,

e=31

L = 2^(5/e) = ~1.94 Semitones
s = 2^(3/e) = ~1.16 Semitones

> LLsLLLs can be seen as both a QCM (like) major scale on a
> circle of 18/31 fifths (taken F to B),

That is,

Assuming e=31, and equivalence of any 2^(d/e) == 3^x,

an '18/31 fifth' is calculated as:

3^(x+1) == 2^((mod_e(d+18))/e)

Giving a 'circle of fifths' of:

e=31

~Semitones Pythagorean equivalent

B 2^(28/e) 10.84 3^ 5
E 2^(10/e) 3.87 3^ 4
A 2^(23/e) 8.90 3^ 3
D 2^( 5/e) 1.94 3^ 2
G 2^(18/e) 6.97 3^ 1
C 2^( 0/e) 0.00 3^ 0
F 2^(13/e) 5.03 3^-1

Note that an '18/31 fifth' can only be divided evenly in
this scale into either (3L + s) or (6s).

So analyzing the former, 2^(18/31)
= 3L + s
= 2^((3*5 + 3)/e) 'tuning units' (tuning unit = 2^(1/e)
= 2^((15 + 3)/e)
= 2^(18/e).

> and as a syntonic major taken (log(N)-log(D))*(e/log(2))

That is,

N = numerator of implied ratio
D = denominator of implied ratio
N/D == 2^(d/e), where d = int((log(N)-log(D))*(e/log(2)))

Giving a 'syntonic major' scale of:

e=31

JI equiv. ~Semitones 2^(d/e) ~Semitones

C 2/1 12.00 == 2^(31=0/e) 12.00
\ \
16:15 2^(3/e)
1.12 S 1.16 S
/ /
B 15/8 10.88 == 2^( 28/e) 10.84
\ \
9:8 2^(5/e)
2.04 S 1.94 S
/ /
A 5/3 8.84 == 2^( 23/e) 8.90
\ \
10:9 2^(5/e)
1.82 S 1.94 S
/ /
G 3/2 7.02 == 2^( 18/e) 6.97
\ \
9:8 2^(5/e)
2.04 S 1.94 S
/ /
F 4/3 4.98 == 2^( 13/e) 5.03
\ \
16:15 2^(3/e)
1.12 S 1.16 S
/ /
E 5/4 3.86 == 2^( 10/e) 3.87
\ \
10:9 2^(5/e)
1.82 S 1.94 S
/ /
D 9/8 2.04 == 2^( 5/e) 1.94
\ \
9:8 2^(5/e)
2.04 S 1.94 S
/ /
C 1/1 0.00 == 2^( 0/e) 0.00

The pattern [LLsLLLs]:

e=31

L L s L L L s
2^(5/e) 2^(5/e) 2^(3/e) 2^(5/e) 2^(5/e) 2^(5/e) 2^(3/e)
1.94 1.94 1.16 1.94 1.94 1.94 1.16

can be seen by comparing with my diagram.

Note that

e=31
for 9/8, d (not rounded) = ~5.267675045
for 10/9, d (not rounded) = ~4.712095897

and int() of both of them = 5. They are both therefore
represented in 31-tET by 2^(5/31).

Thus, as Dan said,

> where 10/9=9/8, and the 81/80 disappears...

In terms of d, the lattice appears thus:

>
> 05----23----10----28
> \ /\ /\ /\
> \ / \ / \ / \
> \/ \/ \/ \
> 13----00----18----05
>

Which I would prefer to notate more completely:

2^(5/31)----2^(23/31)----2^(10/31)----2^(28/31)
\ / \ / \ / \
\ / \ / \ / \
\ / \ / \ / \
2^(13/31)----2^(0/31)----2^(18/31)----2^(5/31)

> But supposing you take a Pythagorean major the same
> (log(N)-log(D))*(e/log(2)) way, then you'd have
> 0 5 11 13 18 23 29 31,

That is,

e=31

Pyth.equiv. ~Semitones 2^(d/e) ~Semitones

C 2/1 12.00 == 2^(31=0/e) 12.00
\ \
256:243 2^(2/e)
0.90 S 0.77 S
/ /
B 243/128 11.10 == 2^( 29/e) 11.23
\ \
9:8 2^(6/e)
2.04 S 2.32 S
/ /
A 27/16 9.06 == 2^( 23/e) 8.90
\ \
9:8 2^(5/e)
2.04 S 1.94 S
/ /
G 3/2 7.02 == 2^( 18/e) 6.97
\ \
9:8 2^(5/e)
2.04 S 1.94 S
/ /
F 4/3 4.98 == 2^( 13/e) 5.03
\ \
256:243 2^(2/e)
0.90 S 0.77 S
/ /
E 81/64 4.08 == 2^( 11/e) 4.26
\ \
9:8 2^(6/e)
2.04 S 2.32 S
/ /
D 9/8 2.04 == 2^( 5/e) 1.94
\ \
9:8 2^(5/e)
2.04 S 1.94 S
/ /
C 1/1 0.00 == 2^( 0/e) 0.00

> or say
>
> 13----00----18----05----23~~~~11----29
>
> where it's easier to note the break, or "inconsistency" that
> occurs between the 81st and the 27th harmonic when this 31e
> Pythagorean major is transferred onto a (one dimensional)
> sequence of 3/2s.

e=31

for 243/128, d (not rounded) = ~28.66918761
for 81/64, d (not rounded) = ~10.53535009
for 27/16, d (not rounded) = ~23.40151257

Thus, d(81/64) is almost exactly between 10 and 11, and this
pinpoints the break in consistency, which simply occurs because
an integer value (designating the degree in the ET scale) must
be chosen, and 11 is only slighter more appropriate than 10
for this ratio.

(Doubling the size of the ET, that is, 62-tET, would give
an integer value [i.e., a 'd'] that would be a very good
representation of 81/64, and thus would preserve consistency
to a greater extent for this 'chain of fifths'.)

> This scale could then be indexed as a 5L 2s (where L=5&s=3) taken
>
> LL's,LLL's,
>
> The apostrophes would then indicate the number of tuning units
> (in other words 1/e) a step is raised by, and the commas would
> indicate the number of tuning units a step is lowered by.

1 'tuning unit' = 2^(1/31)= ~0.39 Semitone (~38.7097 cents), so

L' = L + ~0.39 = ~2.32 Semitone (error due to rounding)
s, = s - ~0.39 = ~0.77 Semitone

e=31

L L' s, L L L' s,

^(5/e) 2^(6/e) 2^(2/e) 2^(5/e) 2^(5/e) 2^(6/e) 2^(2/e)
1.94 2.32 0.77 1.94 1.94 2.32 0.77

(compare with my diagram above.)

> And I suppose it would also probably be wise to write this
> inside of brackets, say [LL's,LLL's,] for example... that
> should minimize any confusion with standard (grammatical)
> commas and apostrophes and whatnot.

Good idea.

> In 53e the 5L 2s is taken L=9&s=4.

e=53

L = 2^(9/e) = ~2.04 Semitones
s = 2^(4/e) = ~0.91 Semitones

> And a LLsLLLs major scale taken from a circle of 31/53 fifths

That is,

Assuming e=53, and equivalence of any 2^(d/e) == 3^x,

a '31/53 fifth' is calculated as:

3^(x+1) == 2^((mod_e(d+53))/e)

Giving a 'circle of fifths' of:

e=53

~Semitones Pythagorean equivalent

B 2^(49/e) 11.09 3^ 5
E 2^(18/e) 4.08 3^ 4
A 2^(40/e) 9.06 3^ 3
D 2^( 9/e) 2.04 3^ 2
G 2^(31/e) 7.02 3^ 1
C 2^( 0/e) 0.00 3^ 0
F 2^(22/e) 4.98 3^-1

> here is the same as the Pythagorean major taken
> (log(N)-log(D))*(e/log(2)),

That is,

e=53

Pyth.equiv. ~Semitones 2^(d/e) ~Semitones

C 2/1 12.00 == 2^(53=0/e) 12.00
\ \
256:243 2^(4/e)
0.90 S 0.91 S
/ /
B 243/128 11.10 == 2^( 49/e) 11.09
\ \
9:8 2^(9/e)
2.04 S 2.04 S
/ /
A 27/16 9.06 == 2^( 40/e) 9.06
\ \
9:8 2^(9/e)
2.04 S 2.04 S
/ /
G 3/2 7.02 == 2^( 31/e) 7.02
\ \
9:8 2^(9/e)
2.04 S 2.04 S
/ /
F 4/3 4.98 == 2^( 22/e) 4.98
\ \
256:243 2^(4/e)
0.90 S 0.91 S
/ /
E 81/64 4.08 == 2^( 18/e) 4.08
\ \
9:8 2^(9/e)
2.04 S 2.04 S
/ /
D 9/8 2.04 == 2^( 9/e) 2.04
\ \
9:8 2^(9/e)
2.04 S 2.04 S
/ /
C 1/1 0.00 == 2^( 0/e) 0.00

(The ~0.341-cent error between 243/128 and 2^(49/53), which
caused the rounding error bringing about the discrepancy in
my diagrams, can probably be ignored.)

> or:
>
> 22----00----31----09----40----18----49
>
> But 53e recognizes the syntonic comma and it differentiates
> between both the 10/9 and the 9/8, and the 256/243 and the
> 16/15.

e=53

for 9/8, d (not rounded) = ~9.006025076
for 10/9, d (not rounded) = ~8.056163953

for 256/243, d (not rounded) = ~3.984937309
for 16/15, d (not rounded) = ~4.934798433

Thus, the integer values for d, 9 and 8, and 5 and 4 (designating
the 53-tET degrees), are clearly differentiated.

Dan should really have written 'because' here rather than 'but',
as this statement explains *why* the 53-tET scale is notated
with the same degrees, whether it is seen as implying a
'Pythagorean major scale' or a 'circle of fifths'.

> So if you wanted to index this as a "strict" 5L 2s where
> L=9&s=4, then [you] would have to spell it:
>
> LL,s'LL,Ls'

That is,

e=53

L L, s' L L, L s'
2^(9/e) 2^(8/e) 2^(5/e) 2^(9/e) 2^(8/e) 2^(9/e) 2^(5/e)
2.04 1.81 1.13 2.04 1.81 2.04 1.13

I'm afraid Dan lost me on this one. It appears that
with this [LL,s'LL,Ls'] illustration, he was illustrating
the *syntonic* and not the Pythagorean 'major scale':

e=53

JI equiv. ~Semitones 2^(d/e) ~Semitones

C 2/1 12.00 == 2^(53=0/e) 12.00
\ \
16:15 2^(5/e)
1.12 S 1.13 S
/ /
B 15/8 10.88 == 2^( 48/e) 10.87
\ \
9:8 2^(9/e)
2.04 S 2.04 S
/ /
A 5/3 8.84 == 2^( 39/e) 8.83
\ \
10:9 2^(8/e)
1.82 S 1.81 S
/ /
G 3/2 7.02 == 2^( 31/e) 7.02
\ \
9:8 2^(9/e)
2.04 S 2.04 S
/ /
F 4/3 4.98 == 2^( 22/e) 4.98
\ \
16:15 2^(5/e)
1.12 S 1.13 S
/ /
E 5/4 3.86 == 2^( 17/e) 3.85
\ \
10:9 2^(8/e)
1.82 S 1.81 S
/ /
D 9/8 2.04 == 2^( 9/e) 2.04
\ \
9:8 2^(9/e)
2.04 S 2.04 S
/ /
C 1/1 0.00 == 2^( 0/e) 0.00

> While this seems to me a pretty clean and understandable way
> to index these types of scales -- where a variety of step sizes
> are referenced to a two (L&s) step size cardinality -- the
> comparisons can get rather convoluted...
>
>
> I've said before that I considered a 0 3 7 8 12 15 18 20 scale
> in 20e (while it has four different step sizes)

e=20

2^(20/e) 12.00
\
2^(2/e)
1.20 S
/
2^(18/e) 10.80
\
2^(3/e)
1.80 S
/
2^(15/e) 9.00
\
2^(3/e)
1.80 S
/
2^(12/e) 7.20
\
2^(4/e)
2.40 S
/
2^(8/e) 4.80
\
2^(1/e)
0.60 S
/
2^(7/e) 4.20
\
2^(4/e)
2.40 S
/
2^(3/e) 1.80
\
2^(3/e)
1.80 S
/
2^(0/e) 0.00

*** FOR REFERENCE: the entire 20-EDO scale

20 12.00
19 11.40
18 10.80
17 10.20
16 9.60
15 9.00
14 8.40
13 7.80
12 7.20
11 6.60
10 6.00
9 5.40
8 4.80
7 4.20
6 3.60
5 3.00
4 2.40
3 1.80
2 1.20
1 0.60
0 0.00

**************

> to be some variety of a 5L 2s, say a
>
> 15/8
> / \
> / \
> /21/16\
> / \
> 1/1-------3/2-------9/8-------27/16------81/64

e=20

for 15/8, d (not rounded) = ~18.13781191
for 27/16, d (not rounded) = ~15.09775004
for 3/2, d (not rounded) = ~11.69925001
for 21/16, d (not rounded) = ~ 7.846348456
for 81/64, d (not rounded) = ~ 6.797000058
for 9/8, d (not rounded) = ~ 3.398500029
for 1/1, d = 0

> where 20e would have to be seen as the following intervals...
>
> I II III IV V VI VII
> 0 1/1 . . . . . .
> 1 28/27
> 2 16/15
> 3 9/8 [9/8] 10/9
> 4 9/8 8/7
> 5 7/6 32/27 [32/27] 6/5
> 6 5/4
> 7 81/64 9/7
> 8 21/16 4/3 [4/3] [4/3]
> 9 4/3 27/20
> 10 10/7 7/5
> 11 40/27 3/2
> 12 3/2 [3/2] 32/21 [3/2]

> 13 128/81 14/9
> 14 8/5
> 15 27/16 5/3 12/7 [27/16]
> 16 16/9 7/4
> 17 16/9 [16/9] 9/5
> 18 15/8
> 19 27/14
> 20 2/1 . . . . . .
>

I find it more helpful to invert this, so as to use an
orientation that reflects the pitch structure, i.e., higher
pitches at the top:

e=53

2^(d/e) I II III IV V VI VII
20 2/1 . . . . . .
19 27/14
18 15/8
17 16/9 [16/9] 9/5
16 16/9 7/4
15 27/16 5/3 12/7 [27/16]
14 8/5
13 128/81 14/9
12 3/2 [3/2] 32/21 [3/2]
11 40/27 3/2
10 10/7 7/5
9 4/3 27/20
8 21/16 4/3 [4/3] [4/3]
7 81/64 9/7
6 5/4
5 7/6 32/27 [32/27] 6/5
4 9/8 8/7
3 9/8 [9/8] 10/9
2 16/15
1 28/27
0 1/1 . . . . . .
I II III IV V VI VII

> or put another way,
>
> 11------03------15~~~~~~06------18
> /\ /\ /\ 10 /\ /\
> / \ / \ / \ /\ / \ / \
> / 01-\--/-13-\--/05~~\~~/~~16\--/-08 \
> / 16/ 08/ \ / \/ \ / 11/ 03
> 13--05-----17------09----X--20--X--12------04-------15--07
> 16\ 08\ / \ /\ / \ 11\ 03\ /
> \ 12-/--\04~~/~~\~~15/--\-07-/--\-19 /
> \ / \ / \/ \ / \ / \ /
> \/ \/ 10 \/ \/ \/
> 02------14~~~~~~05------17------09
>
>
> You could also describe this scale as a major scale taken
> F to B on a circle of fifths where the fifth size is
> (log(3)-log(2))*(20/log(2)) and each multiple or addition
> of this ~11.7/20 fifth is then taken as the rounded fraction
> of e (and here of course that means a rounded fraction of 20,
> as e=20). This would give an internally consistent mapping
> of 20e as:
>
>
> Ax
> /|\
> / | \
> / | \
> Fx--|---Cx
> /|\ | /|\
> / | \ | / | \
> / | \|/ | \
> D#--|---A#--|---E#
> /|\ | /|\ | /|\
> / | \ | / | \ | / | \
> / | \|/ | \|/ | \
> B---|---F#--|---C#--|---G#
> \ | /|\ | /|\ | /|
> \ | / | \ | / | \ | / |
> \|/ | \|/ | \|/ |
> D---|---A---|---E |
> \ | /|\ | /|\ |
> \ | / | \ | / | \ |
> \|/ | \|/ | \|
> F---|---C---|---G
> |\ | /|\ | /|\
> | \ | / | \ | / | \
> | \|/ | \|/ | \
> | Ab--|---Eb--|---Bb
> | /|\ | /|\ | /|\
> | / | \ | / | \ | / | \
> |/ | \|/ | \|/ | \
> Fb--|---Cb--|---Gb--|--Db
> \ | /|\ | /|\ | /
> \ | / | \ | / | \ | /
> \|/ | \|/ | \|/
> Abb-|---Ebb-|---Bbb
> \ | /|\ | /
> \ | / | \ | /
> \|/ | \|/
> Cbb-|---Gbb
> \ | /
> \ | /
> \|/
> Ebbb
>
>

This lattice is illustrating this scale:

implied ~d ~Semitones
3^x in 2^(d/e) for rounded d

17 18.9 11.40 Ax
( 16 7.2 4.20 Dx )
( 15 15.5 9.00 Gx )
14 3.8 2.40 Cx
13 12.1 7.20 Fx
( 12 0.4 0.00 B# )
11 8.7 5.40 E#
10 17.0 10.20 A#
9 5.3 3.00 D#
8 13.6 8.40 G#
7 1.9 1.20 C#
6 10.2 6.00 F#
5 18.5 10.80 B
4 6.8 4.20 E
3 15.1 9.00 A
2 3.4 1.80 D
1 11.7 7.20 G
0 0.0 0.00 C
- 1 8.3 4.80 F
- 2 16.6 10.20 Bb
- 3 4.9 3.00 Eb
- 4 13.2 7.80 Ab
- 5 1.5 1.20 Db
- 6 9.8 6.00 Gb
- 7 18.1 10.80 Cb
- 8 6.4 3.60 Fb
- 9 14.7 9.00 Bbb
-10 3.0 1.80 Ebb
-11 11.3 6.60 Abb
(-12 19.6 12.00 Dbb )
-13 7.9 4.80 Gbb
-14 16.2 9.60 Cbb
(-15 4.5 3.00 Fbb )
(-16 12.8 7.80 Bbbb)
-17 1.1 0.60 Ebbb

When I questioned how this 'circle of fifths' with two
different-sized 'fifths', 2^(11/20) and 2^(12/20), could
be consistent on a 5-limit lattice, Dan explained:

> Well say we lay the circle down F to B, the ~11.7 fifth
> creates an integer fifth sequence of 12, 12, 11, 12, 12, and
> 11/20, which would then be seen to jive with (ah, I mean be
> consistent with) the lattice where 12/20 is "3/2" and 11/20
> is "40/27."

He's looking at this from a basically Pythagorean perspective,
which I suppose is unavoidable when considering a set of
pitches or intervals as a 'circle of fifths'. But it makes
a lot of sense to me, since 20e doesn't imply 5-limit ratios
very well anyway. The lattice is just a good way to
visualize the kinds of Pythagorean-type relationships
that inhere in this scale.

Dan continued:
>
> The consistency is in the rounded integers, in other words
> a "3/2" is always represented by a 12/20; a "5/4" by a 7/20;
> a "6/5" by a 5/20; and a "25/24" by a 2/20.

Just in case anyone's confused, Dan means the following:

20e implied
interval Semitones ratio Semitones

2^(12/20) 7.20 always implies 3/2 7.02
2^( 7/20) 4.20 always implies 5/4 3.86
2^( 5/20) 3.00 always implies 6/5 3.16
2^( 2/20) 1.20 always implies 25/24 0.71

As I said above, and is obvious from this table, this
scale doesn't imply 5-limit ratios well. For example,
2^(7/20) is much closer to the 'septimal major 3rd' 9/7
[= 4.35 Semitones] as well as the Pythagorean 'major 3rd'
or 'ditone' 81/64 [= 4.08 Semitones], than it is to 5/4.
But if the musical context uses the ratios as they're
presented on the lattice it may possibly 'force' the
implication onto the listener.

>
> Ax
> 10.26
> / | \
> / | \
> Fx--|---Cx
> 6.39 | 1.41
> / | \ | / | \
> / | \|/ | \
> D#--|---A#--|---E#
> 2.53 | 9.55 | 4.57
> / | \ | / | \ | / | \
> / | \|/ | \|/ | \
> B---|---F#--|---C#--|---G#
> 10.67 | 5.69 | 0.71 | 7.73
> \ | / | \ | / | \ | / |
> \|/ | \|/ | \|/ |
> D---|---A---|---E |
> 1.82 | 8.84 | 3.86 |
> \ | / | \ | / | \ |
> \|/ | \|/ | \|
> F---|---C---|---G
> 4.98 | 0.00 | 7.02
> | \ | / | \ | / | \
> | \|/ | \|/ | \
> | Ab--|---Eb--|---Bb
> | 8.14 | 3.16 | 10.18
> | / | \ | / | \ | / | \
> |/ | \|/ | \|/ | \
> Fb--|---Cb--|---Gb--|--Db
> 4.27 | 11.29 | 6.31 | 1.33
> \ | / | \ | / | \ | /
> \|/ | \|/ | \|/
> Abb-|---Ebb-|---Bbb
> 7.43 | 2.45 | 9.47
> \ | / | \ | /
> \|/ | \|/
> Cbb-|---Gbb
> 10.59| 5.61
> \ | /
> \|/
> Ebbb
> 1.74

> where the same 0 3 7 8 12 15 18 20 scale is seen as
>
> B
> \
> D---A---E
> \ / \ / \
> F---C---G
>
> But if you want to consider this scale as a strict index of
> 5L 2s, then L=4&s=0, as 20e sits on the (5e) diagonal border
> of a 5L 2s periodic block...
>
> (5) 3 1 6 4 2 7
> 12 (10) 8 13 11 9 14
> 19 17 (15)(20) etc.
>
> So in other words you can look at the 5L 2s as a linear
> mapping of an interval >4/7 & <3/5:
>
> 3/5 4/7
> 7/12
> 10/17 11/19
> 13/22 17/29 18/31 15/26
> 16/27 23/39 27/46 24/41 25/43 29/50 26/45 19/33
>
> (etc.)

I thought it would be useful to supply Semitone values, and
to notate the EDO fractions as real ratios, with Semitone values
(The last line of this diagram goes to 81 characters, and so
it probably has extra carriage returns that I didn't intend.
If it doesn't look right, save it in a text editor and delete
those carriage returns.):

2^(3/5)
2^(4/7)
7.20
6.86

2^(7/12)
7.00

2^(10/17) 2^(11/19)
7.06 6.95

2^(13/22) 2^(17/29) 2^(18/31) 2^(15/26)
7.09 7.03 6.97 6.92

2^(16/27) 2^(23/39) 2^(27/46) 2^(24/41) 2^(25/43) 2^(29/50) 2^(26/45)
2^(19/33)
7.11 7.08 7.04 7.02 6.98 6.96 6.93
6.91

(etc.)

>
> where the periodic block is moving diagonally towards 5e
> (in other words "L") and vertically towards 7e (in other words
> "L+s"), and any EDO where s=(-n) will not have a fifth size
> inside of this 1/70 space, and therefore will be incapable of
> rendering the 5L 2s scale...

This should read 'any EDO where s=(-n) will not have a fifth size
inside of this 1/35 space'.

>
> So as 20e sits right on the diagonal or "L" border where s=0,
> indexing the four step size 0 3 7 8 12 15 18 20 scale as a
> 5L 2s (where again, L=4&s=0) would require a spelling of:
>
>
>
> L,Ls'LL,L,s''
>
> And while this still doesn't seem overly cumbersome to me,
> the whole process certainly can become somewhat convoluted...
>

[This last illustration had an error in Dan's original post
which he corrected in TD 433.5, and which has been incorporated
here.]

Generalizing all of Dan's notation gives:

EDO = equal division of the 'octave'
e = number of EDO
N = numerator of implied ratio
D = denominator of implied ratio
d = degree of EDO = int((log(N)-log(D))*(e/log(2)))
(L or s)' = (ratio of L or s) * 2^(d/e)
(L or s), = (ratio of L or s) / 2^(d/e)

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------