Lattices with Decimal Notation

Introduction
21 and 31 note MOS scales
The 11-limit tonality diamond
Partch scales and the 41 note MOS
7-limit scales
D'Alessandro

This page uses my 7-limit/neutral third lattice to graph scales based on or approximated to the Miracle temperament. To see them working more clearly, you could try the slide lattice kit.

The Miracle tuning is an MOS with a semitone generator sixth of which make up a perfect fifth. Miracle is an acronym for Multiplicity of Integer Ratio Approximations Constructed Linearly and Evenly. It covers 31-, 41- and 72-equal consistently throughout the 11-limit. The defining matrix is:

```     (10 1)
(16 1)(s)
H' = (23 3)(q)
(28 3)
(35 2)
```

Where s is the semitonal generator and q is the difference between 10s and an octave, roughly a sixth-tone. In 31-equal, s=3 and q=1. In 41-equal, s=4 and q=1. In 72-equal, s=7 and q=2.

This implies the intervals (-7 -1 1 1 1)H, (-1 5 0 0 -2)H and (5 -2 -2 1)H or 385/384, 243/242 and 225/224, being approximated to unisons. These approximations are consistent with my 7-limit/neutral-third lattice. (-1 5 0 0 -1)H is what stops two 11:9 neutral thirds from being a 3:2 fifth. Calling a 7:5 flattened by an 8:7 the same as 11:9 removes the comma (0 0 -1 1 0)H - (3 0 0 -1 0)H - (0 -2 0 0 1)H = (-3 2 -1 2 -1)H. That's the sum of the vectors (-7 -1 1 1 1)H and (-1 5 0 0 2)H and so is itself a unison in this temperament.

Decimal notation is based on the following Miracle-tuned scale:

```0.0   0.1   0.2   0.3   0.4   0.5   0.6   0.7   0.8   0.9   1.0
s     s     s     s     s     s     s     s     s    s+q
```

The top line shows the note names in octave.note format. The number of the note is always the number of steps of s from .0. As the lattices are octave-invariant, this is the only number I'll be using from now on.

The symbol ^ is used to raise a note by q, and v to lower by q. They combine so ^^ raises by 2q and w lowers by 2q; m^ raises by 3q and vw lowers by 3q.

Here is the 21 note Miracle MOS, called Blackjack, on the 7-limit/neutral-third lattice showing decimal notes.

```                                                            0^
/
/
2^----5^
/ \/
/  /\
1^----4^-/--7^-\--0
/ \      /  /    \
/   \    /  /      \
0^----3^----6^----9^----2-----5
/ \/       \/  /
/  /\       /\ /
2^----5^-/--8^-\--1--/--4-----7-----0v
/ \      /  /    \   /
/   \    /  /      \ /
1^----4^----7^----0-----3-----6-----9
/ \/       \/  /
/  /\       /\ /
0^----3^----6^-/--9^ \--2--/--5-----8
/ \      /  /    \   /
/   \    /  /      \ /
5^----8^----1-----4-----7-----0v
\/       \/  /
/\       /\ /
0--\--3--/--6-----9
\   /
\ /
5-----8

0v
```

It loops around bottom-left to top-right because of the unison vector the lattice doesn't automatically cover. Now Canasta, the 31 note MOS.

```                                        0^

2^--5^
/ \
1^  4^/ 7^\ 0
/     \
0^--3^--6^--9^--2---5
/ \     / \     /
2^  5^/ 8^\ 1 / 4 \ 7 / 0v
/     \ /     \ /
1^--4^--7^--0---3---6---9---2v--5v
/ \     / \     / \     /
0^  3^  6^/ 9^\ 2 / 5 \ 8 / 1v\ 4v/ 7v  0w
/     \ /     \ /     \ /
5^--8^--1---4---7---0v--3v--6v--9v
\     / \     / \     /
0 \ 3 / 6 \ 9 / 2v\ 5v/ 8v
\ /     \ /     \ /
5---8---1v--4v--7v--0w
/
0v/ 3v  6v  9v
/
5v--8v

0w
```

Modifying Paul Erlich's diagram, the 11-limit diamond becomes

```      10/11-10/9---------5/3---------5/4
/ \   / \         / \         / \
/   \ /   \       /   \       /   \
/14/11-14/9---------7/6--10/7---7/4 \
/     / \     \   /    \  \/ \/  /    \
/     /   \     \ /      \ /\ /\ /      \
16/11-16/9--12/11--4/3-18/11-\-1/1-/-11/9--3/2--11/6---9/8--11/8
\      / \/ \/ \      / \     \   /     /
\    /  /\ /\  \    /   \     \ /     /
\ 8/7-/-7/5-\12/7-/-----\9/7--X-11/7/
\   /       \   /       \   / \   /
\ /         \ /         \ /   \ /
8/5---------6/5---------9/5--20/11
```

Setting 5 as the 1/1, with added Miracle approximations

```                                                        0^
/
3^^-6^^-----2^/
/   / \     / \
/8^^/ 1^\   / 7^\ 0
/   / /   \ /     \
0^--3^--6^--9^--2---5
/ \ / / /     \     /
3^^-6^^-----2^------8^/       \ 7 / 0v
/   / \     / \   / / \         \ / /
/8^^/ 1^\   / 7^\ 0 / 3 \       / 2v/
/   / /   \ /     \ /     \     / / / \
0^--3^--6^--9^--2---5---8---1v--4v--7v--0w
/ \ / / /     \     / \     / \   / /   /
3^^-6^^-----2^------8^/       \ 7 / 0v\ 3v/   \ 9v/ 2w/
/ \     / \   / / \         \ / /   \ /     \ /   /
8^^-1^\   / 7^\ 0 /-3 \       / 2v------8v------4w--7w
/ /   \ /     \ /     \     / / / \ /
3^--6^--9^--2---5---8---1v--4v--7v--0w
/ /     \     / \     / \   / /   /
8^/       \ 7 / 0v\ 3v/   \ 9v/ 2w/
\         \ / /   \ /     \ /   /
3 \       / 2v------8v------4w--7w
\     / / / \ /
8---1v--4v--7v--0w
/ \   / /   /
3v/   \ 9v/ 2w/
/     \ /   /
8v------4w--7w
```

The Miracle tempering gives more consonances than the diamond on its own. But there are still holes in the lattice.

Harry Partch's famous 43 note scale is designed to fill in the melodic gaps between notes in the 11-limit tonality diamond. So let's see what it looks like with Miracle tempering, where 5 is the 1/1.

```
4m^----
\
\
\
4^^-7^^
/ \
4m^-----0^^-3^^-6^^   \ 2^
\     / \       /     \
\   /   \ 8^^-1^--4^--7^
\ /     \   / \     /
4^^-7^^-0^--3^/ 6^\ 9^/ 2
/     \ /
4m^-----0^^-3^^-6^^-----2^------8^--1---4---7
\     / \     / \     / \     / \     / \
\   /   \8^^/ 1^\ 4^/ 7^\ 0 / 3 \ 6 / 9 \ 2v
\ /     \ /     \ /     \ /     \ /     \
4^^-7^^-0^--3^--6^--9^--2---5---8---1v--4v--7v
/ \     / \     / \     / \     / \     /
0^^-3^^-6^^---/-2^\   / 8^\ 1 / 4 \ 7 / 0v\ 3v/ 6v\ 9v/ 2w
/         /     \ /     \ /     \ /     \ /     \ /
/ 8^^-1^--4^--7^--0---3---6---9---2v------8v------4w--7w
/   \     / \     / \     / \     /
0^  3^\ 6^/ 9^\ 2 / 5 \ 8 / 1v\ 4v/ 7v--0w--3w--6w
\     \ /     \ /     \ /     \ / / \     /
\ 8^--1---4---7---0v--3v--6v--9v--2w\   /
\   / \     / \     / \     / /     \ /
0---3 / 6 \ 9 / 2v\   / 8v\   / 4w--7w--0vw
/     \ /     \ /     \ /
5---8---1v--4v--7v--0w--3w--6w
\     /   / \
0v--3v--6v--9v\ 2w/   /   \
\     /       \ /   /     \
\ 8v------4w--7w--0vw-----6vw
\ /
0w--3w--6w
/ \
/   \
/     \
0vw-----6vw
```

That has fewer holes than the plain tonality diamond, although there are still a few conspicuous gaps. One important feature is that all 43 notes of the scale are distinct in Miracle temperament, and so a notation based on Miracle can be used with no ambiguity. If you take out some notes that are more complex, in Miracle terms, than the tonality diamond and replace them with notes nearby in pitch that fill holes in the lattice you happen to get a slightly lesser known scale Partch gave in "Exposition on Monophony", 1933 (I found it in Manuel Op De Coul's scale archive). You use 45/44, 88/45 instead of 81/80, 160/81 and 15/11, 22/15 instead of 27/20, 40/27.

```                                                    4^^-7^^
/ \
3^^ 6^^ 9^^ 2^
/     \
8^^-1^--4^--7^
/ \     /
4^^-7^^-0^--3^/ 6^\ 9^/ 2
/     \ /
3^^-6^^-9^^-2^--5^--8^--1---4---7
/ \     / \     / \     / \
8^^/ 1^\ 4^/ 7^\ 0 / 3 \ 6 / 9 \ 2v
/     \ /     \ /     \ /     \
4^^-7^^-0^--3^--6^--9^--2---5---8---1v--4v--7v
/ \     / \     / \     / \     / \     /
3^^-6^^-9^^-2^\ 5^/ 8^\ 1 / 4 \ 7 / 0v\ 3v/ 6v\ 9v/ 2w
/         /     \ /     \ /     \ / / \ \ /     \ /
/ 8^^-1^--4^--7^--0---3---6---9---2v--5v--8v--1w--4w--7w
/ / \     / \     / \     / \     / /     \
0^/ 3^\ 6^/ 9^\ 2 / 5 \ 8 / 1v\ 4v/ 7v--0w--3w--6w
/     \ /     \ /     \ /     \ / /
5^--8^--1---4---7---0v--3v--6v--9v--2w
\     / \     / \     / \     / /
0 \ 3 / 6 \ 9 / 2v\ 5v/ 8v\ 1w/ 4w--7w
\ /     \ /     \ /     \ /
5---8---1v--4v--7v--0w--3w--6w
/ \     / \     /
0v/ 3v\ 6v/ 9v\ 2w/
/     \ /     \ /
5v--8v--1w--4w--7w
\ /
0w--3w--6w
```

In decimal terms, this is quite a useful scale. It's contained within 45 notes from a chain of Miracle generators. That span can't be improved on so long as you have both 11/10 and 10/9 and octave complements. However, you can alter it to get a superset of a 41 note MOS (Stud Loco).

```                                                        7^^

(3^^)6^^-9^^-2^
/
5^^/ 8^^-1^--4^--7^
/     / \     /
7^^-0^--3^/ 6^\ 9^/ 2
/     \ /
(3^^)6^^-9^^-2^--5^--8^--1---4---7
/ \     / \     / \     / \
5^^ 8^^/ 1^\ 4^/ 7^\ 0 / 3 \ 6 / 9 \ 2v
/     \ /     \ /     \ /     \
7^^-0^--3^--6^--9^--2---5---8---1v--4v--7v
/ \     / \     / \     / \     / \     /
(3^^)6^^-9^^-2^\ 5^/ 8^\ 1 / 4 \ 7 / 0v\ 3v/ 6v\ 9v/ 2w
/         /     \ /     \ /     \ / / \ \ /     \ /
5^^-8^^-1^--4^--7^--0---3---6---9---2v--5v--8v--1w--4w--7w
/ / \     / \     / \     / \     / /     \
0^/ 3^\ 6^/ 9^\ 2 / 5 \ 8 / 1v\ 4v/ 7v--0w--3w
/     \ /     \ /     \ /     \ / /
5^--8^--1---4---7---0v--3v--6v--9v--2w--5w
\     / \     / \     /         /
0 \ 3 / 6 \ 9 / 2v\ 5v/ 8v--1w--4w-(7w)
\ /     \ /     \ /
5---8---1v--4v--7v--0w--3w
/ \     / \     /
0v/ 3v\ 6v/ 9v\ 2w/ 5w
/     \ /     \ /
5v--8v--1w--4w-(7w)
/
0w/-3w
/
5w
```

Notes in brackets lie outside the MOS.

Here's one of my favourite 7-limit scales I worked out a few years back.

```7^^-----3^
\     / \
\   /   \
\ /     \
0-------6-------2v

2-------8-------4v
```

2 and 2v are considered equivalent notes. 0-2-4v-6-8-0 is close to 5-equal, and looks like it with this notation.

An approximation to one of Paul Erlich's pentachordal decatonics

```        9^^
/ \
8^^-----4^------0
\   / / \ \   / \
\ 6^/ 9^\ 2 / 5 \
\ /   \ \ / /   \
1-----\-7-/-----3v
\ /
2v
```

The pairs (9^,9^^) and (2,2v) are equivalent in 22-equal, so decimal names match decatonic degrees. So you could almost use decimal notation for decatonic music. But in practice you'd probably want a way of specifying the note between 9^ and 9^^. At least you can share manuscript paper for decimal and decatonic music. Here's an approximation to a symmetric decatonic

```    9^^-----5^------1
/ \     / \     /
/ 4^--7^--0---3---6
/ / \ \ / / \ \ / /
6^/ 9^\ 2 / 5 \ 8 /
/     \ /     \ /
1-------7-------3v
```

This scale comes from 1989 paper "D'Alessandro, Like a Hurricane" in Xenharmonikon XII, also available from The Wilson Archive. I'm using the 1989 version from Figure 24.

```                1^ (4^) 7^      3   6   9   2v      8v      4w      0vw

3^------9^------5---8---1v--4v-(7v)-0w--3w--6w-----(2vw)
/ \     / \     /   / \     / \     / \     /
(2^)   / 8^\ 1 / 4 \(7)/ 0v/ 3v\ 6v/ 9v\ 2w/ 5w\   /1vw
/     \ /     \ /   /     \ /     \ /     \ /
4^------0-------6-------2v--5v--8v--1w------7w-(0vw)3vw
```

You can see that some notes, like 6 and 2v, are duplicated in the diagram. That means they cease to be distinct notes in Miracle temperament, and so information is lost in the conversion. Notes in brackets are also bracketed in the diagram. They're outside the strict scale, but added to fill a 31-note keyboard. Apply the last Miracle unison vector, and you get

```                                                              (2^)
/ \
1^--4^/ 7^\ 0
/     /     \
/     9^------5
/     / \     /
(2^)-----8^--1 / 4 \(7)/ 0v
/     \ /
1^--4^--7^--0---3---6---9---2v--5v
/ \     / \     / \   \     / \
/   \ 9^/   \ 5 / 8 \ 1v\ 4v/(7v) 0w
/     \ /     \ /     \   \ /     \
(2^)-----8^--1---4--(7)--0v--3v--6v--9v--2w--5w
/ \     / \     / \     / \     /   / \     /
1^  4^/ 7^\ 0 / 3 \ 6 / 9 \ 2v/ 5v\ 8v/ 1w/ 4w\ 7w/ 0vw
/     \ /     \ /     \ /     \ /   /     \ /
3^------9^------5---8---1v--4v-(7v)-0w--3w--6w----(2vw)
/ \     / \     / \     / \     / \     /
(2^)   / 8^\ 1 / 4 \(7)/ 0v\ 3v/ 6v\ 9v/ 2w\ 5w/    1vw
/     \ /     \ /     \ /     \ /     \ /
4^--7^--0---3---6---9---2v--5v--8v--1w--4w--7w-0vw-3vw
\     / \     / \     / \     / \     /
9^\   / 5 \ 8 / 1v\ 4v/(7v) 0w/ 3w\ 6w/    (2vw)
\ /     \ /     \ /     \ /     \ /
4--(7)--0v--3v--6v--9v--2w--5w------1vw
/ \     / \     / \     /
9 / 2v\ 5v/ 8v\ 1w/ 4w\ 7w/0vw 3vw
/     \ /     \ /     \ /
4v-(7v)-0w--3w--6w----(2vw)
\     / \     /
9v\ 2w/ 5w\   /1vw
\ /     \ /
4w--7w-0vw-3vw

2vw
```

That's close to Stud Loco. This isn't so impressive, as the tuning was designed for a 31 note keyboard and it doesn't even fit our 41 note scale, but then it shouldn't be surprising that it works better in the system it was designed for. You can see the gaps in the lattice, some of which are filled in by extra notes Erv added. I think you need 43 notes from a chain of Miracle generators to contain the whole scale. The extremes are 5.7 (1^) and 3.9.11 (3vw). Next to them are the bonus notes 2^ and 2vw. Remove either of them, and you can get a subset of the 41-note Miracle MOS plus one extra note.

You also get this 1.3.5.7.9.11 eikosany:

```                1^------7^------3---6
\     / \     / \
\ 9^/   \ 5 / 8---1v--4v------0w
\ /     \ /   \ \   / \     / \
4-------0v--3v\-6v/ 9v\   / 5w\
\ /     \ /     \
8v--1w------7w------3vw
```

There are 5 approximate eikosanies Stud Loco.

Now see the page on decimal notation.