# L & s (L = Large, s = small)

[Joe Monzo]

The characteristic feature of diatonic scales is the use of two (or possibly more) different interval sizes between degrees of the scale. In the familiar major scale, for example, all pitches have between them either a tone [= 'whole-step'] or semitone [= 'half-step'].

In tuning theory, many scales have this distinction but the intervals do not correspond closely enough to those two (tones and semitones) for the names to make sense logically, so theorists often use 'L' for 'large' and 's' for 'small' to designate the two different sizes of intervals.

Some scales have 3 distinctive intervals and therefore another letter (i.e., 'm' for 'medium') is designated to mark the third one.

Some composers/theorists who use 'L' and 's' are Charles Lucy, Dan Stearns, and Paul Erlich. Below is a list of ratio log(L)/log(s) for several common tunings:

#### ratio of log(L)/log(s)

 tuning ratio decimal 5-edo infinity 42-edo 8:1 8.0 37-edo 7:1 7.0 32-edo 6:1 6.0 27-edo 5:1 5.0 49-edo 9:2 4.5 22-edo 4:1 4.0 39-edo 7:2 3.5 17-edo 3:1 3.0 46-edo 8:3 2.66... 29-edo 5:2 2.5 41-edo 7:3 2.33... pythagorean ~2.260016753 53-edo 9:4 2.25 12-edo 2:1 2.0 1/6-comma meantone ~1.819203619 55-edo 9:5 1.8 43-edo 7:4 1.75 1/5-comma meantone ~1.748010733 31-edo 5:3 1.66... 1/4-comma meantone ~1.649392797 golden meantone ~1.618033989 (= phi) 50-edo 8:5 1.6 LucyTuning ~1.558617346 19-edo 3:2 1.5 45-edo 7:5 1.4 26-edo 4:3 1.33... 33-edo 5:4 1.25 40-edo 6:5 1.2 47-edo 7:6 1.166... 7-edo 1:1 1.0 23-edo 3:4 0.75 16-edo 2:3 0.66... 9-edo 1:2 0.5 11-edo 1:3 0.33...
. . . . . . . . .
[Charles Lucy, private communication with Joe Monzo]

My first use of L and s, for Large and small intervals was in Pitch, Pi, and Other Musical Paradoxes.

John 'Longitude' Harrison referred to the "Larger" and "lesser" "notes", which I translated as Large (L) and small (s) intervals.

An easy way to visualise the Large and small intervals is to view the white notes on a conventional piano keyboard. The Large intervals are between white notes seperated by a black note. C-D; D-E; F-G; G-A; and A-B. The small intervals are between adjacent white notes. E-F; and B-C.

5L + 2s = one octave.

The principle may be applied to any meantone-type tuning, and most ET systems, using the appropriate values for L and s. All intervals may be described in terms of plus or minus Large and small intervals

The Large interval (L) is the wholetone (IInd). In LucyTuning specifically the ratio is 2^(1/(2*pi) = 1.116633 or "the two pi root of two". That is 1200/(2*pi) cents = 190.9858 cents.

The small interval (s) is half the difference between 5 Large intervals and one octave. The small interval (s) is the "flat second" (bIInd). In LucyTuning specifically the ratio is (2/(2^(1/(2*pi)))^5)^(1/2) = 1.073344. That is s= 122.5354 cents.

Details for the use of L&s in LucyTuning may be found at http://www.harmonics.com/lucy/lsd/chap1.html and the same principle applied to other tuning systems may be found at http://www.harmonics.com/lucy/lsd/meanet.html

. . . . . . . . .
[Kraig Grady, Yahoo tuning group, message 6752 (Tue Dec 7, 1999 10:16 pm)]

[Erv] Wilson uses them to designate large and small. The patterns of MOS create a pattern of L & s's that can be replaced with a constant structure preserving this L-S pattern

. . . . . . . . .
[Dan Stearns, Yahoo tuning group, message 7996]
###### L & s -- INDEXING TWO STEP SIZE CARDINALITY:

In mapping scales with two step size cardinality, I use the first mediant of L&s where L=2 & s=1 as the indexing template, you could look at this as the left-hand side of Erv Wilson's Scale-Tree (or Pierce sequence):

```		1/0
1/1
2/1
```

which could also be seen more broadly in this generalized L & s expression:

```		L                                                        s
L+s
2L+s                           L+2s
3L+s            3L+2s         2L+3s             L+3s
...
```

This indexing template can be seen as a linear mapping of an interval (the variable "i" in the following) that is i/(x+y) where I'm letting "x" be L, and "y" be L+s, so in a 9L & 2s mapping for example, i=11, and i/(x+y) is 11/20, and in an [LLLLsLLLLLs] configuration you have a linear form of -1 0 1 2 3 4 5 6 7 8 9:

```
9---0---11---2---13---4---15---6---17---8---19

```

It's also interesting to note that this could also be illustrated as:

```                4
/|
/ |
13-19
/| /
/ |/
2--8
/| /
/ |/
11-17
/| /
/ |/
0--6
/| /
/ |/
9--15
| /
|/
4
```

Where i/(x+y), i.e. 11/20, is synonymous with the "3/2," and round (11/20)*11 is synonymous with the "5/4."

This 9L & 2s indexing template would be:

```round (e/20)*0
round (e/20)*2
round (e/20)*4
round (e/20)*6
round (e/20)*8
round (e/20)*9
round (e/20)*11
round (e/20)*13
round (e/20)*15
round (e/20)*17
round (e/20)*19
round (e/20)*20
```

where "e" is just a variable that indicates any EDO, and the expressions I've written are synonymous with the general expression of (LOG(2^(n/E))-LOG(1))*(e/LOG(2)) were "n" is any 0,1,2,...E number, and "E" indicates the EDO of the indexing template (i.e., x+y).

So in a 5L & 2s mapping where i=7, an [LLsLLLs] configuration gives a -1 0 1 2 3 4 5 linear form of:

```5---0---7---2---9---4---11
```

And using 16e as an example, you would have a 5L&2s periodic table index of [44-2444-2] -- this can be seen in the following L/s periodic table example:

```(1/0)   1/-1   1/-2   2/-2   2/-3   2/-4   2/-5
2/1   (2/0)   2/-1   3/-1   3/-2   3/-3   3/-4
3/2    3/1   (3/0)  (4/0)   4/-1   4/-2   4/-3
4/3    4/2    4/1    5/1   (5/0)   5/-1   5/-2
5/4    5/3    5/2    6/2    6/1   (6/0)   6/-1
6/5    6/4    6/3    7/3    7/2    7/1   (7/0)
7/6    7/5    7/4    8/4    8/3    8/2    8/1
(8/7)   8/6    8/5    9/5    9/4    9/3    9/2
9/7    9/6   10/6   10/5   10/4   10/3
(10/8)  10/7   11/7   11/6   11/5   11/4
11/8   12/8   12/7   12/6   12/5
(12/9)  13/9   13/8   13/7   13/6
(14/10) 14/9   14/8   14/7
15/10  15/9   15/8
(16/11) 16/10  16/9
17/11  17/10
(18/12) 18/11
19/12
(20/13)
```

However, 16e would also have an index of [3222331] when taken:

```round (e/12)* 0, 2, 4, 5, 7, 9, 11, 12
```

By making the first index match the second you get an index of [4,4,,-2''''4,,4,4,-2'''], where a ['] is always an indication to raise the index by 1, and a [,] is always an indication to lower an index by 1. And as L & s are always synonymous with the first mediant form of:

```round (e/E)*2
round (e/E)*1
```

This in turn could then be reduced to a final index of two step size cardinality:

```[33,1'3,331]
```

These would be the complete indexes for the EDOs in the 5L & 2s mapping that do not have a generating interval that falls between 4/7 and 3/5 and are neither of the ambiguous expansions of L and L+s (5 and 7e in this case, as the horizontal and vertical expansions of the periodic table could be seen as the cases of maximum ambiguity in a given mapping, as s=0 and s=L & L=s are what the horizontal and vertical expansions are working towards, i.e., L and L+s):

```35e  L=6 & s=3  [6636,663]
30e  L=5 & s=3  [5535553,]
28e  L=5 & s=2  [55,2'5,552]
25e  L=4 & s=2  [4424'442]
23e  L=4 & s=2  [4424,442]
21e  L=4 & s=2  [44,24,44,2]
20e  L=3 & s=2  [33'2,3'332]
18e  L=3 & s=2  [3323332,]
16e  L=3 & s=1  [33,1'3,331]
15e  L=3 & s=1  [33,133,31]
14e  L=2 & s=1  [22'122'21]
13e  L=2 & s=1  [2212'221]
11e  L=2 & s=1  [2212,221]
10e  L=2 & s=1  [22,1222,1]
9e  L=2 & s=1  [22,12,22,1]
8e  L=1 & s=1  [11'1,1'111]
6e  L=1 & s=1  [1111111,]
4e  L=1 & s=0  [11,0'1,110]
3e  L=1 & s=0  [11,011,10]
2e  L=0 & s=0  [00'000'00]
1e  L=0 & s=0  [0000'000]
```
. . . . . . . . .

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