Musical Wavelength Space

Suppose you've got a fretless string instrument and you want to place markers on the neck to help you find frequency ratios over open string frequencies. 

If you place a marker far from the sound hole so that 1/9 of the string is close to the tuning pegs and 8/9 of the string is free to vibrate when you pluck or bow or strum near the sound hole, the excited tone will have a frequency 9/8 over the frequency of the open string. If we put a marker there, I'll say that the marker is at 1/9 of the string length, meaning that 8/9 of the string is left free to vibrate.

The string length 8/9 and the frequency ratio 9/8 are reciprocal fractions, and this isn't a coincidence. Delightfully, the function that transforms 1/9 into 9/8 is also an involution, i.e. it can transform in both ways.
    y = x / (x - 1)
    frequency_ratio = (string_division) / (string_division - 1) 
    string_division = (frequency_ratio) / (frequency_ratio - 1)

If we divide the string into simple units, what simple frequency ratios do we get? Here's a little table

25.0 -> 25/24 _ 71c
24.5 -> 49/47 _ 72c
24.0 -> 24/23 _ 74c
23.5 -> 47/45 _ 75c
23.0 -> 23/22 _ 77c
22.5 -> 45/43 _ 79c
22.0 -> 22/21 _ 81c
21.5 -> 43/41 _ 82c
21.0 -> 21/20 _ 84c
20.5 -> 41/39 _ 87c
20.0 -> 20/19 _ 89c
19.8 -> 99/94 _ 90c
19.6 -> 98/93 _ 91c
19.5 -> 39/37 _ 91c
19.4 -> 97/92 _ 92c
19.2 -> 96/91 _ 93c
19.0 -> 19/18 _ 94c
18.8 -> 94/89 _ 95c
18.6 -> 93/88 _ 96c
18.5 -> 37/35 _ 96c
18.4 -> 92/87 _ 97c
18.2 -> 91/86 _ 98c
18.0 -> 18/17 _ 99c
17.8 -> 89/84 _ 100c
17.6 -> 88/83 _ 101c
17.5 -> 35/33 _ 102c
17.4 -> 87/82 _ 102c
17.2 -> 86/81 _ 104c
17.0 -> 17/16 _ 105c
16.8 -> 84/79 _ 106c
16.6 -> 83/78 _ 108c
16.5 -> 33/31 _ 108c
16.4 -> 82/77 _ 109c
16.2 -> 81/76 _ 110c
16.0 -> 16/15 _ 112c
15.8 -> 79/74 _ 113c
15.6 -> 78/73 _ 115c
15.5 -> 31/29 _ 115c
15.4 -> 77/72 _ 116c
15.2 -> 76/71 _ 118c
15.0 -> 15/14 _ 119c
14.8 -> 74/69 _ 121c
14.6 -> 73/68 _ 123c
14.5 -> 29/27 _ 124c
14.4 -> 72/67 _ 125c
14.2 -> 71/66 _ 126c
14.0 -> 14/13 _ 128c
13.8 -> 69/64 _ 130c
13.6 -> 68/63 _ 132c
13.5 -> 27/25 _ 133c
13.4 -> 67/62 _ 134c
13.2 -> 66/61 _ 136c
13.0 -> 13/12 _ 139c
12.8 -> 64/59 _ 141c
12.6 -> 63/58 _ 143c
12.5 -> 25/23 _ 144c
12.4 -> 62/57 _ 146c
12.2 -> 61/56 _ 148c
12.0 -> 12/11 _ 151c
11.8 -> 59/54 _ 153c
11.6 -> 58/53 _ 156c
11.5 -> 23/21 _ 157c
11.4 -> 57/52 _ 159c
11.2 -> 56/51 _ 162c
11.0 -> 11/10 _ 165c
10.8 -> 54/49 _ 168c
10.6 -> 53/48 _ 172c
10.5 -> 21/19 _ 173c
10.4 -> 52/47 _ 175c
10.2 -> 51/46 _ 179c
10.0 -> 10/9 _ 182c
9.9 -> 99/89 _ 184c
9.8 -> 49/44 _ 186c
9.7 -> 97/87 _ 188c
9.6 -> 48/43 _ 190c
9.5 -> 19/17 _ 193c
9.4 -> 47/42 _ 195c
9.3 -> 93/83 _ 197c
9.2 -> 46/41 _ 199c
9.1 -> 91/81 _ 202c
9.0 -> 9/8 _ 204c
8.9 -> 89/79 _ 206c
8.8 -> 44/39 _ 209c
8.7 -> 87/77 _ 211c
8.6 -> 43/38 _ 214c
8.5 -> 17/15 _ 217c
8.4 -> 42/37 _ 219c
8.3 -> 83/73 _ 222c
8.2 -> 41/36 _ 225c
8.1 -> 81/71 _ 228c
8.0 -> 8/7 _ 231c
7.9 -> 79/69 _ 234c
7.8 -> 39/34 _ 238c
7.7 -> 77/67 _ 241c
7.6 -> 38/33 _ 244c
7.5 -> 15/13 _ 248c
7.4 -> 37/32 _ 251c
7.3 -> 73/63 _ 255c
7.2 -> 36/31 _ 259c
7.1 -> 71/61 _ 263c
7.0 -> 7/6 _ 267c
6.9 -> 69/59 _ 271c
6.8 -> 34/29 _ 275c
6.7 -> 67/57 _ 280c
6.6 -> 33/28 _ 284c
6.5 -> 13/11 _ 289c
6.4 -> 32/27 _ 294c
6.3 -> 63/53 _ 299c
6.2 -> 31/26 _ 305c
6.1 -> 61/51 _ 310c
6.0 -> 6/5 _ 316c
5.9 -> 59/49 _ 322c
5.8 -> 29/24 _ 328c
5.7 -> 57/47 _ 334c
5.6 -> 28/23 _ 341c
5.5 -> 11/9 _ 347c
5.4 -> 27/22 _ 355c
5.3 -> 53/43 _ 362c
5.2 -> 26/21 _ 370c
5.1 -> 51/41 _ 378c
5.0 -> 5/4 _ 386c
4.9 -> 49/39 _ 395c
4.8 -> 24/19 _ 404c
4.7 -> 47/37 _ 414c
4.6 -> 23/18 _ 424c
4.5 -> 9/7 _ 435c
4.4 -> 22/17 _ 446c
4.3 -> 43/33 _ 458c
4.2 -> 21/16 _ 471c
4.1 -> 41/31 _ 484c
4.0 -> 4/3 _ 498c
3.9 -> 39/29 _ 513c
3.8 -> 19/14 _ 529c
3.7 -> 37/27 _ 545c
3.6 -> 18/13 _ 563c
3.5 -> 7/5 _ 583c
3.4 -> 17/12 _ 603c
3.3 -> 33/23 _ 625c
3.2 -> 16/11 _ 649c
3.1 -> 31/21 _ 674c
3.0 -> 3/2 _ 702c
2.9 -> 29/19 _ 732c
2.8 -> 14/9 _ 765c
2.7 -> 27/17 _ 801c
2.6 -> 13/8 _ 841c
2.5 -> 5/3 _ 884c
2.4 -> 12/7 _ 933c
2.3 -> 23/13 _ 988c
2.2 -> 11/6 _ 1049c
2.1 -> 21/11 _ 1119c
2.0 -> 2/1 _ 1200c

Now, there's no real reason to use decimal as string divisors. You could just as easily place a marker at 3.1 of a the string length as you could at 22/7 of the string length, but I wanted to see how this looked. I've also hidden any frequency ratios with numerators more than 99.

There are some conspicuous absences on this list. Like the justly tuned minor sixth, 8/5, or the justly tuned major seventh, 15/18, and the justly tuned minor seventh, 9/5. Our involution function certainly has less precision near the octave, so we might try using another decimal digit at the high end:

3.0 -> 3/2 _ 702c
2.96 -> 74/49 _ 714c
2.95 -> 59/39 _ 717c
2.92 -> 73/48 _ 726c
2.9 -> 29/19 _ 732c
2.88 -> 72/47 _ 738c
2.85 -> 57/37 _ 748c
2.84 -> 71/46 _ 751c
2.8 -> 14/9 _ 765c
2.76 -> 69/44 _ 779c
2.75 -> 11/7 _ 782c
2.72 -> 68/43 _ 793c
2.7 -> 27/17 _ 801c
2.68 -> 67/42 _ 809c
2.65 -> 53/33 _ 820c
2.64 -> 66/41 _ 824c
2.6 -> 13/8 _ 841c
2.56 -> 64/39 _ 858c
2.55 -> 51/31 _ 862c
2.52 -> 63/38 _ 875c
2.5 -> 5/3 _ 884c
2.48 -> 62/37 _ 894c
2.45 -> 49/29 _ 908c
2.44 -> 61/36 _ 913c
2.4 -> 12/7 _ 933c
2.36 -> 59/34 _ 954c
2.35 -> 47/27 _ 960c
2.32 -> 58/33 _ 976c
2.3 -> 23/13 _ 988c
2.28 -> 57/32 _ 999c
2.25 -> 9/5 _ 1018c
2.24 -> 56/31 _ 1024c
2.2 -> 11/6 _ 1049c
2.16 -> 54/29 _ 1076c
2.15 -> 43/23 _ 1083c
2.12 -> 53/28 _ 1105c
2.1 -> 21/11 _ 1119c
2.08 -> 52/27 _ 1135c
2.05 -> 41/21 _ 1158c
2.04 -> 51/26 _ 1166c
2.0 -> 2 _ 1200c

This got us our minor seventh, but not the other two chromatic 5-limit frequency ratios. It turns out those don't have finite decimal representations: A string divisor of 15/7 gives us a frequency ratio of 15/18, and a string divisor of 8/3 gives us a frequency ratio of 8/5.

I kind of like this? We've found a procedure which privileges a different set of ratios and intervals compared to normal just intonation.

It's not much of a difference: the 2.68 divisor gives us a frequency ratio of 67/42, which is a perceptually indistinguishable 5 cents flat of 8/5. The sounds are still there. But I still think it's a neat sound space. Maybe don't use the high precision divisors if you want a really distinct sound?

...
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Gamelan scales: Pelog and Slendro

In Indonesia there are musical ensembles, called gamelans, with weird tunings. It's wonderful. They don't have exact octaves. They have precise tuning within an ensemble but they vary widely between ensembles. They use inharmonic hammered metallophones with bell like timbres. Their scales are inharmonic. Everything about it is weird and wonderful and awesome.

I want to deeply understand Gamelan music. I'm going to approach this from many angles until something sticks.

One of the Gamelan scales is called Pelog. Some people will tell you it sounds like a subset of 9-EDO. It does sound like 9-EDO, but it's not 9-EDO. It has a wider octave, for example. Is it based on 9 equal divisions of a stretched octave? I'm not sure. I doubt it, because the intervals would still be equal, just equal at a different size, and they're not. But we'll look into it.

Do Gamelan scales minimize sensory dissonance of inharmonic resonators against each other or against a harmonic instrument? This might be part of it, but the metallophones sound pretty similar from region to region while the scales are noticeably different? But we'll look into it.

Before we look into those things, I just wanted to assume for a moment that the Pelog scale was harmonic, even though it's clearly not. I looked at a bunch of measured tunings for pelog scales from different gamelans, took averages in logarithmic frequency space, and tried to make sense of the scale that showed up that way as if the intervals were harmonic.

Here's my first stab at it:

Pelog scale, relative intervals:

    [ReSbAcM2, Prm2, ReSpAcA2, Prm2, m2, Asm2, AsM2] # [14/13, 13/12, 108/91, 13/12, 16/15, 11/10, 55/48] _ [128c, 139c, 297c, 139c, 112c, 165c, 236c]

Pelog scale, absolute intervals:

    [P1, ReSbAcM2, Sbm3, ReAcA4, P5, m6, AsGrd7, AsAsGrd8] # [1/1, 14/13, 7/6, 18/13, 3/2, 8/5, 44/25, 121/60] _ [0c, 128c, 267c, 563c, 702c, 814c, 979c, 1214c]

This has fairly simple frequency ratios and interval names in both the relative and absolute representations. The "ReSbAcM2" relative interval might not look very simple to you, but it's a Zalzalian neutral seconds from Persian/Ottoman/Arabic music theory, and I like it fine.

If you look at the relative scale degrees, you can see how they're fairly close to one or two steps of 9-EDO at 133 cents and 267 cents respectively. In so far as we could use 9-EDO the relative steps are:

     [1, 1, 2, 1, 1, 2, 1]

And the absolute steps are:

[0, 1, 2, 4, 5, 6, 8, 9]

i.e. we skip steps 3 and 7.

If we gave the scale tones traditional names, they would be 

    [low bem, gulu, dada, pelog, lima, nem, barang, high bem].

The other foundational scale in Gamelan is called Slendro. Some people will tell you that it sounds like 5-EDO. You might wonder, if Pelog sounds like 9-EDO and slendro sounds like 5-EDO, could we play Gamelan music in 9 * 5 = 45-EDO? I'm not sure. You could play something similar. In addition to their pseudo-octave being a stretched wide, I've heard that they don't have a notion of octave equivalence, but they at least have, like, low bem versus high bem. And some of their instruments have like 12 notes, so if the scales only have 5 and 9 notes respectively, there's got to be something like repetition of tones higher up or lower down, right?

Let's look at the tuning of slendro. Here are the measured tunings of 8 slendro scales from different regions, measurements by Jaap Kunst, as presented in "Interval Sizes in Javanese Slendro" by Larry Polansky.

Manisrenga: [219.5, 266.5, 227, 233.5, 258.5]

Kanjutmesem: [224, 253.5, 237.5, 232.5, 264]

Udanriris: [255.5, 256.5, 223.5, 235.5, 234]

Pengawesari: [251.5, 233.5, 233.5, 236, 250]

Rarasrum: [229.5, 227.5, 253, 232, 261.5]

Hardjanagara: [216, 249.5, 216, 262, 261.5]

Madukentir: [268.5, 242, 243, 230, 221]

Surak: [206, 231.5, 238.5, 265, 264.5]

These are all in cents, and they're all somewhat close to the 240 cent step of 5-EDO, but they range from 206 to 268, and I think we can provide a finer-grained analysis than "equal-ish?".

These eight scales are all reach a total of 1204 or 1205 cents, except for Kanjutmesem which reaches 1211.5. I think that's pretty tight agreement.

The first five scales have intervals that, to my eye, seem easily split between a small ~230c and a large ~260c.

Manisrenga: [S, L, S, S, L]

Kanjutmesem: [S, L, S, S, L]

Udanriris: [L, L, S, S, S]

Pengawesari: [L, S, S, S, L]

Rarasrum: [S, S, L, S, L]

And indeed they all have 2 large intervals and 3 small intervals, but not in the same places. The Manisrenga and Kanjutmesem modes are the same in this representation, but the others are all distinct. The other scales have very small intervals of 206c and 216c, and/or medium sized interval around 242c, which sure stand in the way of a nice clean binary classification into 2 interval sizes. I guess I would describe them as

Hardjanagara: [vS, L, vS, L, L]

Madukentir: [L, M, M, S, S]

Surak: [vS, S, M, L, L]

In so much as there is some kind of intervallic structure here, which you could argue against, and in so far as all these scales can be represented as deviations from the previous (2 large, 3 small) structure, I'd guess that we have these identities

[S, S, S] = [vS, L, vS]

[L, S] = [M, M]

[vS, M] = [S, S]

These identifies let us define all the intervals in terms of e.g. the small interval and a comma interval, {c},

vS = S - c

M = S + c

L = S + 2c

We could also define all of them in terms of e.g. the large interval and the comma interval:

M = L - c
S = L - 2c
vS = L - 3c

Here are some cut offs that produce my categorization:

    L if 270 > interval > 247:
    M if 247 > interval > 238:
    S if 238 > interval > 220:
    vS if 220 > interval > 205

This categorization gives us average sizes of

vS ~ 214c
S ~ 231c
M ~ 241c
L ~ 259c

which gives us comma sizes of [17c, 10c, 18c] cents between successively sized intervals. Not so close! From this it looks more like

    S - vS = L - M ~ 18c
    L - S = M - vS ~ 28c

For an {a} sized comma of 18 cents and a {b} sized comma of 28c we would have:
M = L - a
S = L - b
vS = L - a - b

or
     L = S + b
M = S + b - a
vS = S - a

These definitions only work with the previous identities derives from [pseudooctave = 2L + 3S] if 
    
    2a = b

I guess that's approximately true? Like if you only have 8 cent pitch discrimination. But that just brings us back to these definitions

M = L - c
S = L - 2c
vS = L - 3c


I don't know, man. Maybe it's fine. For a pseudo-octave of 12.5 cents a Large interval of 259 cents, this gives us a comma value of 15 cents and these tunings for all the intervals:

L = 259
M = 244
S = 229
vS = 214

Yeah, it's fine. Those are exactly where they should be. No shame.

Here's a funny idea: if they've got a 15 cent resolution on their intervals, perhaps their scales are based on 79 or 80 equal divisions of a 1205 cent pseudo-octave. That is, dear reader, a ridiculous number of divisions to represent a couple of pentatonic scales that are basically all just 5-EDO. But if we go ahead with it, then

    L = 17 steps
    M = 16 steps
    S = 15 steps
    vS = 14 steps

Okay, so we fit a model to some data. Now what? Validate against more data! Polansky presents data for 8 more measured slendro scales, these measurements taken by Gadja Mada. These ones unfortunately aren't associated with regional names:

1. [237, 251, 248, 242, 258] # 1236c
2. [252, 239, 242, 236.5, 253.5] # 1223c
3. [237, 238.5, 232.5, 262, 238] # 1208c
4. [226, 252, 260, 234, 256] # 1228c
5. [232, 239, 248, 232, 259.5] # 1211c
6. [218, 238.5, 244.5, 244.5, 260] # 1206c
7. [238, 230, 257, 243, 250.5] # 1218.5c
8. [232, 234, 249, 251, 257] # 1223c

These have sharper pseudo-octaves, at an average of 1219c.

Using the previous cut-off values for categorizing intervals, we get these descriptions:

     GM_1: [S, L, L, M, L]
GM_2: [L, M, M, S, L]
GM_3: [S, M, S, L, S]
GM_4: [S, L, L, S, L]
GM_5: [S, M, L, S, L]
GM_6: [vS, M, M, M, L]
GM_7: [S, S, L, M, L]
GM_8: [S, S, L, L, L]

these scales are not only larger than the Kunst intervals in absolute tuning, they're larger than the Kunst by whole commas, with the exception of GM_3. The Kunst scales were all 79 steps in total, whereas these are variously [82, 81, 78, 81, 80, 79, 80, 81] steps.

Well, now what?

Now perhaps we admit that if all these regions have scales that differ by by twenty cents or more positionally, and they're all called slendro, then maybe slendro isn't that precise of an object. Maybe you don't need precise scales when you have inharmonic instruments. You can do whatever.

Here's a histogram of relative step sizes from Kunst and GM with 30 bins:


Is there anything to conclude from this? I tried to make a qualitative description and then checked how it looked at 40 bins and a lot of my description stopped being true. It's only a range of 60 cents. I think we could excuse people deviating from a desired interval size by 35 cents in either direction if they've got inharmonic instruments. And maybe some peculiarity of the inharmonicity is why all the octaves are stretched a little. Maybe it's just 5-EDO.

...
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Some 17-limit Intervals

Alphabetical scales are made of 2nd intervals, like m2 and M2. I look up the sizes of second intervals often enough that I want a post for them. Here:

Sbm2 # 28/27 _ 63c
Prd2 # 26/25 _ 68c
SbAcm2 # 21/20 _ 84c
PrDem2 # 104/99 _ 85c
DeSbAcM2 # 35/33 _ 102c
m2 # 16/15 _ 112c
AsSbm2 # 77/72 _ 116c
ReSbAcM2 # 14/13 _ 128c
Acm2 # 27/25 _ 133c
Prm2 # 13/12 _ 139c
AsGrm2 # 88/81 _ 143c
DeAcM2 # 12/11 _ 151c
SbAcM2 # 35/32 _ 155c
Asm2 # 11/10 _ 165c
ReAcM2 # 72/65 _ 177c
M2 # 10/9 _ 182c
PrSpm2 # 39/35 _ 187c
AcM2 # 9/8 _ 204c
ReAsM2 # 44/39 _ 209c
DeAcA2 # 25/22 _ 221c
SpM2 # 8/7 _ 231c
AsM2 # 55/48 _ 236c
ReAcA2 # 15/13 _ 248c
SpAcM2 # 81/70 _ 253c
PrSpM2 # 65/56 _ 258c
DeSpAcA2 # 90/77 _ 270c
AcA2 # 75/64 _ 275c
AsSpM2 # 33/28 _ 284c
....

You know what? I want a list of microtonal unisons too.

P1 # 1 _ 0
DeDeAcAcA1 # 243/242 _ 7
AsSpGrd1 # 176/175 _ 10
PrPrSpGrd1 # 169/168 _ 10
ReDeAcA1 # 144/143 _ 12
AsAsGrd1 # 121/120 _ 14
ReSbAcA1 # 105/104 _ 17
 DeA1 # 100/99 _ 17
Ac1 # 81/80 _ 22
PrDeSp1 # 78/77 _ 22
ReAs1 # 66/65 _ 26
Pr1 # 65/64 _ 27
SpGr1 # 64/63 _ 27
AsGr1 # 55/54 _ 32
PrAsSpGrd1 # 143/140 _ 37
DeAcA1 # 45/44 _ 39
ReA1 # 40/39 _ 44
Sp1 # 36/35 _ 49
GrA1 # 250/243 _ 49
As1 # 33/32 _ 53
PrSpGr1 # 65/63 _ 54
DeDeAcAA1 # 125/121 _ 56
ReAsAsGr1 # 121/117 _ 58
ReReSbAcAA1 # 175/169 _ 60
ReAcA1 # 27/26 _ 65
DeSpA1 # 80/77 _ 66
ReReAsA1 # 176/169 _ 70
A1 # 25/24 _ 71
PrSp1 # 117/112 _ 76
SpSpGr1 # 256/245 _ 76
AsSpGr1 # 22/21 _ 81
ReDeAcAA1 # 150/143 _ 83
DeSpAcA1 # 81/77 _ 88

...

Let's just do a ton of 17-limit intervals with just tunings between 1/1/ and 2/1.

P1 # 1/1 _ 0c
DeA1 # 100/99 _ 17c
AsSpSpGrM0 # 99/98 _ 18c
PrSbd2 # 91/90 _ 19c
ExSpGrA0 # 85/84 _ 20c
Ac1 # 81/80 _ 22c
PrDeSp1 # 78/77 _ 22c
ReAs1 # 66/65 _ 26c
Pr1 # 65/64 _ 27c
SpGr1 # 64/63 _ 27c
DeSbAcm2 # 56/55 _ 31c
AsGr1 # 55/54 _ 32c
HbPrd2 # 52/51 _ 34c
Ex1 # 51/50 _ 34c
SpSpGrA0 # 50/49 _ 35c
SbSbAcm2 # 49/48 _ 36c
DeAcA1 # 45/44 _ 39c
ReA1 # 40/39 _ 44c
AsSbd2 # 77/75 _ 46c
Sp1 # 36/35 _ 49c
HbSbAcm2 # 35/34 _ 50c
ExDeA1 # 34/33 _ 52c
As1 # 33/32 _ 53c
PrSpGr1 # 65/63 _ 54c
PrDeSbAcm2 # 91/88 _ 58c
HbAsd2 # 88/85 _ 60c
Sbm2 # 28/27 _ 63c
ReAcA1 # 27/26 _ 65c
DeSpA1 # 80/77 _ 66c
Prd2 # 26/25 _ 68c
ExSpSpGrA0 # 51/49 _ 69c
A1 # 25/24 _ 71c
ExReA1 # 68/65 _ 78c
AsSpGr1 # 22/21 _ 81c
ExGrA1 # 85/81 _ 83c
SbAcm2 # 21/20 _ 84c
PrDem2 # 104/99 _ 85c
DeSpAcA1 # 81/77 _ 88c
ReSpA1 # 96/91 _ 93c
ReAsA1 # 55/52 _ 97c
HbAcm2 # 18/17 _ 99c
DeSbAcM2 # 35/33 _ 102c
PrSpSpGr1 # 52/49 _ 103c
ExA1 # 17/16 _ 105c
m2 # 16/15 _ 112c
AsSbm2 # 77/72 _ 116c
HbPrSbdd3 # 91/85 _ 118c
SpA1 # 15/14 _ 119c
ReSbAcM2 # 14/13 _ 128c
HbAsm2 # 55/51 _ 131c
ExSpGrA1 # 68/63 _ 132c
Acm2 # 27/25 _ 133c
Prm2 # 13/12 _ 139c
AsGrm2 # 88/81 _ 143c
ReAsSpA1 # 99/91 _ 146c
SbSbd3 # 49/45 _ 147c
ExReAA1 # 85/78 _ 149c
DeAcM2 # 12/11 _ 151c
SbAcM2 # 35/32 _ 155c
HbSbd3 # 56/51 _ 162c
ReSpAA1 # 100/91 _ 163c
Asm2 # 11/10 _ 165c
SpSpA1 # 54/49 _ 168c
HbAcM2 # 75/68 _ 170c
ExDeSpAA1 # 85/77 _ 171c
ReAcM2 # 72/65 _ 177c
M2 # 10/9 _ 182c
DeSbSbAcm3 # 49/44 _ 186c
PrSpm2 # 39/35 _ 187c
Sbd3 # 28/25 _ 196c
ExReSpAA1 # 102/91 _ 198c
AsSpSpGrA1 # 55/49 _ 200c
PrSbGrd3 # 91/81 _ 202c
AcM2 # 9/8 _ 204c
ReAsM2 # 44/39 _ 209c
Hbd3 # 96/85 _ 211c
DeSbm3 # 112/99 _ 214c
HbAsSbd3 # 77/68 _ 215c
ExM2 # 17/15 _ 217c
DeAcA2 # 25/22 _ 221c
PrSbd3 # 91/80 _ 223c
SpM2 # 8/7 _ 231c
DeSbAcm3 # 63/55 _ 235c
AsM2 # 55/48 _ 236c
HbPrd3 # 39/34 _ 238c
HbSbSbAcdd4 # 98/85 _ 246c
ReAcA2 # 15/13 _ 248c
PrGrd3 # 52/45 _ 250c
SpAcM2 # 81/70 _ 253c
ExDeAcA2 # 51/44 _ 256c
PrSpM2 # 65/56 _ 258c
Dem3 # 64/55 _ 262c
HbAsd3 # 99/85 _ 264c
Sbm3 # 7/6 _ 267c
DeSpAcA2 # 90/77 _ 270c
AcA2 # 75/64 _ 275c
AsGrd3 # 88/75 _ 277c
Hbm3 # 20/17 _ 281c
AsSpM2 # 33/28 _ 284c
ExA2 # 85/72 _ 287c
PrDem3 # 13/11 _ 289c
ReAsSbm3 # 77/65 _ 293c
Grm3 # 32/27 _ 294c
ReSpAcA2 # 108/91 _ 297c
SpA2 # 25/21 _ 302c
HbAcm3 # 81/68 _ 303c
DeSbAcM3 # 105/88 _ 306c
PrSpSpM2 # 117/98 _ 307c
m3 # 6/5 _ 316c
ExDeSbM3 # 119/99 _ 319c
AsSbm3 # 77/64 _ 320c
PrGrm3 # 65/54 _ 321c
ReAsSpA2 # 110/91 _ 328c
SbSbd4 # 98/81 _ 330c
ReSbAcM3 # 63/52 _ 332c
DeM3 # 40/33 _ 333c
PrSbdd4 # 91/75 _ 335c
ExSpA2 # 17/14 _ 336c
Prm3 # 39/32 _ 342c
AsGrm3 # 11/9 _ 347c
HbPrdd4 # 104/85 _ 349c
SpSpA2 # 60/49 _ 351c
SbSbAcd4 # 49/40 _ 351c
DeAcM3 # 27/22 _ 355c
ReM3 # 16/13 _ 359c
GrM3 # 100/81 _ 365c
AsAsSpSpGrM2 # 121/98 _ 365c
HbSbAcd4 # 21/17 _ 366c
ExDeM3 # 68/55 _ 367c
Asm3 # 99/80 _ 369c
PrSpGrm3 # 26/21 _ 370c
ExSbM3 # 119/96 _ 372c
Sbd4 # 56/45 _ 379c
ReAcM3 # 81/65 _ 381c
DeSpM3 # 96/77 _ 382c
M3 # 5/4 _ 386c
Hbd4 # 64/51 _ 393c
ReSbSbAc4 # 49/39 _ 395c
AsSpGrm3 # 44/35 _ 396c
ExGrM3 # 34/27 _ 399c
SbAcd4 # 63/50 _ 400c
AsAsGrm3 # 121/96 _ 401c
DeA3 # 125/99 _ 404c
PrSbd4 # 91/72 _ 405c
AcM3 # 81/64 _ 408c
ReAsM3 # 33/26 _ 413c
SpGrM3 # 80/63 _ 414c
HbAcd4 # 108/85 _ 415c
DeSbAc4 # 14/11 _ 418c
HbPrd4 # 65/51 _ 420c
ExM3 # 51/40 _ 421c
SpSpAA2 # 125/98 _ 421c
d4 # 32/25 _ 427c
ReA3 # 50/39 _ 430c
AsSbd4 # 77/60 _ 432c
PrGrd4 # 104/81 _ 433c
SpM3 # 9/7 _ 435c
ExDeA3 # 85/66 _ 438c
ReSbAc4 # 84/65 _ 444c
De4 # 128/99 _ 445c
HbAsd4 # 22/17 _ 446c
Sb4 # 35/27 _ 449c
DeSpA3 # 100/77 _ 452c
Prd4 # 13/10 _ 454c
A3 # 125/96 _ 457c
SpSpGrM3 # 64/49 _ 462c
SbSbdd5 # 98/75 _ 463c
ExReA3 # 17/13 _ 464c
DeAc4 # 72/55 _ 466c
AsSpGrM3 # 55/42 _ 467c
SbAc4 # 21/16 _ 471c
PrDe4 # 130/99 _ 472c
HbSbdd5 # 112/85 _ 478c
ReSpA3 # 120/91 _ 479c
Asd4 # 33/25 _ 481c
ExSb4 # 119/90 _ 484c
HbAc4 # 45/34 _ 485c
ExDeSpA3 # 102/77 _ 487c
PrSpSpGrM3 # 65/49 _ 489c
ExA3 # 85/64 _ 491c
PrDeAc4 # 117/88 _ 493c
ReAsAsSpGrM3 # 121/91 _ 493c
P4 # 4/3 _ 498c
HbPrSbdd5 # 91/68 _ 504c
SpA3 # 75/56 _ 506c
AsAsGrd4 # 121/90 _ 512c
ReSbAcA4 # 35/26 _ 515c
AsSpSpGrM3 # 66/49 _ 516c
ExSpGrA3 # 85/63 _ 519c
Ac4 # 27/20 _ 520c
PrDeSp4 # 104/77 _ 520c
ExDeSbAcA4 # 119/88 _ 522c
ReAs4 # 88/65 _ 524c
Pr4 # 65/48 _ 525c
AsGr4 # 110/81 _ 530c
Ex4 # 34/25 _ 532c
SbSbd5 # 49/36 _ 534c
DeAcA4 # 15/11 _ 537c
Sp4 # 48/35 _ 547c
HbSbd5 # 70/51 _ 548c
ReSpAA3 # 125/91 _ 550c
ExDeA4 # 136/99 _ 550c
As4 # 11/8 _ 551c
HbPrdd5 # 117/85 _ 553c
SpSpA3 # 135/98 _ 555c
PrDeSbd5 # 91/66 _ 556c
SbGrd5 # 112/81 _ 561c
ReAcA4 # 18/13 _ 563c
PrGrdd5 # 104/75 _ 566c
ExSpSpGrA3 # 68/49 _ 567c
A4 # 25/18 _ 569c
PrSp4 # 39/28 _ 574c
AsSpGr4 # 88/63 _ 579c
Sbd5 # 7/5 _ 583c
DeSpAcA4 # 108/77 _ 586c
AcA4 # 45/32 _ 590c
ReSpA4 # 128/91 _ 591c
ReAsA4 # 55/39 _ 595c
Hbd5 # 24/17 _ 597c
DeSb5 # 140/99 _ 600c
AsSp4 # 99/70 _ 600c
ExA4 # 17/12 _ 603c
PrDed5 # 78/55 _ 605c
DeAcAA4 # 125/88 _ 608c
PrSbd5 # 91/64 _ 609c
Grd5 # 64/45 _ 610c
HbAsAsGrdd5 # 121/85 _ 611c
AsSbGrd5 # 77/54 _ 614c
SpA4 # 10/7 _ 617c
DeSbAc5 # 63/44 _ 621c
ReSb5 # 56/39 _ 626c
d5 # 36/25 _ 631c
AsAsSpGr4 # 121/84 _ 632c
HbSbSbAcd6 # 49/34 _ 633c
ReAcAA4 # 75/52 _ 634c
PrGrd5 # 13/9 _ 637c
SpAcA4 # 81/56 _ 639c
ReAsSpA4 # 132/91 _ 644c
De5 # 16/11 _ 649c
HbAsd5 # 99/68 _ 650c
ExSpA4 # 51/35 _ 652c
Sb5 # 35/24 _ 653c
PrAsSpSpGr4 # 143/98 _ 654c
Prd5 # 117/80 _ 658c
AsGrd5 # 22/15 _ 663c
ExSbGr5 # 119/81 _ 666c
SpSpA4 # 72/49 _ 666c
Hb5 # 25/17 _ 668c
DeAc5 # 81/55 _ 670c
Re5 # 96/65 _ 675c
PrDe5 # 65/44 _ 676c
ReAsSb5 # 77/52 _ 680c
Gr5 # 40/27 _ 680c
HbSbAcd6 # 126/85 _ 681c
ReSpAcAA4 # 135/91 _ 683c
DeSbSbAcm6 # 49/33 _ 684c
PrSpGrd5 # 52/35 _ 685c
ExSb5 # 119/80 _ 687c
SpAA4 # 125/84 _ 688c
PrAsGrd5 # 143/96 _ 690c
Sbd6 # 112/75 _ 694c
AsAsGrGrd5 # 121/81 _ 695c
ExReSpAA4 # 136/91 _ 696c
P5 # 3/2 _ 702c
Hbd6 # 128/85 _ 709c
ReSbSbAcm6 # 98/65 _ 711c
HbAsSbd6 # 77/51 _ 713c
ExGr5 # 68/45 _ 715c
AsAsGrd5 # 121/80 _ 716c
DeA5 # 50/33 _ 719c
PrSbd6 # 91/60 _ 721c
ExSpAA4 # 85/56 _ 722c
PrDeSp5 # 117/77 _ 724c
ReAs5 # 99/65 _ 728c
SpGr5 # 32/21 _ 729c
ExReSbA5 # 119/78 _ 731c
DeSbAcm6 # 84/55 _ 733c
AsGr5 # 55/36 _ 734c
HbPrd6 # 26/17 _ 736c
SpSpAA4 # 75/49 _ 737c
SbSbAcm6 # 49/32 _ 738c
DeAcA5 # 135/88 _ 741c
ReA5 # 20/13 _ 746c
AsSbd6 # 77/50 _ 748c
Sp5 # 54/35 _ 751c
GrA5 # 125/81 _ 751c
HbSbAcm6 # 105/68 _ 752c
ExDeA5 # 17/11 _ 754c
As5 # 99/64 _ 755c
PrSpGr5 # 65/42 _ 756c
ReAsAsGr5 # 121/78 _ 760c
HbAsd6 # 132/85 _ 762c
Sbm6 # 14/9 _ 765c
ReAcA5 # 81/52 _ 767c
DeSpA5 # 120/77 _ 768c
Prd6 # 39/25 _ 770c
ExSpSpAA4 # 153/98 _ 771c
A5 # 25/16 _ 773c
Hbm6 # 80/51 _ 779c
ExReA5 # 102/65 _ 780c
AsSpGr5 # 11/7 _ 782c
ExGrA5 # 85/54 _ 785c
SbAcm6 # 63/40 _ 786c
PrDem6 # 52/33 _ 787c
Grm6 # 128/81 _ 792c
ReSpA5 # 144/91 _ 795c
ExSbm6 # 119/75 _ 799c
SpGrA5 # 100/63 _ 800c
HbAcm6 # 27/17 _ 801c
PrAsGrd6 # 143/90 _ 802c
DeSbAcM6 # 35/22 _ 804c
PrSpSpGr5 # 78/49 _ 805c
ExA5 # 51/32 _ 807c
m6 # 8/5 _ 814c
ReAA5 # 125/78 _ 816c
AsSbm6 # 77/48 _ 818c
PrGrm6 # 130/81 _ 819c
SpA5 # 45/28 _ 821c
AsAsGrd6 # 121/75 _ 828c
ReSbAcM6 # 21/13 _ 830c
DeM6 # 160/99 _ 831c
HbAsm6 # 55/34 _ 833c
ExSpGrA5 # 34/21 _ 834c
Acm6 # 81/50 _ 835c
DeSpAA5 # 125/77 _ 839c
Prm6 # 13/8 _ 841c
AsGrm6 # 44/27 _ 845c
SpSpGrA5 # 80/49 _ 849c
SbSbd7 # 49/30 _ 849c
ExReAA5 # 85/52 _ 851c
DeAcM6 # 18/11 _ 853c
SbAcM6 # 105/64 _ 857c
ReM6 # 64/39 _ 858c
HbSbd7 # 28/17 _ 864c
ReSpAA5 # 150/91 _ 865c
Asm6 # 33/20 _ 867c
PrSpGrm6 # 104/63 _ 868c
ExSbM6 # 119/72 _ 870c
SpSpA5 # 81/49 _ 870c
PrDeSbd7 # 91/55 _ 872c
ReAcM6 # 108/65 _ 879c
DeSpM6 # 128/77 _ 880c
M6 # 5/3 _ 884c
DeSbSbAcm7 # 147/88 _ 888c
PrSpm6 # 117/70 _ 889c
ExGrM6 # 136/81 _ 897c
Sbd7 # 42/25 _ 898c
AsAsGrm6 # 121/72 _ 899c
ExReSpAA5 # 153/91 _ 900c
HbPrAsGrdd7 # 143/85 _ 901c
AsSpSpGrA5 # 165/98 _ 902c
PrSbGrd7 # 91/54 _ 903c
AcM6 # 27/16 _ 906c
PrDeSpM6 # 130/77 _ 907c
ReAsM6 # 22/13 _ 911c
Hbd7 # 144/85 _ 913c
DeSbm7 # 56/33 _ 916c
ExM6 # 17/10 _ 919c
PrAsSpGrm6 # 143/84 _ 921c
DeAcA6 # 75/44 _ 923c
Grd7 # 128/75 _ 925c
PrPrDeGrd7 # 169/99 _ 926c
AsSbGrd7 # 77/45 _ 930c
SpM6 # 12/7 _ 933c
ExDeA6 # 170/99 _ 936c
AsM6 # 55/32 _ 938c
HbPrd7 # 117/68 _ 939c
ReSbm7 # 112/65 _ 942c
PrPrSpSpGrm6 # 169/98 _ 943c
HbAsGrd7 # 88/51 _ 944c
SbGrm7 # 140/81 _ 947c
AsAsSpGrm6 # 121/70 _ 947c
HbSbSbAcdd8 # 147/85 _ 948c
ReAcA6 # 45/26 _ 950c
PrGrd7 # 26/15 _ 952c
ExSpSpGrAA5 # 85/49 _ 954c
A6 # 125/72 _ 955c
ExDeAcA6 # 153/88 _ 958c
ExReA6 # 68/39 _ 962c
Dem7 # 96/55 _ 964c
AsSpGrM6 # 110/63 _ 965c
Sbm7 # 7/4 _ 969c
DeSpAcA6 # 135/77 _ 972c
ReSpA6 # 160/91 _ 977c
AsGrd7 # 44/25 _ 979c
PrPrGrd7 # 169/96 _ 979c
Hbm7 # 30/17 _ 983c
PrAsGrGrd7 # 143/81 _ 984c
ExDeSpA6 # 136/77 _ 985c
DeSbM7 # 175/99 _ 986c
AsSpM6 # 99/56 _ 986c
ExA6 # 85/48 _ 989c
PrDem7 # 39/22 _ 991c
Grm7 # 16/9 _ 996c
HbAsAsGrd7 # 121/68 _ 998c
ReSpAcA6 # 162/91 _ 998c
DeSbSbAcd8 # 98/55 _ 1000c
HbPrSbdd8 # 91/51 _ 1002c
SpA6 # 25/14 _ 1004c
PrAsGrd7 # 143/80 _ 1006c
ReSbM7 # 70/39 _ 1013c
AsSpSpGrM6 # 88/49 _ 1014c
m7 # 9/5 _ 1018c
ExDeSbM7 # 119/66 _ 1021c
PrGrm7 # 65/36 _ 1023c
HbAsSbdd8 # 154/85 _ 1029c
ReAsSpA6 # 165/91 _ 1030c
ExGrm7 # 136/75 _ 1030c
SbSbd8 # 49/27 _ 1032c
DeM7 # 20/11 _ 1035c
PrSbdd8 # 91/50 _ 1037c
ExSpA6 # 51/28 _ 1038c
SbM7 # 175/96 _ 1039c
Prm7 # 117/64 _ 1044c
SpGrm7 # 64/35 _ 1045c
ExReSbM7 # 119/65 _ 1047c
AsGrm7 # 11/6 _ 1049c
HbPrdd8 # 156/85 _ 1051c
SpSpA6 # 90/49 _ 1053c
SbSbAcd8 # 147/80 _ 1053c
HbM7 # 125/68 _ 1054c
PrDeSbd8 # 182/99 _ 1054c
DeAcM7 # 81/44 _ 1057c
ReM7 # 24/13 _ 1061c
GrM7 # 50/27 _ 1067c
HbSbAcd8 # 63/34 _ 1068c
ExDeM7 # 102/55 _ 1069c
PrSpGrm7 # 13/7 _ 1072c
ExSbM7 # 119/64 _ 1074c
ReAsAsGrm7 # 121/65 _ 1076c
Sbd8 # 28/15 _ 1081c
ExReSpAA6 # 170/91 _ 1082c
DeSpM7 # 144/77 _ 1084c
M7 # 15/8 _ 1088c
PrPrGrdd8 # 169/90 _ 1091c
Hbd8 # 32/17 _ 1095c
ReSbSbAc8 # 49/26 _ 1097c
AsSpGrm7 # 66/35 _ 1098c
ExGrM7 # 17/9 _ 1101c
AsAsGrm7 # 121/64 _ 1103c
PrDed8 # 104/55 _ 1103c
DeA7 # 125/66 _ 1106c
PrSbd8 # 91/48 _ 1107c
AsSbGrd8 # 154/81 _ 1112c
ReAsM7 # 99/52 _ 1115c
SpGrM7 # 40/21 _ 1116c
HbAcd8 # 162/85 _ 1117c
PrAsGrdd8 # 143/75 _ 1117c
ExAsSpSpGrA6 # 187/98 _ 1119c
DeSbAc8 # 21/11 _ 1119c
HbPrd8 # 65/34 _ 1122c
ExM7 # 153/80 _ 1123c
d8 # 48/25 _ 1129c
PrPrDed8 # 169/88 _ 1130c
AsAsSpGrGrm7 # 121/63 _ 1130c
HbSbSbAcd9 # 98/51 _ 1131c
ReA7 # 25/13 _ 1132c
AsSbd8 # 77/40 _ 1134c
PrGrd8 # 52/27 _ 1135c
SpM7 # 27/14 _ 1137c
ExDeA7 # 85/44 _ 1140c
ReAsSpGrM7 # 176/91 _ 1142c
ReSbAc8 # 126/65 _ 1146c
De8 # 64/33 _ 1147c
HbAsd8 # 33/17 _ 1148c
ExSpGrM7 # 68/35 _ 1150c
Sb8 # 35/18 _ 1151c
ExAsGrM7 # 187/96 _ 1154c
DeSpA7 # 150/77 _ 1154c
Prd8 # 39/20 _ 1156c
A7 # 125/64 _ 1159c
AsGrd8 # 88/45 _ 1161c
SpSpGrM7 # 96/49 _ 1164c
SbSbAcd9 # 49/25 _ 1165c
Hb8 # 100/51 _ 1166c
ExReA7 # 51/26 _ 1166c
DeAc8 # 108/55 _ 1168c
AsSpGrM7 # 55/28 _ 1169c
SbAc8 # 63/32 _ 1173c
Re8 # 128/65 _ 1173c
PrDe8 # 65/33 _ 1174c
ReAsSb8 # 77/39 _ 1178c
Gr8 # 160/81 _ 1178c
HbSbAcd9 # 168/85 _ 1180c
ReSpA7 # 180/91 _ 1181c
DeSbSbAcm9 # 196/99 _ 1182c
Asd8 # 99/50 _ 1183c
ExSb8 # 119/60 _ 1186c
SpGrA7 # 125/63 _ 1186c
HbAc8 # 135/68 _ 1187c
PrAsGrd8 # 143/72 _ 1188c
ExDeSpA7 # 153/77 _ 1189c
HbPrPrdd9 # 169/85 _ 1190c
DeSbAcA8 # 175/88 _ 1190c
PrSpSpGrM7 # 195/98 _ 1191c
P8 # 2/1 _ 1200c

Now that's useful.
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