# LucyTuning

### and 88-ed2 / 88-edo / 88-ET / 88-tone equal-temperament, and 3/10-comma meantone

[Joe Monzo]

LucyTuning is a tuning invented by John 'Longitude' Harrison in the 1700s and rediscovered and promoted currently by Charles Lucy. It is a type of meantone composed of two different step sizes (designated L and s) such that 5L + 2s = one octave, as in many familiar diatonic scales.

The Large interval (L) is the "whole-tone" (IInd), with the ratio 2π√2 ("the two pi root of two") or 2(1 / 2π) = ~1.116633. That is 1200 / 2π cents = ~190.9858 cents.

The small interval (s) is half the difference between 5 Large intervals and one octave, i.e. (8ve - 5L) / 2 , and is the "flat second" (bIInd), with the ratio (2 / 2(5 / 2π))(1/2) = ~1.073344. That is s = ~122.5354 cents.

LucyTuning has a "5th" or generator at the "low end" of the meantone spectrum, with an interval size between that of 1/3-comma meantone and Zarlino's 2/7-comma meantone (see the right side of the graph at the bottom of the meantone entry).

This generator or "5th" is composed of three Large (3L) plus one small note (s), i.e. (3L+s) = (~190.986*3) + (~122.535) = ~695.493 cents or ratio of

2(3 / 2π) * (2 / 2(5 / 2π))(1/2) = 2(2π + 1) / 4π, or 2(1/2 + 1/4π) = ~1.49441151.

Note that the size of the LucyTuning "5th" can also be expressed logarithmically as 600+(300/π) cents.

The LucyTuning "major-3rd" has the ratio 2(1/π) = ~1.246868989 = ~381.9718634 cents.

Below is a table and graph showing a 25-tone LucyTuning chain of 5ths, with all the intervals normally found in common-practice harmony labeled.

. . . . . . . . .
###### 3/10-comma meantone

The LucyTuning generator "5th" is audibly indistinguishable from that of 3/10-comma meantone:

```
22/10         * 3-2/10 * 53/10       3/10-comma meantone "5th"
- 2(2π+1)/4π                          Lucytuning "5th"
---------------------------------
2(-12π-10)/40π * 3-2/10 * 53/10   =   ~0.010148131 cent = ~1/99 cent
```
. . . . . . . . .
###### 88-edo

Carrying the LucyTuning meantone chain out to the 88th generator results in a pitch ~3.380995252 cents higher than the starting pitch. If the chain is centered on the origin (i.e., divided as equally as possible on the "flat" and "sharp" sides), and carried out to 12, 19, or 31 generators (typical sizes of meantone chains), the maximum differences of 88-edo from LucyTuning are respectively ~2/9, ~1/3, and ~4/7 of a cent. Therefore, for most purposes LucyTuning and 88-edo are essentially the same.

```
2(2π+1)/4π           Lucytuning "5th"
- 251/88              88edo "5th"
----------------
2(-7π+22)/88π   =     ~0.038420401 cent = ~1/26 cent
```

Even a 101-tone LucyTuning chain of 5ths will still present less than 2 cents error from 88edo, as shown in the table below:

```         ---------------- cents ---------------------
generator   LucyTuning     88-edo           error

50    1174.648293    1172.727273     - 1.92102003
49     479.1553269    477.2727273    - 1.882599629
48     983.662361     981.8181818    - 1.844179228
47     288.1693952    286.3636364    - 1.805758828
46     792.6764293    790.9090909    - 1.767338427
45      97.18346348    95.45454545   - 1.728918027
44     601.6904976    600.0000000    - 1.690497626
43    1106.197532    1104.545455     - 1.652077225
42     410.7045659    409.0909091    - 1.613656825
41     915.2116001    913.6363636    - 1.575236424
40     219.7186342    218.1818182    - 1.536816024
39     724.2256684    722.7272727    - 1.498395623
38      28.7327025     27.27272727   - 1.459975222
37     533.2397366    531.8181818    - 1.421554822
36    1037.746771    1036.363636     - 1.383134421
35     342.2538049    340.9090909    - 1.344714021
34     846.7608391    845.4545455    - 1.30629362
33     151.2678732    150.0000000    - 1.26787322
32     655.7749074    654.5454545    - 1.229452819
31    1160.281942    1159.090909     - 1.191032418
30     464.7889757    463.6363636    - 1.152612018
29     969.2960098    968.1818182    - 1.114191617
28     273.8030439    272.7272727    - 1.075771217
27     778.3100781    777.2727273    - 1.037350816
26      82.81711223    81.81818182   - 0.998930415
25     587.3241464    586.3636364    - 0.960510015
24    1091.831181    1090.909091     - 0.922089614
23     396.3382147    395.4545455    - 0.883669214
22     900.8452488    900.0000000    - 0.845248813
21     205.352283     204.5454545    - 0.806828412
20     709.8593171    709.0909091    - 0.768408012
19      14.36635125    13.63636364   - 0.729987611
18     518.8733854    518.1818182    - 0.691567211
17    1023.38042     1022.727273     - 0.65314681
16     327.8874537    327.2727273    - 0.614726409
15     832.3944878    831.8181818    - 0.576306009
14     136.901522     136.3636364    - 0.537885608
13     641.4085561    640.9090909    - 0.499465208
12    1145.91559     1145.454545     - 0.461044807
11     450.4226244    450.0000000    - 0.422624407
10     954.9296586    954.5454545    - 0.384204006
9     259.4366927    259.0909091    - 0.345783605
8     763.9437268    763.6363636    - 0.307363205
7      68.45076099    68.18181818   - 0.268942804
6     572.9577951    572.7272727    - 0.230522404
5    1077.464829    1077.272727     - 0.192102003
4     381.9718634    381.8181818    - 0.153681602
3     886.4788976    886.3636364    - 0.115261202
2     190.9859317    190.9090909    - 0.076840801
1     695.4929659    695.4545455    - 0.038420401
0       0.0000000      0.0000000      0.000000000
-1     504.5070341    504.5454545    + 0.038420401
-2    1009.014068    1009.090909     + 0.076840801
-3     313.5211024    313.6363636    + 0.115261202
-4     818.0281366    818.1818182    + 0.153681602
-5     122.5351707    122.7272727    + 0.192102003
-6     627.0422049    627.2727273    + 0.230522404
-7    1131.549239    1131.818182     + 0.268942804
-8     436.0562732    436.3636364    + 0.307363205
-9     940.5633073    940.9090909    + 0.345783605
-10     245.0703414    245.4545455    + 0.384204006
-11     749.5773756    750.0000000    + 0.422624407
-12      54.08440974    54.54545455   + 0.461044807
-13    .558.5914439    559.0909091    + 0.499465208
-14    1063.098478    1063.636364     + 0.537885608
-15     367.6055122    368.1818182    + 0.576306009
-16     872.1125463    872.7272727    + 0.614726409
-17     176.6195805    177.2727273    + 0.65314681
-18     681.1266146    681.8181818    + 0.691567211
-19    1185.633649    1186.363636     + 0.729987611
-20     490.1406829    490.9090909    + 0.768408012
-21     994.647717     995.4545455    + 0.806828412
-22     299.1547512    300.0000000    + 0.845248813
-23     803.6617853    804.5454545    + 0.883669214
-24     108.1688195    109.0909091    + 0.922089614
-25     612.6758536    613.6363636    + 0.960510015
-26    1117.182888    1118.181818     + 0.998930415
-27     421.6899219    422.7272727    + 1.037350816
-28     926.1969561    927.2727273    + 1.075771217
-29     230.7039902    231.8181818    + 1.114191617
-30     735.2110243    736.3636364    + 1.152612018
-31      39.71805849    40.90909091   + 1.191032418
-32     544.2250926    545.4545455    + 1.229452819
-33    1048.732127    1050.000000     + 1.26787322
-34     353.2391609    354.5454545    + 1.30629362
-35     857.7461951    859.0909091    + 1.344714021
-36     162.2532292    163.6363636    + 1.383134421
-37     666.7602634    668.1818182    + 1.421554822
-38    1171.267298    1172.727273     + 1.459975222
-39     475.7743316    477.2727273    + 1.498395623
-40     980.2813658    981.8181818    + 1.536816024
-41     284.7883999    286.3636364    + 1.575236424
-42     789.2954341    790.9090909    + 1.613656825
-43      93.80246823    95.45454545   + 1.652077225
-44     598.3095024    600.0000000    + 1.690497626
-45    1102.816537    1104.545455     + 1.728918027
-46     407.3235707    409.0909091    + 1.767338427
-47     911.8306048    913.6363636    + 1.805758828
-48     216.337639     218.1818182    + 1.844179228
-49     720.8446731    722.7272727    + 1.882599629
-50      25.35170724    27.27272727   + 1.92102003
```

88-edo is the lowest cardinality EDO which gives a very good approximation to LucyTuning. 1420-edo is an extremely good approximation, as can be seen in the graph below, which shows the error of the EDO 5th from the LucyTuning 5th as a percentage of that EDO's step size, for several EDOs which give a better approximation than 88-edo.