"Standard" ets for which h(25/24)=0, h(49/48)=1 7,17

h(25/24)=1 h(49/48)=0 5,9,15,19,25,29

h(50/49)=0 h(25/24)=1 2,12,16,18,22,26,32

Linear temperaments with 25/24 a comma and geometric badness below 5000:

25/24

[0, 0, 4, 9, -6, 0] [[4, 6, 9, 0], [0, 0, 0, 1]]

complexity 6.647907 rms 63.402645 badness 2802.058941

generators [300.0000000, 3306.069668]

[0, 0, 7, 16, -11, 0] [[7, 11, 16, 0], [0, 0, 0, 1]]

complexity 11.633838 rms 25.354978 badness 3431.699321

generators [171.4285715, 3348.926812]

[2, 1, -4, -12, 12, -3] [[1, 1, 2, 4], [0, 2, 1, -4]]

complexity 9.849244 rms 25.068246 badness 2431.810367

generators [1200., 358.0334333]

[0, 0, 3, 7, -5, 0] [[3, 5, 7, 0], [0, 0, 0, 1]]

complexity 4.985930 rms 61.312549 badness 1524.199406

generators [400.0000000, 3406.069668]

[4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]]

complexity 10.574200 rms 23.945252 badness 2677.407574

generators [600.0000000, 950.9775006]

[2, 1, 6, 11, -4, -3] [[1, 1, 2, 1], [0, 2, 1, 6]]

complexity 9.849244 rms 26.099830 badness 2531.881840

generators [1200., 360.2272895]

[2, 1, 3, 4, 1, -3] [[1, 1, 2, 2], [0, 2, 1, 3]]

complexity 6.245166 rms 48.926006 badness 1908.216791

generators [1200., 322.2119006]

[2, 1, -1, -5, 7, -3] [[1, 1, 2, 3], [0, 2, 1, -1]]

complexity 6.245166 rms 53.747748 badness 2096.274853

generators [1200., 318.5700997]

Ditto with 49/48:

[4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]]

complexity 10.574200 rms 23.945252 badness 2677.407574

generators [600.0000000, 950.9775006]

[6, 5, 3, -7, 12, -6] [[1, 0, 1, 2], [0, 6, 5, 3]]

complexity 16.383068 rms 12.273810 badness 3294.350648

generators [1200., 316.6640534]

[4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, 4, -3, 2]]

complexity 14.729697 rms 12.188571 badness 2644.480844

generators [1200., -125.4687958]

[2, 3, 1, -6, 4, 0] [[1, 0, 0, 2], [0, 2, 3, 1]]

complexity 6.691597 rms 34.566097 badness 1547.782446

generators [1200., 929.2070233]

[6, 0, 3, 7, 12, -14] [[3, 0, 7, 6], [0, 2, 0, 1]]

complexity 17.012788 rms 16.786584 badness 4858.624067

generators [400.0000000, 956.3327071]

[0, 5, 0, -14, 0, 8] [[5, 8, 0, 14], [0, 0, 1, 0]]

complexity 10.254281 rms 15.815352 badness 1662.988582

generators [240.0000000, 2789.386744]

[2, 8, 1, -20, 4, 8] [[1, 0, -4, 2], [0, 2, 8, 1]]

complexity 15.298626 rms 12.690078 badness 2970.086938

generators [1200., 947.2576877]

[2, -6, 1, 19, 4, -14] [[1, 0, 7, 2], [0, 2, -6, 1]]

complexity 15.298626 rms 18.261110 badness 4273.975544

generators [1200., 939.2948284]

Ditto with 50/49:

[4, 4, 4, -2, 5, -3] [[4, 0, 3, 5], [0, 1, 1, 1]]

complexity 11.405897 rms 19.136993 badness 2489.617178

generators [300.0000000, 1885.698206]

[0, 2, 2, -1, -3, 3] [[2, 3, 0, 1], [0, 0, 1, 1]]

complexity 4.275602 rms 59.723378 badness 1091.789278

generators [600.0000000, 2726.592310]

[6, -2, -2, 1, 20, -17] [[2, 2, 5, 6], [0, 3, -1, -1]]

complexity 19.126831 rms 11.798337 badness 4316.252447

generators [600.0000000, 231.2978354]

[2, -4, -4, 2, 12, -11] [[2, 0, 11, 12], [0, 1, -2, -2]]

complexity 11.925109 rms 10.903178 badness 1550.521640

generators [600.0000000, 1908.814331]

[2, 6, 6, -3, -4, 5] [[2, 0, -5, -4], [0, 1, 3, 3]]

complexity 11.925109 rms 18.863889 badness 2682.600333

generators [600.0000000, 1928.512337]

[2, 8, 8, -4, -7, 8] [[2, 0, -8, -7], [0, 1, 4, 4]]

complexity 15.871133 rms 11.218941 badness 2825.971103

generators [600.0000000, 1893.651026]

[8, 6, 6, -3, 13, -9] [[2, 1, 3, 4], [0, 4, 3, 3]]

complexity 21.576275 rms 10.132266 badness 4716.930929

generators [600.0000000, 325.6113679]

[4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]]

complexity 10.574200 rms 23.945252 badness 2677.407574

generators [600.0000000, 950.9775006]

i don't know if the usual badness measure was appropriate in this

case -- see my results and then you might get a better feel for it.

the first key to interpreting this is to note that the chromatic

unison vector's size in generators will equal the cardinality of the

scale (per period) . . .

let's cap the rms error at 20-22 cents . . .

> Ditto with 49/48:

>

> [6, 5, 3, -7, 12, -6] [[1, 0, 1, 2], [0, 6, 5, 3]]

> complexity 16.383068 rms 12.273810 badness 3294.350648

> generators [1200., 316.6640534]

25:24 chroma = 10 - 4 = 7 generators -> 4 note scale

graham complexity = 6

X

> [4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, 4, -3, 2]]

> complexity 14.729697 rms 12.188571 badness 2644.480844

> generators [1200., -125.4687958]

25:24 chroma = -6 - 4 = -10 generators -> 10 note scale

graham complexity = 7 -> 6 tetrads

> [6, 0, 3, 7, 12, -14] [[3, 0, 7, 6], [0, 2, 0, 1]]

> complexity 17.012788 rms 16.786584 badness 4858.624067

> generators [400.0000000, 956.3327071]

25:24 chroma = 0 - 2 = -2 generators -> 6 note scale

graham complexity = 2*3=6

X

> [0, 5, 0, -14, 0, 8] [[5, 8, 0, 14], [0, 0, 1, 0]]

> complexity 10.254281 rms 15.815352 badness 1662.988582

> generators [240.0000000, 2789.386744]

blackwood-10

25:24 chroma = 2 - 0 = 2 generators -> 10 note scale

graham complexity = 1*5=5 -> 10 tetrads

> [2, 8, 1, -20, 4, 8] [[1, 0, -4, 2], [0, 2, 8, 1]]

> complexity 15.298626 rms 12.690078 badness 2970.086938

> generators [1200., 947.2576877]

25:24 chroma = 16 - 2 = 14 generators -> 14 note scale

graham complexity = 8 -> 12 tetrads

> [2, -6, 1, 19, 4, -14] [[1, 0, 7, 2], [0, 2, -6, 1]]

> complexity 15.298626 rms 18.261110 badness 4273.975544

> generators [1200., 939.2948284]

25:24 chroma = -12 - 2 = -14 generators -> 14 note scale

graham complexity = 8 -> 12 tetrads

> Ditto with 50/49:

>

> [4, 4, 4, -2, 5, -3] [[4, 0, 3, 5], [0, 1, 1, 1]]

> complexity 11.405897 rms 19.136993 badness 2489.617178

> generators [300.0000000, 1885.698206]

25:24 chroma = 2 - 1 = 1 generator -> 4 note scale

graham complexity = 1*4 = 4

X

> [6, -2, -2, 1, 20, -17] [[2, 2, 5, 6], [0, 3, -1, -1]]

> complexity 19.126831 rms 11.798337 badness 4316.252447

> generators [600.0000000, 231.2978354]

25:24 chroma = -2 - 3 = 5 generators -> 10 note scale

graham complexity = 8 -> 4 tetrads

> [2, -4, -4, 2, 12, -11] [[2, 0, 11, 12], [0, 1, -2, -2]]

> complexity 11.925109 rms 10.903178 badness 1550.521640

> generators [600.0000000, 1908.814331]

pajara-10

25:24 chroma = -4 - 1 = 5 generators -> 10 note scale

graham complexity = 3*2 = 6 -> 8 tetrads

> [2, 6, 6, -3, -4, 5] [[2, 0, -5, -4], [0, 1, 3, 3]]

> complexity 11.925109 rms 18.863889 badness 2682.600333

> generators [600.0000000, 1928.512337]

25:24 chroma = 6 - 1 = 5 generators -> 10 note scale

graham complexity = 3*2 = 6 -> 8 tetrads

> [2, 8, 8, -4, -7, 8] [[2, 0, -8, -7], [0, 1, 4, 4]]

> complexity 15.871133 rms 11.218941 badness 2825.971103

> generators [600.0000000, 1893.651026]

injera-14

25:24 chroma = 8 - 1 = 7 generators -> 14 note scale

graham complexity = 4*2 = 8 -> 12 tetrads

> [8, 6, 6, -3, 13, -9] [[2, 1, 3, 4], [0, 4, 3, 3]]

> complexity 21.576275 rms 10.132266 badness 4716.930929

> generators [600.0000000, 325.6113679]

25:24 chroma = 6 - 4 = 2 generators -> 4 note scale

graham complexity = 4*2 = 8

X

p.s. note that this one:

> [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]]

> complexity 10.574200 rms 23.945252 badness 2677.407574

> generators [600.0000000, 950.9775006]

appeared in all three lists . . .

Thank you, Gene and Paul! Great job!

How is complexity calculated and why? Badness is somewhere around

complexity^2*rms, what's the idea behind this?

I guess everything is in the archives, any old messages you'd like me

to read?

Kalle

--- In tuning-math@yahoogroups.com, "Kalle Aho <kalleaho@m...>" <kalleaho@m...> wrote:

> How is complexity calculated and why?

I calculated complexity using a rather complicated formula, but Paul tells me that Graham complexity would have been better; evidentally you are interested only in complete tetrads. It is easy to compute: take the range in generator steps of the representations of

{1,3,5,7,5/3,7/3,7/5} and multiply by the number of periods in an octave.

Badness is somewhere around

> complexity^2*rms, what's the idea behind this?

It makes the badness measure "log-flat", roughly meaning about as many temperaments appear in one size range as in another.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> --- In tuning-math@yahoogroups.com, "Kalle Aho

<kalleaho@m...>" <kalleaho@m...> wrote:

>

> > How is complexity calculated and why?

>

> I calculated complexity using a rather complicated formula, but

>Paul tells me that Graham complexity would have been better;

actually the representation of the chromatic unison vector comes

into it too.

>evidentally you are interested only in complete tetrads. It is

easy >to compute: take the range in generator steps of the

>representations of

> {1,3,5,7,5/3,7/3,7/5} and multiply by the number of periods in an

>octave.

that's graham's complexity. but you still haven't calculated the

number of complete tetrads.

to do that, you also need to calculate the number of notes in the

scale. first calculate the cardinality per period, by determining

(from the mapping) the number of generators comprising a

49:48 if 25:24 is tempered out, a 25:24 if 49:48 is tempered out,

or either (they'll be the same) if 50:49 is tempered out. then

multiply by the number of periods in an octave.

finally, the number of tetrads will be the graham complexity

minus the number of notes in the scale -- times two (since both

otonal and utonal tetrads count).