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Message: 11150 - Contents - Hide Contents

Date: Tue, 29 Jun 2004 07:21:27

Subject: Re: An early reference to 13-lemba

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:


> This is the same mapping for 13-lemba that I posted to the tuning list. 
> It only took a year and a half for me to notice it!


I'm surprised you even found it, now that Yahoo has screwed up the
search archive feature.

Message: 11151 - Contents - Hide Contents

Date: Tue, 29 Jun 2004 08:32:21

Subject: Re: Some 13-limit temperaments

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:

> Here are a few temperaments found by the brute-force approach (try all 
> possible generators and periods until something interesting comes up).
> 
> No. 1: Coendou? (a South American porcupine)

{55/54, 65/64, 100/99, 105/104}

> 7L + 15s (period=1201.0c, generator=166.2c)
> [[1, 2, 3, 1, 4, 3], [0, -3, -5, 13, -4, 5]]
> <<3, 5, -13, 4, -5, 1, -29, -4, -19, -44, -8, -30, 56, 34, -32]]
> 
> No. 2: Roman? (26 letters in the Roman alphabet)
> 3L + 23s (period=1200.6c, generator=414.5c)
> [[1, 4, 3, -1, 0, 3], [0, -7, -2, 11, 10, 2]]
> <<7, 2, -11, -10, -2, -13, -37, -40, -29, -31, -30, -12, 10, 35, 30]]
> 
> No. 3: Leapday? (Feb. 29) Eh?

{91/90, 121/120, 169/168, 352/351}

> 17L + 12s (period=1200.0c, generator=495.7c)
> [[1, 2, 11, 9, 8, 7], [0, -1, -21, -15, -11, -8]]
> <<1, 21, 15, 11, 8, 31, 21, 14, 9, -24, -47, -59, -21, -33, -13]]
> 
> No. 4: Myna / Pangolin (formerly Nonkleismic)
> 27L + 4s (period=1198.5c, generator=310.0c)
> [[1, -1, 0, 1, -3, 5], [0, 10, 9, 7, 25, -5]]
> <<10, 9, 7, 25, -5, -9, -17, 5, -45, -9, 27, -45, 46, -40, -110]]
> 
> No. 5:

{144/143, 351/350, 364/363, 441/440}
Another extension of HTT to a linear temperament.

> 26L + 6s (period=599.6c, generator=185.8c)
> [[2, 1, 0, 5, 6, 4], [0, 7, 15, 2, 3, 11]]
> <<14, 30, 4, 6, 22, 15, -33, -39, -17, -75, -90, -60, 3, 47, 54]]
> 
> No. 6:
> 29L + 5s (period=1199.9c, generator=247.8c)
> [[1, 2, 5, 9, 8, 7], [0, -2, -13, -30, -22, -16]]
> <<2, 13, 30, 22, 16, 16, 42, 28, 18, 33, 6, -11, -42, -66, -26]]
> 
> No. 7: Hitchcock? (39 steps) This is in the amity family; it has the

same TOP tuning as the 11-limit reduction, but there are other
11-limit versions of amity which accord better with 5 and 7 limit
amity (adjustments of the 11-limit reduction by +53 or -46.)


> 7L + 32s (period=1201.1c, generator=339.7c)
> [[1, 3, 6, -2, 6, 2], [0, -5, -13, 17, -9, 6]]
> <<5, 13, -17, 9, -6, 9, -41, -3, -28, -76, -24, -62, 84, 46, -54]]
> 
> No. 8: Supersupermajor
> 5L + 36s (period=1200.1c, generator=234.5c)
> [[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]]
> <<3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58,
-28]]
> 
> No. 9: Cassandra 1

I've always hated this name. It's either cassandra or it isn't, but no
subscripts, please! My suggestion of "septishis" for this is pretty
lousy, but what about just calling it cassandra? The 11 and 13
versions have the same TOP tuning, and the 7-limit is close. Schismic
would be a different deal, what I once called counterschismic.

> 12L + 29s (period=1200.2c, generator=498.0c)
> [[1, 2, -1, -3, 13, 12], [0, -1, 8, 14, -23, -20]]
> <<1, -8, -14, 23, 20, -15, -25, 33, 28, -10, 81, 76, 113, 108, -16]]
> 
> No. 10: Hemififths
> 17L + 24s (period=1198.9c, generator=351.2c)
> [[1, 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]]
> <<2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22]]
> 
> No. 11:
> 17L + 26s (period=1201.0c, generator=140.3c)
> [[1, 1, -2, 0, 1, 3], [0, 5, 37, 24, 21, 6]]
> <<5, 37, 24, 21, 6, 47, 24, 16, -9, -48, -79, -123, -24, -72, -57]]
> 
> No. 12:
> 4L + 42s (period=600.5c, generator=287.8c)
> [[2, 7, 8, 8, 5, 5], [0, -8, -7, -5, 4, 5]]
> <<16, 14, 10, -8, -10, -15, -29, -68, -75, -16, -67, -75, -57, -65, -5]]
> 
> No. 13: Diaschismic

Number 15 is closer in tuning to 5-limit diaschismic

> 12L + 34s (period=599.6c, generator=103.6c)
> [[2, 3, 5, 7, 9, 10], [0, 1, -2, -8, -12, -15]]
> <<2, -4, -16, -24, -30, -11, -31, -45, -55, -26, -42, -55, -12, -25,
-15]]


I tagged the name unidec onto the 11 and 13 limit temperament a long
time ago, so the 7-limit version should have been unidec also.

> No. 14: (Number 33 from the 7-limit list)
> 26L + 20s (period=600.5c, generator=183.3c)
> [[2, 5, 8, 5, 6, 8], [0, -6, -11, 2, 3, -2]]
> <<12, 22, -4, -6, 4, 7, -40, -51, -38, -71, -90, -72, -3, 26, 36]]
> 
> No. 15:

Closer to 5-limit diaschismic than number 13

> 34L + 12s (period=599.7c, generator=104.9c)
> [[2, 3, 5, 3, 5, 6], [0, 1, -2, 15, 11, 8]]
> <<2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26]]
> 
> No. 16: Vulture

{176/175, 351/350, 540/539, 640/637}

> 5L + 43s (period=1199.3c, generator=475.4c)
> [[1, 0, -6, 4, -12, -7], [0, 4, 21, -3, 39, 27]]
> <<4, 21, -3, 39, 27, 24, -16, 48, 28, -66, 18, -15, 120, 87, -51]]
> 
> No: 17: (Number 76 from the 7-limit list)
> 3L + 47s (period=1200.8c, generator=408.0c)
> [[1, 6, 3, 13, -3, 2], [0, -13, -2, -30, 19, 5]]
> <<13, 2, 30, -19, -5, -27, 11, -75, -56, 64, -51, -19, -157, -125, 53]]
> 
> No. 18:
> 26L + 25s (period=1201.8c, generator=46.7c)
> [[1, 1, 1, 3, 4, 3], [0, 15, 34, -5, -14, 18]]
> <<15, 34, -5, -14, 18, 19, -50, -74, -27, -107, -150, -84, -22, 69,
114]]
> 
> No. 19:
> 29L + 22s (period=1199.9c, generator=165.0c)
> [[1, 2, 7, 9, 8, 7], [0, -3, -34, -45, -33, -24]]
> <<3, 34, 45, 33, 24, 47, 63, 42, 27, 9, -41, -70, -63, -99, -39]]
> 
> No. 20:
> 5L + 48s (period=1199.4c, generator=475.8c)
> [[1, 0, 17, 4, 11, 16], [0, 4, -37, -3, -19, -31]]
> <<4, -37, -3, -19, -31, -68, -16, -44, -64, 97, 84, 65, -43, -76, -37]]
> 
> No. 21: Catakleismic (Hanson)
> 19L + 34s (period=1200.9c, generator=317.0c)
> [[1, 0, 1, -3, 9, 0], [0, 6, 5, 22, -21, 14]]
> <<6, 5, 22, -21, 14, -6, 18, -54, 0, 37, -66, 14, -135, -42, 126]]

Message: 11152 - Contents - Hide Contents

Date: Tue, 29 Jun 2004 12:47:12

Subject: Re: Paul's nifty fifty

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> 

>> So, like I said, call one "simple kleismic" and the 
other "complex 
>> kleismic", or even "simple-7 kleismic" and "complex-7 kleismic".
> 

> So would "dominant sevenths" become "simple meantone" and 
(septimal)
> meantone become "complex meantone"?
> 

>> It doesn't make sense to have a completely unrelated name for 
the 
>> same temperament at a higher limit.
> 

> That's what I'm trying to avoid, by giving the name (as in the 
case of
> meantone) to the one keeping the tuning, rather than the least 
complex
> (which would be dominant sevenths in this case.)


You have a good point there. No I wouldn't call "dominant sevenths" 
any kind of meantone since I don't consider anything with fifths 
wider than those of 12-ET to be meantone. But I note that "simple-7 
kleismic" does far less violence to the 5-limit ratios of the parent 
5-limit temperament than does "dominant sevenths".

Your proposed system does have a precedent in the naming of rivers 
where, as you work your way up-river and you come to forks, the 
widest one gets to keep the name and the other is referred to as a 
tributary. But sometimes it's a tough call. What would correspond to 
tributary wideness here? Shouldn't it be some kind of badness 
measure (combining error and complexity, rather than just error as 
you and Paul seem to be proposing)?

You know me. I like systems. And since you're proposing a system 
here, I'll be happy. Meyers-Briggs personality type INTP = "the 
architect". :-)

When there is no system to the naming, the name just becomes one 
more thing you have to memorise (or look up every time), rather than 
an aid to memory.

Message: 11153 - Contents - Hide Contents

Date: Tue, 29 Jun 2004 17:17:18

Subject: Re: Paul's nifty fifty

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:


> Your proposed system does have a precedent in the naming of rivers 
> where, as you work your way up-river and you come to forks, the 
> widest one gets to keep the name and the other is referred to as a 
> tributary. But sometimes it's a tough call. What would correspond to 
> tributary wideness here? Shouldn't it be some kind of badness 
> measure (combining error and complexity, rather than just error as 
> you and Paul seem to be proposing)?


It would need to be something like that; what I've been doing is
picking the closest tuning in any system which doesn't seem so bad it
isn't worth considering, but that's not formalized.




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Message: 11154 - Contents - Hide Contents

Date: Wed, 30 Jun 2004 21:17:30

Subject: Semihemidemi help!

From: Gene Ward Smith

After proposing 198 as competition for 152 and 176, I decided to check
out 11-limit microtemperaments it supports. I was run down by a crazed
mob of semis and hemis.

I mentione hemigamera and semigamera below; the corresponding 7-limit
99-et system would of course be gamera: <<23 40 1 10 -63 -110||. The
names for systems with 8/7 generators tend to be boring, so I thought
I'd start the ball rolling with Japanese movie monsters. Hemifourths,
supermajor seconds, and supersupermajor could become godzilla, mothra
and rodan if that sounds cool, leaving hystrix and beep as is.
 
hemiennealimmal
[36, 54, 36, 18, 2, -44, -96, -68, -145, -74] [[18, 28, 41, 50, 62],
[0, 2, 3, 2, 1]]
72 .049207 61.317611

hemiamity
[10, 26, -34, -28, 18, -82, -79, -152, -155, 39] [[2, 1, -1, 13, 13],
[0, 5, 13, -17, -14]]
60 .276106 253.899159

hemigamera
[46, 80, 2, -10, 20, -126, -175, -220, -300, -35] [[2, 12, 20, 6, 5],
[0, -23, -40, -1, 5]]
102 .148153 329.896268

semihemiwuerschmidt
[32, 4, 10, 49, -68, -74, -33, 12, 100, 103] [[1, 15, 4, 7, 24], [0,
-32, -4, -10, -49]]
64 .333461 341.464500

icosidillic
[22, -22, 44, -22, -86, 8, -111, 164, 25, -214] [[22, 35, 51, 62, 76],
[0, -1, 1, -2, 1]]
66 .337818 364.129624

semiparakleismic
[26, 28, 70, 46, -16, 38, -17, 84, 10, -113] [[2, -3, -2, -11, -4],
[0, 13, 14, 35, 23]]
70 .311091 369.871482

semifourths-hemififths
[4, 50, 26, 68, 70, 30, 94, -80, -15, 101] [[2, 4, 15, 11, 21], [0,
-2, -25, -13, -34]]
68 .335467 380.041765

semihemififths
[4, 50, 26, -31, 70, 30, -63, -80, -245, -177] [[1, 1, -5, -1, 8], [0,
4, 50, 26, -31]]
81 .299647 454.379188

[14, 76, -8, 40, 88, -52, 15, -232, -170, 140] [[2, 6, 20, 4, 15], [0,
-7, -38, 4, -20]]
84 .317362 511.313408

semigamera
[46, 80, 2, 89, 20, -126, -18, -220, -70, 243] [[1, 6, 10, 3, 12], [0,
-46, -80, -2, -89]]
92 .289283 542.378831

Message: 11155 - Contents - Hide Contents

Date: Wed, 30 Jun 2004 00:14:00

Subject: The 81/80 and 31 mob

From: Gene Ward Smith

The TM basis for 31 is {81/80, 126/125, 1029/1024} which we can use to
find mobs as large as we like for 31 and a chosen comma; however only
finitely many will make even minimal sense. Below I cut the mob off at
eight members, which seems to be plenty. I give the Graham complexity,
the TOP error and the corresponding badness on the second line. Note
that all except for a rather useless version of meantone have the
exact same TOP tuning!

meantone
[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
10 1.698521 169.852100

supermajor seconds
[3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
13 1.698521 287.050049

squares
[4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
16 1.698521 434.821376

semififths
[2, 8, -11, 8, -23, -48] [[1, 1, 0, 6], [0, 2, 8, -11]]
19 1.698521 613.166081

[5, 20, 19, 20, 16, -12] [[1, 4, 12, 12], [0, -5, -20, -19]]
20 1.698521 679.408400

-31 nexial adjusted meantone
[1, 4, -21, 4, -36, -60] [[1, 2, 4, -6], [0, -1, -4, 21]]
25 1.762367 1101.479671

[7, 28, 8, 28, -7, -60] [[1, 0, -4, 1], [0, 7, 28, 8]]
28 1.698521 1331.640464

[6, 24, 29, 24, 29, 0] [[1, 1, 0, 0], [0, 6, 24, 29]]
29 1.698521 1428.456161

Message: 11156 - Contents - Hide Contents

Date: Wed, 30 Jun 2004 00:32:11

Subject: The 126/125 and 31 mob

From: Gene Ward Smith

myna/pangolin/nonkleismic
[10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
10 1.171542 117.154200

valentine
[9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
12 1.049791 151.169908

meantone
[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
10 1.698521 169.852100

[19, 14, 4, -22, -47, -30] [[1, -7, -4, 1], [0, 19, 14, 4]]
19 .989846 357.334453

"Number 67"
[11, 13, 17, -5, -4, 3] [[1, 3, 4, 5], [0, -11, -13, -17]]
17 1.485250 429.237250

-31 nexially adjusted wuerschmidt
[8, 1, -13, -17, -43, -33] [[1, -1, 2, 7], [0, 8, 1, -13]]
21 1.116045 492.175800

[21, 22, 24, -14, -21, -6] [[1, 7, 8, 9], [0, -21, -22, -24]]
24 1.335885 769.470278

[28, 19, 1, -35, -77, -51] [[1, 7, 6, 3], [0, -28, -19, -1]]
28 1.012789 794.026781

[29, 23, 11, -31, -64, -39] [[1, 10, 9, 6], [0, -29, -23, -11]]
29 1.052506 885.157546

-31 nexially adjusted orwell
[7, -3, -23, -21, -56, -45] [[1, 0, 3, 8], [0, 7, -3, -23]]
30 1.151535 1036.381617

[17, 6, -16, -30, -73, -54] [[1, -5, 0, 9], [0, 17, 6, -16]]
33 1.088818 1185.722753

[31, 31, 31, -23, -38, -15] [[31, 49, 72, 87], [0, 1, 1, 1]]
31 1.282877 1232.844322

[12, 17, 27, -1, 9, 15] [[1, 7, 10, 15], [0, -12, -17, -27]]
27 1.762368 1284.766030

Message: 11157 - Contents - Hide Contents

Date: Wed, 30 Jun 2004 22:26:31

Subject: Re: 24 13-limit temperaments supported by 46

From: Herman Miller

Gene Ward Smith wrote:

> We've been on a 13-limit kick, and George thinks 46 may be his fav, so
> I'm giving this. Note that valentine and semisixths have the same
> Graham complexity and TOP error, so they are tied at the 7.5th place
> by the criterion I used to sort this list. In terms of TOP tuning,
> they share a tuning for 5, 7, and 13. Leapday came out #1; it's been
> kicking around a while without a name so it's good Herman named it,
> even if I still havn't figured out what 2/29 has to do with it.


Its smallest MOS with full 13-limit hexads has 29 steps. I hadn't 
intended any meaning for the "2" from February.

"Hitchcock", although named in reference to the 39-step MOS, does have a 
smaller 32-step MOS with full 13-limit hexads, but my program was set to 
ignore scales with large steps more than 3 times the size of the small 
steps (which tend to be less useful musically), so I didn't notice it.

Message: 11158 - Contents - Hide Contents

Date: Wed, 30 Jun 2004 01:16:37

Subject: the 2401/2400 and 171 gang

From: Gene Ward Smith

A ruthless bunch. Neptune has a trident, except that this one has a
tritone. No, it doesn't make much sense, but you can remember it.

ennealimmal
[18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
27 .036378 26.519210

tertiaseptal
[22, -5, 3, -59, -57, 21] [[1, 3, 2, 3], [0, -22, 5, -3]]
27 .073527 53.601055

neptune
[40, 22, 21, -58, -79, -13] [[1, 21, 13, 13], [0, -40, -22, -21]]
40 .053350 85.360253

sesquiquartififths
[4, -32, -15, -60, -35, 55] [[1, 1, 7, 5], [0, 4, -32, -15]]
36 .066120 85.691520

[14, 59, 33, 61, 13, -89] [[1, -3, -17, -8], [0, 14, 59, 33]]
59 .037361 130.053641

[58, 49, 39, -57, -101, -47] [[1, -13, -10, -7], [0, 58, 49, 39]]
58 .048083 161.751212

[62, 17, 24, -117, -136, 8] [[1, 15, 6, 8], [0, -62, -17, -24]]
62 .063269 243.206036

[32, 86, 51, 62, -9, -123] [[1, 13, 33, 21], [0, -32, -86, -51]]
86 .035267 260.836522

[76, 76, 57, -56, -123, -81] [[19, 31, 45, 54], [0, -4, -4, -3]]
76 .045310 261.710560

[26, -37, -12, -119, -92, 76] [[1, -1, 6, 4], [0, 26, -37, -12]]
63 .070288 278.973072

[10, 91, 48, 121, 48, -144] [[1, 1, -3, 0], [0, 10, 91, 48]]
91 .051198 423.970638

[50, 113, 69, 63, -31, -157] [[1, 1, 1, 2], [0, 50, 113, 69]]
113 .034174 436.367806

[94, 103, 75, -55, -145, -115] [[1, -27, -29, -20], [0, 94, 103, 75]]
103 .043600 462.552400

[98, 71, 60, -115, -180, -60] [[1, -3, -1, 0], [0, 98, 71, 60]]
98 .050233 482.435179

[84, 12, 27, -176, -193, 29] [[3, -9, 5, 4], [0, 28, 4, 9]]
84 .068943 486.459716

[48, -42, -9, -178, -149, 97] [[3, 7, 5, 8], [0, -16, 14, 3]]
90 .071266 577.254600

[102, 39, 45, -175, -215, -5] [[3, -3, 4, 5], [0, 34, 13, 15]]
102 .058596 609.629747

[68, 140, 87, 64, -53, -191] [[1, 33, 67, 43], [0, -68, -140, -87]]
140 .033502 656.645658

[30, -69, -27, -179, -127, 131] [[3, 3, 11, 10], [0, 10, -23, -9]]
99 .069139 677.633056

[46, 145, 84, 123, 4, -212] [[1, 11, 32, 20], [0, -46, -145, -84]]
145 .036119 759.401975

[106, 7, 30, -235, -250, 50] [[1, 27, 4, 10], [0, -106, -7, -30]]
106 .072262 811.935832

[6, 123, 63, 181, 83, -199] [[3, 5, 12, 11], [0, -2, -41, -21]]
123 .057923 876.317067

[86, 167, 105, 65, -75, -225] [[1, -13, -26, -15], [0, 86, 167, 105]]
167 .033048 921.675672

[70, -47, -6, -237, -206, 118] [[1, 11, -4, 2], [0, -70, 47, 6]]
117 .071704 981.556056

Message: 11159 - Contents - Hide Contents

Date: Wed, 30 Jun 2004 03:57:37

Subject: The 49/48 and 19 mob

From: Gene Ward Smith

The 81/80 and 31 mob has a lot of common TOP errors, and another case
like that connects kleismic and negri, which are in the 49/48 and 19
mob. This also has some common tunings; kleismic, negri, and the
fourth, seventh and eigth temperaments listed all have the same TOP
error. Kleismic and negri and the other three temperaments all have
the same TOP tuning for 2, 3, and 7, and all but negri have the same
NOT tuning for these as well.

1. kleismic
[6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
6 3.187309 114.743119

2. negri
[4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
7 3.187309 156.178134

3. hemifourths
[2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
8 3.668842 234.805888

4.
[10, 2, 5, -20, -20, 6] [[1, 0, 2, 2], [0, 10, 2, 5]]
10 3.187309 318.730900

5.
[8, 13, 4, 2, -16, -27] [[1, 2, 3, 3], [0, -8, -13, -4]]
13 3.427735 579.287215

6.
[2, -11, 1, -22, -4, 33] [[1, 2, 0, 3], [0, -2, 11, -1]]
13 3.676560 621.338643

7.
[14, -1, 7, -34, -28, 19] [[1, 6, 2, 5], [0, -14, 1, -7]]
15 3.187309 717.144490

8.
[16, 7, 8, -26, -32, -1] [[1, -6, -1, -1], [0, 16, 7, 8]]
16 3.187309 815.951104

9.
[14, 18, 7, -4, -28, -34] [[1, 6, 8, 5], [0, -14, -18, -7]]
18 3.320608 1075.876992

10.
[0, 19, 0, 30, 0, -53] [[19, 30, 44, 53], [0, 0, -1, 0]]
19 3.833866 1384.025626

Message: 11160 - Contents - Hide Contents

Date: Wed, 30 Jun 2004 05:12:41

Subject: The 50/49 and 22 mob

From: Gene Ward Smith

The TOP tunings of pajara, Number 43, and the fifth and eigth
temperaments listed are the same for 2, 5, and 7. The NOT tunings of
doublewide, Number 43, and the seventh and ninth listed are also the
same for 2, 5, and 7.


1. pajara 
[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
6 3.106578 111.836820

2. doublewide
[8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
8 3.268439 209.180089

3. "Number 43"
[6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
10 3.106578 310.657834

4.
[10, 2, 2, -20, -25, -1] [[2, 5, 5, 6], [0, -5, -1, -1]]
10 3.276635 327.663549

5.
[4, 14, 14, 13, 11, -7] [[2, 3, 4, 5], [0, 2, 7, 7]]
14 3.106578 608.889355

6.
[12, -2, -2, -31, -37, 1] [[2, 1, 5, 6], [0, 6, -1, -1]]
14 3.219949 631.109957

7.
[14, 16, 16, -7, -14, -8] [[2, 0, 1, 2], [0, 7, 8, 8]]
16 3.253803 832.973502

8.
[2, 18, 18, 24, 23, -9] [[2, 3, 3, 4], [0, 1, 9, 9]]
18 3.106578 1006.531383

9.
[18, 8, 8, -29, -38, -4] [[2, 4, 5, 6], [0, -9, -4, -4]]
18 3.273831 1060.721373

10.
[14, -6, -6, -42, -49, 3] [[2, 0, 6, 7], [0, 7, -3, -3]]
20 3.149020 1259.608029

Message: 11161 - Contents - Hide Contents

Date: Wed, 30 Jun 2004 16:55:53

Subject: Re: The 50/49 and 22 mob

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> The TOP tunings of pajara, Number 43, and the fifth and eigth
> temperaments listed are the same for 2, 5, and 7. The NOT tunings of
> doublewide, Number 43, and the seventh and ninth listed are also the
> same for 2, 5, and 7.
> 
> 
> 1. pajara 
> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> 6 3.106578 111.836820
> 
> 2. doublewide
> [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
> 8 3.268439 209.180089
> 
> 3. "Number 43"
> [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
> 10 3.106578 310.657834


We've been calling the latter "hedgehog".

-the absent one

Message: 11162 - Contents - Hide Contents

Date: Wed, 30 Jun 2004 17:02:50

Subject: 24 13-limit temperaments supported by 46

From: Gene Ward Smith

We've been on a 13-limit kick, and George thinks 46 may be his fav, so
I'm giving this. Note that valentine and semisixths have the same
Graham complexity and TOP error, so they are tied at the 7.5th place
by the criterion I used to sort this list. In terms of TOP tuning,
they share a tuning for 5, 7, and 13. Leapday came out #1; it's been
kicking around a while without a name so it's good Herman named it,
even if I still havn't figured out what 2/29 has to do with it. It's a 
system with a sharp (about 2.4 cents) fifth as generator and 5 with
the highest complexity; it therefore seems perfect for Margo, who may
have mentioned it.

leapday
[1, 21, 15, 11, 8, 31, 21, 14, 9, -24, -47, -59, -21, -33, -13] [[1,
2, 11, 9, 8, 7], [0, -1, -21, -15, -11, -8]]
21 1.589486 254.071241

unidec
[12, 22, -4, -6, 4, 7, -40, -51, -38, -71, -90, -72, -3, 26, 36] [[2,
5, 8, 5, 6, 8], [0, -6, -11, 2, 3, -2]]
30 .899546 260.550493

hitchcock/amity
[5, 13, -17, 9, -6, 9, -41, -3, -28, -76, -24, -62, 84, 46, -54] [[1,
3, 6, -2, 6, 2], [0, -5, -13, 17, -9, 6]]
30 1.176228 340.690750

supersuper
[3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58,
-28] [[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]]
39 .958363 429.837463

twothirdtonic
[13, -3, 11, 5, 12, -35, -19, -37, -29, 34, 22, 39, -24, -7, 23] [[1,
3, 2, 4, 4, 5], [0, -13, 3, -11, -5, -12]]
29 1.697662 464.709140

srutar
[4, -8, 14, -2, -14, -22, 11, -17, -37, 55, 23, -3, -54, -91, -41]
[[2, 3, 5, 5, 7, 8], [0, 2, -4, 7, -1, -7]]
28 1.845166 476.393162

valentine
[9, 5, -3, 7, 26, -13, -30, -20, 8, -21, -1, 42, 30, 84, 64] [[1, 1,
2, 3, 3, 2], [0, 9, 5, -3, 7, 26]]
29 1.769869 484.474653

semisixths
[7, 9, 13, -15, 10, -2, 1, -48, -10, 5, -66, -10, -87, -20, 90] [[1,
-1, -1, -2, 9, 0], [0, 7, 9, 13, -15, 10]]
29 1.769869 484.474653

[6, -12, -2, 20, 2, -33, -20, 11, -19, 29, 88, 49, 63, 13, -67] [[2,
2, 7, 6, 3, 7], [0, 3, -6, -1, 10, 1]]
32 1.646850 531.174551

[16, 14, 10, -8, -10, -15, -29, -68, -75, -16, -67, -75, -57, -65, -5]
[[2, 7, 8, 8, 5, 5], [0, -8, -7, -5, 4, 5]]
42 1.278944 649.033350

valentino
[9, 5, -3, 7, -20, -13, -30, -20, -65, -21, -1, -65, 30, -45, -95]
[[1, 1, 2, 3, 3, 5], [0, 9, 5, -3, 7, -20]]
38 1.548127 664.935019

shrutar
[4, -8, 14, -2, 32, -22, 11, -17, 36, 55, 23, 104, -54, 38, 118] [[2,
3, 5, 5, 7, 6], [0, 2, -4, 7, -1, 16]]
40 1.425313 666.822498

[8, -16, -18, -4, 18, -44, -51, -34, -1, 3, 46, 101, 51, 117, 77] [[2,
3, 5, 6, 7, 7], [0, 4, -8, -9, -2, 9]]
36 1.704422 668.982422

superduper
[3, 17, -1, -13, 24, 20, -10, -31, 27, -50, -89, -7, -33, 71, 131]
[[1, 1, -1, 3, 6, -1], [0, 3, 17, -1, -13, 24]]
37 1.802476 740.523851

[2, -4, -16, 22, 16, -11, -31, 28, 18, -26, 65, 52, 117, 104, -26]
[[2, 3, 5, 7, 5, 6], [0, 1, -2, -8, 11, 8]]
38 1.769869 760.175249

[11, 1, -19, -17, -4, -24, -61, -65, -47, -47, -43, -13, 18, 59, 49]
[[1, -2, 2, 9, 9, 5], [0, 11, 1, -19, -17, -4]]
41 1.561845 761.396598

[18, 10, -6, 14, 6, -26, -60, -40, -57, -42, -2, -23, 60, 39, -31]
[[2, 2, 4, 6, 6, 7], [0, 9, 5, -3, 7, 3]]
42 1.535699 779.329982

[12, 22, -4, -6, -42, 7, -40, -51, -111, -71, -90, -179, -3, -103,
-123] [[2, 5, 8, 5, 6, 1], [0, -6, -11, 2, 3, 21]]
66 .748762 807.080623

[10, -20, 12, 18, -12, -55, -9, -6, -56, 84, 111, 46, 9, -78, -108]
[[2, 1, 9, 3, 3, 10], [0, 5, -10, 6, 9, -6]]
40 1.735774 812.069301

[23, 23, 23, 23, 0, -17, -28, -43, -85, -11, -26, -85, -15, -85, -85]
[[23, 36, 53, 64, 79, 85], [0, 1, 1, 1, 1, 0]]
46 1.533497 905.619955

[22, 2, 8, 12, 38, -48, -49, -57, -21, 13, 21, 81, 6, 77, 87] [[2, 7,
5, 7, 9, 14], [0, -11, -1, -4, -6, -19]]
44 1.667793 914.597575

[14, 18, -20, 16, -26, -4, -71, -23, -93, -97, -25, -127, 114, 1,
-149] [[2, 5, 7, 3, 9, 4], [0, -7, -9, 10, -8, 13]]
54 1.219228 940.606327

[16, 14, 10, -8, 36, -15, -29, -68, -2, -16, -67, 32, -57, 64, 154]
[[2, 7, 8, 8, 5, 16], [0, -8, -7, -5, 4, -18]]
44 1.769869 970.574458

[8, -16, -18, -4, -28, -44, -51, -34, -74, 3, 46, -6, 51, -12, -82]
[[2, 3, 5, 6, 7, 8], [0, 4, -8, -9, -2, -14]]
44 1.820926 998.573790

Message: 11163 - Contents - Hide Contents

Date: Wed, 30 Jun 2004 18:36:38

Subject: Leapday

From: Gene Ward Smith

This is the 13-limit temperament with wedgie

<<1 21 15 11 8 31 21 14 9 -24 -47 -59 -21 -33 -13||

and mapping to primes

[<1 2 11 9 8 7|, <0 -1 -21 -15 -11 -8|]

The generator is a fourth or fifth, and if we want the completem
temperament, including fives, then in terms of equal temperaments
there is not a lot of point in tuning it to anything but 46; 27/46 is
a poptimal generator, and the next one I found was 254/615; hence
leapday is firmly wedded to 46. 

The TM basis is {91/90, 121/120, 169/168, 352/351}, compatible as we
have seen with HTT. The no-fives version of leapday has a TM basis 
{169/168, 352/351, 364/363}, which adds 169/168 to the commas of HTT.
It has a different TOP tuning, and we may want to use the 63-equal
tuning in place of 46 equal if we have no interest in 5, because this
gives better results for the other primes excepting 3, and is close to
the no-fives rms optimal value.

We have MOS of size 17 and 29, and a white-black analysis gives the 46
equal tuning as 29+17; the 63 equal tuning has a white-black of
46+17.

Here's a comparison of 46 and 63, using 21/63 = 400 cents in place of
20/63.

46 vs 63 sharpness of primes

3: 2.39 2.81
5: 4.99 13.67
7: -3.61 2.60
11: -3.49 1.06
13: -5.75 -2.43





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Message: 11164 - Contents - Hide Contents

Date: Thu, 01 Jul 2004 22:20:46

Subject: Gene's mail server

From: Paul Erlich

Hi Gene,

Your mail server rejected my e-mail response to you as "probable 
SPAM".

Assuming you disagree with your server's assessment, let me know how 
I should proceed.

-Paul

Message: 11165 - Contents - Hide Contents

Date: Thu, 01 Jul 2004 00:58:31

Subject: Simultaneous temperament approximation

From: Gene Ward Smith

If we have several temperaments we want to approximate, we can treat
it as a modified multiple diophantine approximation problem, modified
since we step through the lcm of the periods. Hence, to approximate
meantone and pajara simultaneously, we look for even numbers n which
give a good simultaneous approximation to both fifths. Below I give
the maximum relative error, with the rms tuning treated as optimal,
times the square root of n, where 2n is the approximating et; and list
solutions under 1000 with this badness figure less than 0.6. We can,
of course, do this for more temperaments than just two. We've
discussed 198, and Paul mentioned 88. That 12 should be good is of
course obvious.


12 .215906 -.033520 .088143
88 .580526 .087518 -.020282
100 .479848 .053997 .067861
198 .528192 -.053085 -.045636
298 .271293 .000912 .022225

Message: 11166 - Contents - Hide Contents

Date: Thu, 01 Jul 2004 22:35:37

Subject: Re: Wedgies and generators

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
> wrote:
> 

>> Put it all together and you get an antisymmetric matrix, the upper
>> right corner of which is the wedgie. I think Herman was the one who
>> suggested writing it in the form
>> 
>> <<1 4 10
>> 4 13
>> 12||
> 

> When I asked you, some time ago, about these triangles of numbers and 
> their great similarity to the upper triangle of certain matrix of 
> vanishing commas, you had no response.


Sorry. I did discuss it fairly extensively when I first introduced the
wedge product, if I recall correctly, in connection with the commutator.

Message: 11167 - Contents - Hide Contents

Date: Thu, 01 Jul 2004 02:12:57

Subject: my paper nears completion

From: Paul Erlich

I'd appreciate any comments or corrections . . . note that it's 
incomplete, and the 46 horagrams are not included -- 

Yahoo groups: /tuning/files/perlich/coyotepape... * [with cont.] 

Message: 11168 - Contents - Hide Contents

Date: Thu, 01 Jul 2004 22:42:20

Subject: The 7-limit using 3 and 5

From: Gene Ward Smith

3 and 5 are a possible pair of generators for ennealimmal, and tuning
them justly is a possible tuning. As a consequence, you get a {3,5}
version of the 7-limit which is wafso-just if I recall correctly what
that means. In place of 2, you would use (27/25)^9, and in the place
of 7, (3^17/5^11)^2. While ennealimmal is accurate enough that this
will work, it isn't a very good tuning, relatively speaking, and you
can do much better by using a tempered 3 and 5.

Message: 11169 - Contents - Hide Contents

Date: Thu, 01 Jul 2004 22:45:07

Subject: Re: Gene's mail server

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> Hi Gene,
> 
> Your mail server rejected my e-mail response to you as "probable 
> SPAM".
> 
> Assuming you disagree with your server's assessment, let me know how 
> I should proceed.


Considering the huge quantity of spam which makes it through, this is
kind of depressing. Maybe upload it, or post it if it is postable. I
suppose I should get an alternative email address.

Message: 11171 - Contents - Hide Contents

Date: Thu, 01 Jul 2004 23:25:48

Subject: Extending temperaments using TOP tuning

From: Gene Ward Smith

I havn't made use of the "brute force" approach Dave and Herman have
had success with as a temperament finding algorithm, being an
algebraist by training. However, it occurs to me that if I want to
find higher limit tunings which accord with a lower-limit tuning and
hence deserve the same name, I could as an alternative to my algebraic
approach simply use the TOP tuned generators and search for good
approximations to the higher-limit primes. If I use the 5-limit TOP
values for 2 and 3, and if I search for exponents for each in the
range -100 to 100, I get only one value within 2 cents for 7, namely
3^10/2^13 (leading to 7-limit meantone), and only one value for 11
within 2 cents, 2^24/3^13, leading to 11-limit meantone, or "meanpop".
I find nothing for 13 under 2 cents, and it hardly makes sense to
increase the range of the exponents past a complexity of 100. Pushing
the error limit up to 9 cents gives 3^46/2^69, which isn't going to
break any records for wonderfulness.

The wedgie for this 13-limit meantone is

<<1 4 10 -13 46 4 13 -24 69 12 -44 92 -71 92 207||

The mapping is

[<1 2 4 7 -2 23|, <0 -1 -4 -10 13 -46|]

The TM basis is {81/80, 126/125, 385/384, 1573/1568}

The TOP generators are only slightly different. 112 equal is one
tuning choice, but the temperament really makes no sense, since you
may as well simply use 31.

Message: 11172 - Contents - Hide Contents

Date: Thu, 01 Jul 2004 00:24:33

Subject: Re: my paper nears completion

From: Herman Miller

Paul Erlich wrote:


> I'd appreciate any comments or corrections . . . note that it's 
> incomplete, and the 46 horagrams are not included -- 
> 
> Yahoo groups: /tuning/files/perlich/coyotepape... * [with cont.] 


Is the "Sagittal" font available for download? (Be sure to send a copy 
of it with the paper.)

Augmented should be [7 0 -3> in the table at the end.

The mathematical parts looked easy enough to understand -- some of it 
seemed a little too obvious to me, but for the average reader who hasn't 
been following the tuning-math list it would be more of a challenge. A 
whole paper could be written just on the musical applications of 
wedgies, so it's probably just as well that they're not mentioned. But 
you should at least have some description of what a "bivector" is if 
you're going to include them in the tables.

A couple of brief notated musical examples would be nice, like a typical 
octatonic chord progression you might find in 12-ET music to illustrate 
the 648;625 comma (the A minor - C minor - Eb minor - F# minor cycle on 
my diminished temperament page for instance).

Message: 11173 - Contents - Hide Contents

Date: Thu, 01 Jul 2004 06:09:55

Subject: Re: my paper nears completion

From: Paul Erlich

Thanks, Dante . . .

Well, each temperament will be represented by a nice horagram 
(floragrams not ready in time, unfortunately). Other than that, we'll 
see if there's room and time for more individualized 
discussions . . . 

--- In tuning-math@xxxxxxxxxxx.xxxx "Dante Rosati" <dante@i...> wrote:

> Wow, Paul, its looking great! Very clearly laid out and developed. 
Is there
> going to be any discussion of the various temperaments at the end, 
their
> characteristics & quirks? Or would that be for another different 
paper?
> 
> Dante
> 
> 
>> -----Original Message-----

>> From: Paul Erlich [mailto:perlich@a...]
>> Sent: Wednesday, June 30, 2004 10:13 PM
>> To: tuning-math@xxxxxxxxxxx.xxx
>> Subject: [tuning-math] my paper nears completion
>> 
>> 
>> I'd appreciate any comments or corrections . . . note that it's
>> incomplete, and the 46 horagrams are not included --
>> 
>> 
Yahoo groups: /tuning/files/perlich/coyotepape... * [with cont.] 

Message: 11174 - Contents - Hide Contents

Date: Thu, 01 Jul 2004 21:18:07

Subject: Re: my paper nears completion

From: Herman Miller

Paul Erlich wrote:


> If you could provide such notated examples for me to include, I'd be 
> extremely grateful. Dave Keenan provided the lattices that are in 
> there and one more that will be soon. I'll be sure to thank you both, 
> and Gene too.


Well, I don't have any good notation software, but I managed to put a 
couple of examples together with Voyetra Digital Orchestrator and some 
cutting and pasting in Paint Shop Pro. Unfortunately I couldn't figure 
out how to tell it to use sharps instead of flats, if it can even do 
that (probably not, since it's a MIDI editor).

But here's a notated version of my octatonic chord progression in 12-ET:

ftp://ftp.io.com/pub/usr/hmiller/music/octatonic.gif
(MIDI at http://www.io.com/~hmiller/music/ex/dim12.mid - Type Ok * [with cont.]  (Wayb.))

and the porcupine chord progression in 12-ET, which illustrates the 
250;243 comma (which of course doesn't vanish in 12-ET, but does in 
porcupine temperament):

ftp://ftp.io.com/pub/usr/hmiller/music/porcupine.gif
(MIDI at http://www.io.com/~hmiller/midi/porcupine-12.mid - Type Ok * [with cont.]  (Wayb.))

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