Something new for Christmas

Hi again.

Well, finally, I decided to make a small Christmas present for all of you and eventually anyone else who would like to know more about my non-12-EDO tuning experiences.
I've divided the "invisible but audible present" into two parts and I also used two different tunings, one for each part. Although they can sound a bit like 5-limit or 7-limit JI, neither of them is actually JI, both are particular kinds of temperament. For now, I'm not going to tell you which temperaments they are. -- Okay, here we go.
Part I: http://download.yousendit.com/1B3DDF7E47ED2E3B
Part II: http://download.yousendit.com/E0BFDEA832613A23

Have a great day and all the best in the new year of 2008.

Petr

Petr Pa��zek wrote:
> Hi again.
> > Well, finally, I decided to make a small Christmas present for all of you > and eventually anyone else who would like to know more about my non-12-EDO > tuning experiences.
> I've divided the "invisible but audible present" into two parts and I also > used two different tunings, one for each part. Although they can sound a bit > like 5-limit or 7-limit JI, neither of them is actually JI, both are > particular kinds of temperament. For now, I'm not going to tell you which > temperaments they are. -- Okay, here we go.
> Part I: http://download.yousendit.com/1B3DDF7E47ED2E3B
> Part II: http://download.yousendit.com/E0BFDEA832613A23
> > Have a great day and all the best in the new year of 2008.

Well done, I especially like the second one. I like how it goes off into all kinds of harmonic places, and just when it seems to be getting back to the tonic, does an unexpected turn and ends up in a different key. The first modulation around 0:57 seems a little jarring at first, but I'm getting used to it. Do these temperaments have anything to do with the improvisation you posted earlier this month? Some of the progressions sound similar, but the character of the music is quite different.

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote:
> Well, finally, I decided to make a small Christmas present for all
> of you and eventually anyone else who would like to know more
> about my non-12-EDO tuning experiences.
//
> Okay, here we go.
> Part I: http://download.yousendit.com/1B3DDF7E47ED2E3B
> Part II: http://download.yousendit.com/E0BFDEA832613A23
>
> Have a great day and all the best in the new year of 2008.
>
> Petr

Thanks! Nice pieces. These seem to continue a trend away from
the safety of soundscapes toward (apparently improvised works of)
more 'notey' structures.

-Carl

Herman wrote:

> Well done, I especially like the second one. I like how it goes off into
> all kinds of harmonic places, and just when it seems to be getting back
> to the tonic, does an unexpected turn and ends up in a different key.
> The first modulation around 0:57 seems a little jarring at first, but
> I'm getting used to it. Do these temperaments have anything to do with
> the improvisation you posted earlier this month? Some of the
> progressions sound similar, but the character of the music is quite
> different.

Glad you like it.
I'll give you a tiny clu. Yes, the temperaments do have something in common with the one I used earlier -- but it's nothing more than the fact that they are linear temperaments. Instead of having a half-octave period, both of these have the usual period of one octave. So, ... Okay, do you wanna try to think about the possible generators? Or leave it untill I tell you what they are?

Petr

Petr Pa��zek wrote:
> Herman wrote:
> >> Well done, I especially like the second one. I like how it goes off into
>> all kinds of harmonic places, and just when it seems to be getting back
>> to the tonic, does an unexpected turn and ends up in a different key.
>> The first modulation around 0:57 seems a little jarring at first, but
>> I'm getting used to it. Do these temperaments have anything to do with
>> the improvisation you posted earlier this month? Some of the
>> progressions sound similar, but the character of the music is quite
>> different.
> > Glad you like it.
> I'll give you a tiny clu. Yes, the temperaments do have something in common > with the one I used earlier -- but it's nothing more than the fact that they > are linear temperaments. Instead of having a half-octave period, both of > these have the usual period of one octave. So, ... Okay, do you wanna try to > think about the possible generators? Or leave it untill I tell you what they > are?

My first thought was that instead of dividing the octave in half, you might have divided your 162.7 cent generator in half. I also thought it might be Alpha (one of Wendy Carlos' non-octave scales) with octaves added. But I couldn't get either of those to work out.

They both seem 22-ish, but that's not much to go on. I can play along with a 22-ET keyboard, but the notes don't quite match. I suspected porcupine for the second one, but I couldn't find a porcupine tuning to match. Superpyth, magic, and orwell are the most likely possibilities.

Orwell has a generator around 271 cents; the 3/1 is divided into 7 equal parts. Magic has a major third as a generator. The other one that might fit with either of these two is superpyth. The generator is a slightly flat fourth (or sharp fifth). But orwell and magic are more accurate temperaments. I suspect if you didn't like hedgehog for tempering out 50/49, superpyth (which tempers out 64/63) is probably also not one that you'd pick.

Herman wrote:

> They both seem 22-ish, but that's not much to go on. I can play along
> with a 22-ET keyboard, but the notes don't quite match. I suspected
> porcupine for the second one, but I couldn't find a porcupine tuning to
> match. Superpyth, magic, and orwell are the most likely possibilities.

For one thing, you've made a very interesting point. If I heard the music without any instructions, I would think the first part could be some kind of modified 31 and that the second part would be some kind of modified 19. For another thing, you've perfectly guessed the first part -- it's really an Orwell with the generator being (24/5)^(1/10). The generator of the second tuning is the one which is so audible in those specific harmonic progressions -- to be precise, (50/3)^(1/11).

OTOH, I found some similarities between hedgehog and 22-EDO, which I never thought about earlier.

PS: What is porcupine?

Petr

Petr Pa��zek wrote:
> Herman wrote:
> >> They both seem 22-ish, but that's not much to go on. I can play along
>> with a 22-ET keyboard, but the notes don't quite match. I suspected
>> porcupine for the second one, but I couldn't find a porcupine tuning to
>> match. Superpyth, magic, and orwell are the most likely possibilities.
> > For one thing, you've made a very interesting point. If I heard the music > without any instructions, I would think the first part could be some kind of > modified 31 and that the second part would be some kind of modified 19. For > another thing, you've perfectly guessed the first part -- it's really an > Orwell with the generator being (24/5)^(1/10). The generator of the second > tuning is the one which is so audible in those specific harmonic > progressions -- to be precise, (50/3)^(1/11).

This seems to be semisixths temperament (a.k.a. sensipent). The generator is half a major sixth, and it does make sense to describe it as a kind of modified 19. Now that you mention it, one of the things I tried was keemun, which is another kind of modified 19 (with a minor third as a generator). The scale in the first melody is one that fits keemun fairly well, but I couldn't get the harmonic progressions to work. Magic is also a modified 19, which may be why magic seemed like a better fit.

> OTOH, I found some similarities between hedgehog and 22-EDO, which I never > thought about earlier.
> > PS: What is porcupine?

Porcupine tempers out 250/243. The generator (around 162 cents) divides the minor sixth into five equal parts.

Here's a generator mapping for semisixths (the 5-limit version or sensipent) and porcupine for comparison:

sensipent [<1, -1, -1], <0, 7, 9]>
porcupine [<1, 2, 3], <0, -3, -5]>

What this does is describe the mapping of the three prime intervals: 2/1, 3/1, and 5/1; from these you can map any rational interval into a tempered interval. To keep it simple, we can ignore the octaves and look at the last two numbers: it takes 7 generators up to reach a 3/1 (or a 3/2) in sensi, and 9 generators up to reach 5/1 (or 5/4). Porcupine on the other hand has 3 generators down for a 3/2 and 5 generators down for a 5/4.

Herman wrote:

> This seems to be semisixths temperament (a.k.a. sensipent). The
> generator is half a major sixth, and it does make sense to describe it
> as a kind of modified 19. Now that you mention it, one of the things I
> tried was keemun, which is another kind of modified 19 (with a minor
> third as a generator). The scale in the first melody is one that fits
> keemun fairly well, but I couldn't get the harmonic progressions to
> work.

Is that the same as Hanson?

> Magic is also a modified 19, which may be why magic seemed like a
> better fit.

I was also looking at magic the other day but I haven't tried it out yet. I was a bit disappointed by the amount of mistuning which was much larger than in semisixth even if I used the softest tempering possible. I will certainly try it out soon, I only don't know when I'll get to making a new piece in it.

> sensipent [<1, -1, -1], <0, 7, 9]>
> porcupine [<1, 2, 3], <0, -3, -5]>

Okay, thanks for clarifiing that the tuning I was experimenting with, after I made the hedgehog scale, was actually porcupine. Unfortunately, I wasn't finally able to tune it the way I wished as the XG retuning format is not good for this; but I definitely want to try at least a smaller porcupine scale one day and I'll surely find a way how to tune it even with the limitations that the XG format has.

Petr

Petr Pa��zek wrote:
> Herman wrote:
> >> This seems to be semisixths temperament (a.k.a. sensipent). The
>> generator is half a major sixth, and it does make sense to describe it
>> as a kind of modified 19. Now that you mention it, one of the things I
>> tried was keemun, which is another kind of modified 19 (with a minor
>> third as a generator). The scale in the first melody is one that fits
>> keemun fairly well, but I couldn't get the harmonic progressions to
>> work.
> > Is that the same as Hanson?

Hanson is the 5-limit kleismic temperament; there's more than one 7-limit version.

hanson [<1, 0, 1], <0, 6, 5]>
TOP P = 1200.291038, G = 317.069381

keemun [<1, 0, 1, 2], <0, 6, 5, 3]>
TOP P = 1203.187309, G = 317.834461

catakleismic [<1, 0, 1, -3], <0, 6, 5, 22]>
TOP P = 1200.536355, G = 316.906396

and a few others that don't have names.

>> Magic is also a modified 19, which may be why magic seemed like a
>> better fit.
> > I was also looking at magic the other day but I haven't tried it out yet. I > was a bit disappointed by the amount of mistuning which was much larger than > in semisixth even if I used the softest tempering possible. I will certainly > try it out soon, I only don't know when I'll get to making a new piece in > it.

The usual 7-limit semisixths (sensisept) is mistuned even a bit more than magic, but I looked around and found this temperament, which oddly enough is 19&84 (19-ET combined with 84-ET). Considering that 19 steps of 84-ET is an orwell generator, you could do both of these pieces as subsets of 84-ET.

Unnamed temperament 19&84
[<1, -1, -1, -9], <0, 7, 9, 32]>
TOP P = 1200.796867, G = 443.069838
TOP-RMS P = 1200.643572, G = 442.992533

But that may be too complex for what it's worth, since there are temperaments like miracle which are both simpler and more accurate. The good temperaments have a balance between accuracy and complexity -- the simpler ones tend to be less accurate, and the more accurate ones tend to require more notes.

Herman wrote:

> The usual 7-limit semisixths (sensisept) is mistuned even a bit more
> than magic,

Is there anywhere any kind of list of rank 2 temperaments? I was trying to find something on Monz's website but there was not a word about hedgehog or keemun.

> but I looked around and found this temperament, which oddly
> enough is 19&84 (19-ET combined with 84-ET). Considering that 19 steps
> of 84-ET is an orwell generator, you could do both of these pieces as
> subsets of 84-ET.

My goodness, I'll have to look at 84-ET more carefully, I've never thought it could work. :-D

Petr

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote:
> Is there anywhere any kind of list of rank 2 temperaments? I was
trying to
> find something on Monz's website but there was not a word about
hedgehog or
> keemun.

Hi Petr,

See Paul Erlich's paper.
http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

Monz,

Do you have links to Paul's paper from appropriate places in your
wonderful Encyclopedia?

By the way, I've just been having a wonderful time learning to play a
22-EDO guitar (Choob) thanks to Carl Lumma who commissioned me to
build it for him. I'll have to reread Paul's 22-EDO paper too.

-- Dave Keenan

Petr Pa��zek wrote:
> Herman wrote:
> >> The usual 7-limit semisixths (sensisept) is mistuned even a bit more
>> than magic,
> > Is there anywhere any kind of list of rank 2 temperaments? I was trying to > find something on Monz's website but there was not a word about hedgehog or > keemun.

It's been a long time since I posted a reply to this, and it hasn't shown up, so I'll have to assume it was blocked by some filter on Yahoo. Probably didn't like the URL with the numbers in it or something.

In any case, the page I mentioned can be found by googling for +hedgehog +keemun. The name of the page is "Seven-limit temperament names", or "Seven Limit Named Temperaments". I also mentioned Paul Erlich's paper (which Dave Keenan mentioned yesterday). Most of the best 5- and 7-limit rank 2 temperaments are listed in Paul's paper, with the generator mappings and other useful data, but there are a few others that, even though they might not be as good, still have interesting musical uses. Gene's page, on the other hand, while it lists a huge number of temperaments, only provides the wedgie, which isn't much to go on. There's a handful of 11-limit and higher temperaments that have been named, but I don't know if there's a convenient list of them anywhere.

It happens that most useful rank 2 temperaments can be described as combinations of ET's. Graham Breed has a page which takes two ET's and finds a generator mapping (http://x31eq.com/temper/twoet.html). I've gone through Gene's big list and looked up all the wedgies, and most of them were already in a list I have of ET combinations. An ET combination isn't a unique identifier like the wedgie (many temperaments have more than one possible combination of ETs that generates them), but it's easier to get a feel for what the temperament might be useful for. About the only useful thing you can get out of a wedgie without a lot of complicated manipulation of the numbers is a general idea of the temperament's complexity.

So here's a list of ET combinations for each of the named temperaments in Gene's list, with asterisks by those that weren't in my list. (Keep in mind that some of these may have more than one name.) You can then draw a line on an ET chart like the one on the Tonalsoft web site (http://tonalsoft.com/enc/e/equal-temperament.aspx) or the one in Paul Erlich's paper, or plug the numbers into Graham Breed's temperament finder to get a generator mapping and suggested size for the generator.

46&53 Amity <<5 13 -17 9 -41 -76||
12&15 Augene <<3 0 -6 -7 -18 -14||
9&12 Augie <<3 0 6 -7 1 14||
10&27 Beatles <<2 -9 -4 -19 -12 16||
4&5 Beep <<2 3 1 0 -4 -6||
* Bipelog <<2 -6 -6 -14 -15 3||
22&118 Bisupermajor <<16 -10 34 -53 9 107||
5&10 Blackwood <<0 5 0 8 0 -14||
41&49 Bohpier <<13 19 23 0 0 0||
19&53 Catakleismic <<6 5 22 -6 18 37||
19&49 Clyde <<12 10 25 -12 6 30||
53&87 Countercata <<6 5 -31 -6 -66 -86||
4&6 Decimal <<4 2 2 -6 -8 -1||
12&46 Diaschismic <<2 -4 -16 -11 -31 -26||
* Dicot <<2 1 6 -3 4 11||
4&12 Diminished <<4 4 4 -3 -5 -2||
5&12 Dominant <<1 4 -2 4 -6 -16||
4&18 Doublewide <<8 6 6 -9 -13 -3||
12&36 Duodec <<0 0 12 0 19 28||
19&152 Enneadecal <<19 19 57 -14 37 79||
27&45 Ennealimmal <<18 27 18 1 -22 -34||
* Father <<1 -1 3 -4 2 10||
19&26 Flattone <<1 4 -9 4 -17 -32||
26&73 Gamera <<23 40 1 10 -63 -110||
12&29 Garibaldi <<1 -8 -14 -15 -25 -10||
* Gidorah <<3 2 -1 -4 -10 -8||
5&19 Godzilla <<2 8 1 8 -4 -20||
5&16 Gorgo <<3 7 -1 4 -10 -22||
12&77 Grackle <<1 -8 -26 -15 -44 -38||
31&59 Grendel <<23 -1 13 -55 -44 33||
36&41 Guiron <<3 -24 -1 -45 -10 65||
58&72 Harry <<12 34 20 26 -2 -49||
* Hedgehog <<6 10 10 2 -1 -5||
41&58 Hemififths <<2 25 13 35 15 -40||
15&53 Hemikleismic <<12 10 -9 -12 -48 -49||
12&118 Hemischismic <<2 -16 -40 -30 -69 -48||
31&56 Hemithirds <<15 -2 -5 -38 -50 -6||
6&31 Hemiwuer <<16 2 5 -34 -37 6||
22&77 Hendecatonic <<11 -11 22 -43 4 82||
6&12 Hexe <<6 0 0 -14 -17 0||
* Hystrix <<3 5 1 1 -7 -12||
12&26 Injera <<2 8 8 8 7 -4||
* Jamesbond <<0 0 7 0 11 16||
4&15 Keemun <<6 5 3 -6 -12 -7||
22&56 Keen <<2 -4 18 -11 23 53||
41&70 Kwai <<1 33 27 50 40 -30||
10&16 Lemba <<6 -2 -2 -17 -20 1||
19&36 Liese <<3 12 11 12 9 -8||
19&22 Magic <<5 1 12 -10 5 25||
* Marvo <<6 17 46 13 56 59||
* Mavila <<1 -3 -4 -7 -9 -1||
12&19 Meantone <<1 4 10 4 13 12||
10&31 Miracle <<6 -7 -2 -25 -20 15||
12&87 Misty <<3 -12 -30 -26 -56 -36||
46&125 Mitonic <<17 35 -21 16 -81 -147||
* Mother <<1 -1 -2 -4 -6 -2||
5&26 Mothra <<3 12 -1 12 -10 -36||
16&19 Muggles <<5 1 -7 -10 -25 -19||
84&87 Mutt <<21 3 -36 -44 -116 -92||
4&27 Myna <<10 9 7 -9 -17 -9||
15&29 Nautilus <<6 10 3 2 -12 -21||
9&10 Negri <<4 -3 2 -14 -8 13||
35&68 Neptune <<40 22 21 -58 -79 -13||
31&70 Nusecond <<11 13 17 -5 -4 3||
27&41 Octacot <<8 18 11 10 -5 -25||
10&18 Octokaidecal <<2 6 6 5 4 -3||
9&22 Orwell <<7 -3 8 -21 -7 27||
10&12 Pajara <<2 -4 -4 -11 -12 2||
19&80 Parakleismic <<13 14 35 -8 19 42||
9&16 Pelogic <<1 -3 5 -7 5 20||
* Penta <<3 2 4 -4 -2 4||
53&118 Pontiac <<1 -8 39 -15 59 113||
15&22 Porcupine <<3 5 -6 1 -18 -28||
26&27 Quartonic <<11 18 5 3 -23 -39||
31&177 Quasiorwell <<38 -3 8 -93 -94 27||
* Quasisuper <<1 -13 -2 -23 -6 32||
* Ragimetric <<88 151 183 35 43 1||
5&41 Rodan <<3 17 -1 20 -10 -50||
* Schism <<1 -8 -2 -15 -6 18||
* Semiaug <<6 0 15 -14 7 35||
* Semififths <<2 8 -11 8 -23 -48||
18&31 Semisept <<17 6 15 -30 -24 18||
19&27 Sensi <<7 9 13 -2 1 5||
41&89 Sesquiquartififths <<4 -32 -15 -60 -35 55||
* Sharptone <<1 4 3 4 2 -4||
22&46 Shrutar <<4 -8 14 -22 11 55||
* Sidi <<4 2 9 -6 3 15||
31&63 Slender <<13 -10 6 -46 -27 42||
31&45 Squares <<4 16 9 16 3 -24||
15&26 Superkleismic <<9 10 -3 -5 -30 -35||
80&91 Supermajor <<37 46 75 -13 15 45||
* Superpelog <<2 -6 1 -14 -4 19||
5&22 Superpyth <<1 9 -2 12 -6 -30||
12&159 Term <<3 -24 -54 -45 -94 -58||
31&78 Tertiaseptal <<22 -5 3 -59 -57 21||
15&57 Trikleismic <<18 15 -6 -18 -60 -56||
* Triton <<3 -7 -8 -18 -21 1||
29&31 Tritonic <<5 -11 -12 -29 -33 3||
29&70 Undecental <<1 -37 -43 -61 -71 4||
26&46 Unidec <<12 22 -4 7 -40 -71||
15&16 Valentine <<9 5 -3 -13 -30 -21||
12&60 Waage <<0 12 24 19 38 22||
22&50 Wizard <<12 -2 20 -31 -2 52||
* Wuerschmidt <<8 1 18 -17 6 39||

Ragimetric is also an ET combination, but I didn't have it in my list; it appears to be 171&1757. As for the others (some of which are better known as 5-limit temperaments), a generator mapping is probably the best way to characterize them.

Bipelog [<2, 3, 5, 6], <0, 1, -3, -3]>
TOP P = 603.274132, G = 81.753849
TOP-RMS P = 603.810389, G = 81.711065

Dicot [<1, 1, 2, 1], <0, 2, 1, 6]>
TOP P = 1204.048159, G = 356.399825
TOP-RMS P = 1202.700337, G = 358.743395

Father [<1, 2, 2, 4], <0, -1, 1, -3]>
TOP P = 1185.869125, G = 447.386341
TOP-RMS P = 1181.065349, G = 450.771422

Gidorah [<1, 1, 2, 3], <0, 3, 2, -1]>
TOP P = 1183.266011, G = 230.283085
TOP-RMS P = 1193.239850, G = 229.462249

Hedgehog [<2, 4, 6, 7], <0, -3, -5, -5]>
TOP P = 598.446711, G = 162.315961
TOP-RMS P = 599.619018, G = 164.247626

Hystrix [<1, 2, 3, 3], <0, -3, -5, -1]>
TOP P = 1187.933715, G = 161.100895
TOP-RMS P = 1188.142146, G = 157.297886

Jamesbond [<7, 11, 16, 20], <0, 0, 0, -1]>
TOP P = 172.775916, G = 86.692412 (e.g.)
TOP-RMS P = 172.811781, G = 87.409713

Marvo [<1, -1, -5, -17], <0, 6, 17, 46]>
TOP P = 1200.528119, G = 516.941012
TOP-RMS P = 1200.631099, G = 516.966133

Mavila [<1, 2, 1, 1], <0, -1, 3, 4]>
TOP P = 1209.734056, G = 532.941225
TOP-RMS P = 1210.494099, G = 531.756780

Mother [<1, 2, 2, 2], <0, -1, 1, 2]>
TOP P = 1179.240917, G = 476.032979
TOP-RMS P = 1187.933729, G = 473.620593

Penta [<1, 1, 2, 2], <0, 3, 2, 4]>
TOP P = 1179.240917, G = 238.016490
TOP-RMS P = 1187.760073, G = 237.529105

Quasisuper [<1, 2, -3, 2], <0, -1, 13, 2]>
TOP P = 1197.567789, G = 490.259205
TOP-RMS P = 1196.995367, G = 490.530245

Ragimetric [<1, -19, -33, -40], <0, 88, 151, 183]>
TOP P = 1200.003174, G = 280.704709
TOP-RMS P = 1200.000118, G = 280.704046

Schism [<1, 2, -1, 2], <0, -1, 8, 2]>
TOP P = 1195.155395, G = 496.239889
TOP-RMS P = 1197.420963, G = 497.372595

Semiaug [<3, 5, 7, 9], <0, -2, 0, -5]>
TOP P = 399.020013, G = 44.470842 (e.g.)
TOP-RMS P = 398.933869, G = 44.998030

Semififths [<1, 1, 0, 6], <0, 2, 8, -11]>
TOP P = 1201.698520, G = 348.782195
TOP-RMS P = 1200.822776, G = 348.653786

Sharptone [<1, 2, 4, 4], <0, -1, -4, -3]>
TOP P = 1214.253642, G = 509.401230
TOP-RMS P = 1204.750303, G = 501.838993

Sidi [<1, 3, 3, 6], <0, -4, -2, -9]>
TOP P = 1208.170435, G = 428.584377
TOP-RMS P = 1207.317244, G = 429.813424

Superpelog [<1, 2, 1, 3], <0, -2, 6, -1]>
TOP P = 1206.548265, G = 260.760141
TOP-RMS P = 1208.917150, G = 261.883710

Triton [<1, 3, -1, -1], <0, -3, 7, 8]>
TOP P = 1202.900537, G = 570.447951
TOP-RMS P = 1203.405329, G = 570.479707

Wuerschmidt [<1, -1, 2, -3], <0, 8, 1, 18]>
TOP P = 1201.135545, G = 387.584136
TOP-RMS P = 1199.978801, G = 387.375695

Petr Pa��zek wrote:
> Herman wrote:
> >> The usual 7-limit semisixths (sensisept) is mistuned even a bit more
>> than magic,
> > Is there anywhere any kind of list of rank 2 temperaments? I was trying to > find something on Monz's website but there was not a word about hedgehog or > keemun.

Well, a google for +hedgehog +keemun does bring up this link, but it only makes sense if you know how to use the wedgies. I prefer using a generator mapping -- you can easily get the wedgie from that if that's what you need, and the generator mapping is more useful for actual tuning. Deriving a generator mapping from a wedgie is technically possible, but it's much easier to go the other way.

http://66.98.148.43/~xenharmo/sevnames.htm

Paul Erlich's "Middle Path" paper includes most of the "better" 5- and 7-limit rank 2 temperaments (the ones that have the best tradeoff of complexity vs. error). It also has useful charts including horograms for each of these temperaments, along with a lot of good background material, and there's a copy of it online.

http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

Some of the temperaments mentioned in Paul's paper have other names that have been used: e.g., luna is also called "hemithirds", cynder is also called "mothra", including some names that are just as well forgotten (e.g. "nonkleismic" for myna). And there are a few temperaments that have been used in music that aren't in the list. But the "Middle Path" paper is the best place to start.