monzieurs,

someone let me know if anything is wrong or missing . . .

25/24 ("neutral thirds"?)

generators [1200., 350.9775007]

ets 3 4 7 10 11 13 17

81/80 (3)^4/(2)^4/(5) meantone

generators [1200., 696.164845]

ets 5 7 12 19 31 50

128/125 (2)^7/(5)^3 augmented

generators [400.0000000, 91.20185550]

ets 3 9 12 15 27 39 66

135/128 (3)^3*(5)/(2)^7 pelogic

generators [1200., 677.137655]

ets 7 9 16 23

250/243 (2)*(5)^3/(3)^5 porcupine

generators [1200., 162.9960265]

ets 7 8 15 22 37

256/243 (2)^8/(3)^5 quintal (blackwood?)

generators [240.0000000, 84.66378778]

ets 5 10 15 25

648/625 (2)^3*(3)^4/(5)^4 diminished

generators [300.0000000, 94.13435693]

ets 4 8 12 16 20 28 32 40 52 64

2048/2025 (2)^11/(3)^4/(5)^2 diaschismic

generators [600.0000000, 105.4465315]

ets 10 12 34 46 80

3125/3072 (5)^5/(2)^10/(3) magic

generators [1200., 379.9679493]

ets 3 13 16 19 22 25

15625/15552 (5)^6/(2)^6/(3)^5 kleismic

generators [1200., 317.0796753]

ets 4 11 15 19 34 53 87

16875/16384 negri

generators [1200., 126.2382718]

ets 9 10 19 28 29 47 48 66 67 85 86

20000/19683 (2)^5*(5)^4/(3)^9 quadrafifths

generators [1200., 176.2822703]

ets 7 13 20 27 34 41 48 61 75 95

32805/32768 (3)^8*(5)/(2)^15 shismic

generators [1200., 701.727514]

ets 12 17 29 41 53 65

78732/78125 (2)^2*(3)^9/(5)^7 hemisixths

generators [1200., 442.9792975]

ets 8 11 19 27 46 65 84

393216/390625 (2)^17*(3)/(5)^8 wuerschmidt

generators [1200., 387.8196733]

ets 3 28 31 34 37 40

531441/524288 (3)^12/(2)^19 pythagoric (NOT pythagorean)/aristoxenean?

generators [100.0000000, 14.66378756]

ets 12 48 60 72 84 96

1600000/1594323 (2)^9*(5)^5/(3)^13 amt

generators [1200., 339.5088256]

ets 7 11 18 25 32

2109375/2097152 (3)^3*(5)^7/(2)^21 orwell

generators [1200., 271.5895996]

ets 9 13 22 31 53 84

4294967296/4271484375 (2)^32/(3)^7/(5)^9 septathirds

generators [1200., 55.27549315]

ets 22 43 65 87

thanks, paul! i'll add it to my "linear temperaments"

definition when i get a chance.

because of the tunings used in some of my favorites

of Herman Miller's _Pavane for a warped princess_,

there's a family of equal-temperaments which i've become

interested in lately, which all temper out the apotome,

{2,3}-vector [-11 7], ratio 2187:2048, ~114 cents:

14-, 21-, and 28-edo.

i noticed that these EDOs all have cardinalities which

are multiples of the exponent of 3 of the "vanishing comma".

looking at the lattices on my "bingo-card-lattice" definition

/tuning-math/files/dict/bingo.htm

i can see it works the same way for 10-, 15-, and 20-edo,

which all temper out the _limma_, {2,3}-vector [8 -5] = ~90 cents.

so apparently, at least in these few cases (but my guess

is that it happens in many more), there is some relationship

between the logarithmic division of 2 which creates the

EDO and the exponent of 3 of a comma that's tempered out.

has anyone noted this before? any further comments on it?

is it possible that for these two "commas" it's just

a coincidence?

-monz

----- Original Message -----

From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

To: <tuning-math@yahoogroups.com>

Sent: Friday, October 18, 2002 6:40 PM

Subject: [tuning-math] for monzoni: bloated list of 5-limit linear

temperaments

> monzieurs,

>

> someone let me know if anything is wrong or missing . . .

>

> 25/24 ("neutral thirds"?)

> generators [1200., 350.9775007]

> ets 3 4 7 10 11 13 17

>

> 81/80 (3)^4/(2)^4/(5) meantone

> generators [1200., 696.164845]

> ets 5 7 12 19 31 50

>

> 128/125 (2)^7/(5)^3 augmented

> generators [400.0000000, 91.20185550]

> ets 3 9 12 15 27 39 66

>

> 135/128 (3)^3*(5)/(2)^7 pelogic

> generators [1200., 677.137655]

> ets 7 9 16 23

>

> 250/243 (2)*(5)^3/(3)^5 porcupine

> generators [1200., 162.9960265]

> ets 7 8 15 22 37

>

> 256/243 (2)^8/(3)^5 quintal (blackwood?)

> generators [240.0000000, 84.66378778]

> ets 5 10 15 25

>

> 648/625 (2)^3*(3)^4/(5)^4 diminished

> generators [300.0000000, 94.13435693]

> ets 4 8 12 16 20 28 32 40 52 64

>

> 2048/2025 (2)^11/(3)^4/(5)^2 diaschismic

> generators [600.0000000, 105.4465315]

> ets 10 12 34 46 80

>

> 3125/3072 (5)^5/(2)^10/(3) magic

> generators [1200., 379.9679493]

> ets 3 13 16 19 22 25

>

> 15625/15552 (5)^6/(2)^6/(3)^5 kleismic

> generators [1200., 317.0796753]

> ets 4 11 15 19 34 53 87

>

> 16875/16384 negri

> generators [1200., 126.2382718]

> ets 9 10 19 28 29 47 48 66 67 85 86

>

> 20000/19683 (2)^5*(5)^4/(3)^9 quadrafifths

> generators [1200., 176.2822703]

> ets 7 13 20 27 34 41 48 61 75 95

>

> 32805/32768 (3)^8*(5)/(2)^15 shismic

> generators [1200., 701.727514]

> ets 12 17 29 41 53 65

>

> 78732/78125 (2)^2*(3)^9/(5)^7 hemisixths

> generators [1200., 442.9792975]

> ets 8 11 19 27 46 65 84

>

> 393216/390625 (2)^17*(3)/(5)^8 wuerschmidt

> generators [1200., 387.8196733]

> ets 3 28 31 34 37 40

>

> 531441/524288 (3)^12/(2)^19 pythagoric (NOT pythagorean)/aristoxenean?

> generators [100.0000000, 14.66378756]

> ets 12 48 60 72 84 96

>

> 1600000/1594323 (2)^9*(5)^5/(3)^13 amt

> generators [1200., 339.5088256]

> ets 7 11 18 25 32

>

> 2109375/2097152 (3)^3*(5)^7/(2)^21 orwell

> generators [1200., 271.5895996]

> ets 9 13 22 31 53 84

>

> 4294967296/4271484375 (2)^32/(3)^7/(5)^9 septathirds

> generators [1200., 55.27549315]

> ets 22 43 65 87

oh, and of course, your list already shows that this

also happens with the "Pythagoric" temperaments, which

all temper out the Pythagorean comma, {2,3}-vector [-19 12],

and which all have cardinalities which are multiples of 12.

so it seems that any EDO which tempers out a 3-limit

"comma" has a cardinality (= logarithmic division of 2)

which is a multiple of the exponent of 3 in that "comma".

interesting. looks to me like there's some kind of

"bridge between incommensurable primes" going on here.

-monz

----- Original Message -----

From: "monz" <monz@attglobal.net>

To: <tuning-math@yahoogroups.com>

Sent: Saturday, October 19, 2002 1:47 AM

Subject: Re: [tuning-math] for monzoni: bloated list of 5-limit linear

temperaments

> thanks, paul! i'll add it to my "linear temperaments"

> definition when i get a chance.

>

> because of the tunings used in some of my favorites

> of Herman Miller's _Pavane for a warped princess_,

> there's a family of equal-temperaments which i've become

> interested in lately, which all temper out the apotome,

> {2,3}-vector [-11 7], ratio 2187:2048, ~114 cents:

> 14-, 21-, and 28-edo.

>

> i noticed that these EDOs all have cardinalities which

> are multiples of the exponent of 3 of the "vanishing comma".

>

> looking at the lattices on my "bingo-card-lattice" definition

> /tuning-math/files/dict/bingo.htm

> i can see it works the same way for 10-, 15-, and 20-edo,

> which all temper out the _limma_, {2,3}-vector [8 -5] = ~90 cents.

>

>

> so apparently, at least in these few cases (but my guess

> is that it happens in many more), there is some relationship

> between the logarithmic division of 2 which creates the

> EDO and the exponent of 3 of a comma that's tempered out.

>

> has anyone noted this before? any further comments on it?

> is it possible that for these two "commas" it's just

> a coincidence?

>

> -monz

--- In tuning-math@y..., "monz" <monz@a...> wrote:

> thanks, paul! i'll add it to my "linear temperaments"

> definition when i get a chance.

>

> because of the tunings used in some of my favorites

> of Herman Miller's _Pavane for a warped princess_,

> there's a family of equal-temperaments which i've become

> interested in lately, which all temper out the apotome,

> {2,3}-vector [-11 7], ratio 2187:2048, ~114 cents:

> 14-, 21-, and 28-edo.

>

> i noticed that these EDOs all have cardinalities which

> are multiples of the exponent of 3 of the "vanishing comma".

>

> looking at the lattices on my "bingo-card-lattice" definition

> /tuning-math/files/dict/bingo.htm

> i can see it works the same way for 10-, 15-, and 20-edo,

> which all temper out the _limma_, {2,3}-vector [8 -5] = ~90 cents.

>

>

> so apparently, at least in these few cases (but my guess

> is that it happens in many more), there is some relationship

> between the logarithmic division of 2 which creates the

> EDO and the exponent of 3 of a comma that's tempered out.

>

> has anyone noted this before? any further comments on it?

> is it possible that for these two "commas" it's just

> a coincidence?

>

> -monz

examine the table below -- you'll note that certain commas vanishing

force the generator to be a fraction of an octave (600 cents, 400

cents, 300 cents, 240 cents) instead of a full octave . . .

the reason i posted this is that i wanted to see you fill out the

list on the eqtemp page . . . also lots of e-mails and post on the

tuning list awaiting your attention . . .

>

>

> ----- Original Message -----

> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>

> To: <tuning-math@y...>

> Sent: Friday, October 18, 2002 6:40 PM

> Subject: [tuning-math] for monzoni: bloated list of 5-limit linear

> temperaments

>

>

> > monzieurs,

> >

> > someone let me know if anything is wrong or missing . . .

> >

> > 25/24 ("neutral thirds"?)

> > generators [1200., 350.9775007]

> > ets 3 4 7 10 11 13 17

> >

> > 81/80 (3)^4/(2)^4/(5) meantone

> > generators [1200., 696.164845]

> > ets 5 7 12 19 31 50

> >

> > 128/125 (2)^7/(5)^3 augmented

> > generators [400.0000000, 91.20185550]

> > ets 3 9 12 15 27 39 66

> >

> > 135/128 (3)^3*(5)/(2)^7 pelogic

> > generators [1200., 677.137655]

> > ets 7 9 16 23

> >

> > 250/243 (2)*(5)^3/(3)^5 porcupine

> > generators [1200., 162.9960265]

> > ets 7 8 15 22 37

> >

> > 256/243 (2)^8/(3)^5 quintal (blackwood?)

> > generators [240.0000000, 84.66378778]

> > ets 5 10 15 25

> >

> > 648/625 (2)^3*(3)^4/(5)^4 diminished

> > generators [300.0000000, 94.13435693]

> > ets 4 8 12 16 20 28 32 40 52 64

> >

> > 2048/2025 (2)^11/(3)^4/(5)^2 diaschismic

> > generators [600.0000000, 105.4465315]

> > ets 10 12 34 46 80

> >

> > 3125/3072 (5)^5/(2)^10/(3) magic

> > generators [1200., 379.9679493]

> > ets 3 13 16 19 22 25

> >

> > 15625/15552 (5)^6/(2)^6/(3)^5 kleismic

> > generators [1200., 317.0796753]

> > ets 4 11 15 19 34 53 87

> >

> > 16875/16384 negri

> > generators [1200., 126.2382718]

> > ets 9 10 19 28 29 47 48 66 67 85 86

> >

> > 20000/19683 (2)^5*(5)^4/(3)^9 quadrafifths

> > generators [1200., 176.2822703]

> > ets 7 13 20 27 34 41 48 61 75 95

> >

> > 32805/32768 (3)^8*(5)/(2)^15 shismic

> > generators [1200., 701.727514]

> > ets 12 17 29 41 53 65

> >

> > 78732/78125 (2)^2*(3)^9/(5)^7 hemisixths

> > generators [1200., 442.9792975]

> > ets 8 11 19 27 46 65 84

> >

> > 393216/390625 (2)^17*(3)/(5)^8 wuerschmidt

> > generators [1200., 387.8196733]

> > ets 3 28 31 34 37 40

> >

> > 531441/524288 (3)^12/(2)^19 pythagoric (NOT

pythagorean)/aristoxenean?

> > generators [100.0000000, 14.66378756]

> > ets 12 48 60 72 84 96

> >

> > 1600000/1594323 (2)^9*(5)^5/(3)^13 amt

> > generators [1200., 339.5088256]

> > ets 7 11 18 25 32

> >

> > 2109375/2097152 (3)^3*(5)^7/(2)^21 orwell

> > generators [1200., 271.5895996]

> > ets 9 13 22 31 53 84

> >

> > 4294967296/4271484375 (2)^32/(3)^7/(5)^9 septathirds

> > generators [1200., 55.27549315]

> > ets 22 43 65 87

--- In tuning-math@y..., "monz" <monz@a...> wrote:

> oh, and of course, your list already shows that this

> also happens with the "Pythagoric" temperaments, which

> all temper out the Pythagorean comma, {2,3}-vector [-19 12],

> and which all have cardinalities which are multiples of 12.

i hope you'll update your eqtemp page -- it currently claims that 12-

equal acts as a pythagorean tuning (with a link to 3-limit JI), but

what you actually mean is "pythagoreic" or "aristoxenean" or whatever

the vanishing of the pythagorean comma is called.

> so it seems that any EDO which tempers out a 3-limit

> "comma" has a cardinality (= logarithmic division of 2)

> which is a multiple of the exponent of 3 in that "comma".

it doesn't have to be 3 -- it can be any prime or composite (product

and/or ratio) of primes. diesic, for example, tempers out 5^3, and so

divides the octave into 3 equal parts. diminished tempers out (3/5)

^4, so 4 equal parts. blackwood tempers out 3^5, so 5 equal parts.

the famous ennealimmal tempers out (3^3/5^2)^9, so 9 equal parts.

once you get beyond the 5-limit, a linear temperament will have

several vanishing commas, so things aren't as simple . . .

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Saturday, October 19, 2002 9:22 AM

> Subject: [tuning-math] Re: for monzoni: bloated list of 5-limit linear

temperaments

>

>

> i hope you'll update your eqtemp page -- it currently claims that 12-

> equal acts as a pythagorean tuning (with a link to 3-limit JI), but

> what you actually mean is "pythagoreic" or "aristoxenean" or whatever

> the vanishing of the pythagorean comma is called.

thanks.

i decided to go with "aristoxenean" in honor of Aristoxenos.

see the new Dictionary entry:

/tuning-math/files/dict/aristox.htm

-monz

"all roads lead to n^0"

hi monz,

if you could add a footnote in carl lumma's (orphaned) table

(on /tuning-math/files/dict/eqtemp.htm again)

to the pythagorean comma entry, referencing and linking to

aristoxenean temperament, i'd love you forever! (do anyway ;) )

-paul

--- In tuning-math@y..., "monz" <monz@a...> wrote:

>

> hi paul,

>

> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>

> > To: <tuning-math@y...>

> > Sent: Saturday, October 19, 2002 9:22 AM

> > Subject: [tuning-math] Re: for monzoni: bloated list of 5-limit

linear

> temperaments

> >

> >

> > i hope you'll update your eqtemp page -- it currently claims that

12-

> > equal acts as a pythagorean tuning (with a link to 3-limit JI),

but

> > what you actually mean is "pythagoreic" or "aristoxenean" or

whatever

> > the vanishing of the pythagorean comma is called.

>

>

>

> thanks.

> i decided to go with "aristoxenean" in honor of Aristoxenos.

>

> see the new Dictionary entry:

> /tuning-math/files/dict/aristox.htm

>

>

>

>

> -monz

> "all roads lead to n^0"

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

wrote:

> --- In tuning-math@y..., "monz" <monz@a...> wrote:

> > thanks, paul! i'll add it to my "linear temperaments"

> > definition when i get a chance.

> >

> > because of the tunings used in some of my favorites

> > of Herman Miller's _Pavane for a warped princess_,

> > there's a family of equal-temperaments which i've become

> > interested in lately, which all temper out the apotome,

> > {2,3}-vector [-11 7], ratio 2187:2048, ~114 cents:

> > 14-, 21-, and 28-edo.

> >

> > i noticed that these EDOs all have cardinalities which

> > are multiples of the exponent of 3 of the "vanishing comma".

> >

> > looking at the lattices on my "bingo-card-lattice" definition

> > /tuning-math/files/dict/bingo.htm

> > i can see it works the same way for 10-, 15-, and 20-edo,

> > which all temper out the _limma_, {2,3}-vector [8 -5] = ~90 cents.

> >

> >

> > so apparently, at least in these few cases (but my guess

> > is that it happens in many more), there is some relationship

> > between the logarithmic division of 2 which creates the

> > EDO and the exponent of 3 of a comma that's tempered out.

> >

> > has anyone noted this before? any further comments on it?

> > is it possible that for these two "commas" it's just

> > a coincidence?

> >

> > -monz

>

> examine the table below -- you'll note that certain commas

vanishing

> force the generator to be a fraction of an octave (600 cents, 400

> cents, 300 cents, 240 cents) instead of a full octave . . .

in fact, your "limma" example is just the blackwood temperament

below . . .

> the reason i posted this is that i wanted to see you fill out the

> list on the eqtemp page . . .

specifically, pelogic (135/128 -- 7, 9, 16, 23 -tET) and blackwood

(256/243 -- 5, 10, 15, 25 -tET) are entirely missing from the

(carl's) list, and the names are missing for negri (16875/16834 -- 9,

10, 19, 28, 29, 47, 48, 66, 67, 85, 86 -tET) and hemisixths

(78732/78125 -- 8, 11, 19, 27, 46, 65, 84 -tET).

a few of the more complex 5-limit temperaments, such as ennealimmal,

might be good to show on some of the "zooms" if you wish . . .