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

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 . . .