Notating phi temperaments (lemba, hendec, semimiracle, semivalentine and semimothra)

רועיסיני · Post by **רועיסיני** » Mon Jul 17, 2023 6:03 am

For the past few months I was wondering how I could notate the phi temperaments I talked about in this facebook post. I was putting researching it off because I thought this discussion about metalic scales had something to do with it and it linked to a Facebook discussion I didn't find the time to read... Anyway, last weekend I looked at the posts (which didn't really help me) and started figuring out how to notate these temperaments.

These are all restrictions of one rank-3 temperament, that is a bit off-topic in this subforum (and some may say that the phi temperaments themselves, having half octave periods, are off topic, but I think they are closer in spirit to this sub forum than to the "Notations for other tunings" one and if one of the mods wants he can probably move it there, as well as my discussion about the rank-3 pental temperament that also has an octave-fraction period, or just change this subforum's name to "Regulat Temperament notations"?) but I think a good starting point (and an objective in and of itself) would be to a phi chain in this tuning, with major thirds in either side of it. This tuning has 3 generators to a perfect fifth, and with a half octave and 3 chains this means we'll need floor(3 * 2 * 3 / 2) = floor(18 / 2) = 9 symbols (without the natural). This rank-3 tuning is also Keenanismic, which already suggests using

,

and $:/|\:$ which are all useful at this point. $:/|\:$ comes in a bundle with

, which, when subtracting

gives

, which is useful as well, and we've already have everything we need for the 11-limit intervals. Another thing I found useful is to use

(which usually has the 7-limit meaning of 5120/5103) as

, the 13/11k or 352/351, which can also be combined with

to get

for the 13/7 S-diesis, or 1701/1664 (which is

in Olympian).

However, here I got a bit stuck. First of all, in this tuning the Pythagorean comma is both negative and splits in half, which means we need a sagittal for the negative half pythagorean comma, or [9.5, -6⟩, which, if the golden ratio is tuned pure, is 2^12.5/ϕ¹⁸, but there is no secondary comma that maps to it. However, there are two intervals with n2d3p9 of less than 307 which map to it: 208/207 which is an alternative meaning for

and 1024/1023 which is an alternative meaning of

, so both

and

can technically be used for this interval. The next one to consider is 5ϕ/8, which has 2 sagittals mapped to it:

(144/143) which is technically fine and

(65/64) which is rejected because

is already taken. There is also

that can take its place, but we already have

and

and they sum to something else so that is rejected too, at least for now.

is also not ideal because actually this comma is greater than the alteration notated by

, and is actually the sum of it and the negative half pythagorean comma. However, these can't be notated by the same flag, since the only Prometheans that use <

> are

itself,

which is already used, $:(|\:$ which is rejected because the value is not $:/|\:$ -

+

and

which is rejected because it comes in a bundle with $:|\\:$ which doesn't get the value of 2( $:/|\:$ -

). The third hole is for 5√2/ϕ⁴, which has many secondaries that are rejected for similar reasons, except for

. However, these three cannot be used together, since the last one gives a value to the flag <

> which is equal to neither the one of the minus half pythagorean comma nor the one gotten from it and the 5ϕ/8 if we decide they get

and

respectuvely. Also, if the 5s are tuned pure, this has the value of almost 54 cents, which is more than half an apotome, so notating it with a single shaft sagittal and its complement with a double shaft seems wrong to me.

I have also run the same search for 5ϕ/8 and got 640/637, 78/77, 120/119, 135/133, 190/189, 85/84, 144/143, 324/319, 2025/1984, 145/144, 65/64, 189/187, 1701/1672, 133/132, 357/352, 105/104, 945/928 and 217/216, and for 5√2/ϕ⁴ and got 12160/11907, 1360/1323, 1024/1001, 580/567, 65/63, 192/187, 216/209, 304/297, 34/33, 40/39, 30/29, 248/243, 91/88, 35/34, 315/304, 665/648, 595/576, 147/143, 3185/3072, and I guess all of which can be used for these intervals if deemed necessary.

This is a WIP and I just tried to summarize all I have found in this post, but it's a bit late here now so I'll hopefully go back to it tomorrow. In the meantime I'll be very glad to receive help if anyone else is also interested, by searching directions I rejected (or didn't even considered), continuing from here or notating some of the rank-2 tunings with the 6 sagittals I've already established (+tentative placeholders for the 3 others?), and maybe some that are mapped to a chain of fifths a third above and below what I considered, like

.

רועיסיני · Post by **רועיסיני** » Tue Jul 18, 2023 3:29 am

I think the best accidental for 5√2/ϕ⁴ is $:|\\:$ , justified by 3185/3072 (n2d3p9 = 287.5347222) which falls in its Promethean capture zone. That's because I didn't want to cause a size reversal between it and its complement, and this symbol seemed better than any other valid one > $:)/|\:$ from the flag arithmetic point of view, since $:|\:$ is not used on its own and its complement includes <

> which does not appear in any other symbol I chose. I'm also fine, for the moment, with

(with its primary meaning) for 5ϕ/8, since it's not the worst size reversal in its area, that title going to

>

. Also, I think

(justified by 1024/1023) is the better symbol for 4096√2/ϕ¹⁸ (the negative half pythagorean comma), since this is the smallest alteration that is given an accidental in this tuning. So, for now, the symbols, ordered by their size if 5 is pure, are

	~1024/1023
	~81/80
	~45927/45056
	~352/351
	~144/143
	~1701/1664
	~64/63
	~6561/6370
	~729/704
	~33/32
	~3185/3072

If anyone has better suggestions, especially regarding the size reversals between

and the two symbols after it, I'd love to hear them. Also, the order if 11 is pure instead, if anyone is interested, is

$:|\\:$ $:/|\:$ .

Anyway, even if we remove the 3 problematic symbols, this already gives us a notation for phi stacks, by using just the sagittals

,

and

. This way, a MOS of 13 golden ratios, for example, would be

רועיסיני · Post by **רועיסיני** » Fri Jul 28, 2023 10:41 pm

Now that we have the symbols for the rank-3 temperament let's try to notate the rank-2 ones:
Lemba: This is the simplest of the bunch, having phi approximate 8/5 (as well as all higher Fibonacci ratios like the other temperaments), so I think we should use

for ϕ¹³/2⁹, or 13 generators,

, like in the rank-3 temperament, for 2^3.5/ϕ⁵, or -5 generators, and

, again like in the tank-3 temperament, for 18 generators. Another option, since lemba is special and starts one Fibonacci step below the rest, is to make an exception for it and allow

to be used, as 5120/5103, instead.
Semimothra: I think it should use

,

and

. The first two are for the exact same meaning as in the rank-3 temperament, and the last one for the negative half Pythagorean comma, which is one 5-comma less than its meaning in the rank-3 temperament, which is fine because the 5-comma is tempered out in semimothra. This also suggests the use of

to notate 5 generators of mothra.
Semimiracle:

,

and $:/|\:$ are easy choices since they are already used in miracle,

and

are kind of forced because they are both the only commas from the ones considered that are fit for their roles, and

and $:|\\:$ , being apotome complements with a difference of a Pythagorean comma between them, must be included as well. For the last symbol we have a choice between

and

where I think

should be preferred because its value already follows from flag arithmetic.
Hendec: Similarly to semimiracle,

,

and

are forced. Then we have 3 choices, all between 11 or lower limit commas and higher limit commas, which means

, $:/|\:$ and

are taken as well.
Semivalentine: All the symbols notate different alterations and so are all needed. This also suggests using

,

and $:/|\:$ to notate valentine.

To summarize:
Lemba:
Generator: ϕ
The equivalence lattice L is spanned by 2 ~ (0, 2) – no golden ratios and 2 half-octaves, and 3 ~ (3, -1).

	(-18, 25) ≡  (3, 0) mod L
	(-18, 25), alternative
	(13, -18) ≡  (1, 0) mod L
	(-5, 7)   ≡ (-2, 0) mod L

Semimothra:
Generator: ϕ
The equivalence lattice L is the same.

	(-18, 25) ≡  (3, 0) mod L
	(13, -18) ≡  (1, 0) mod L
	(-5, 7)   ≡ (-2, 0) mod L

Semimiracle:
Generator: a secor, tuned as √(ϕ/√2)
L is spanned by 2 ~ (0, 2) and 3/2 ~ (6, 0), or 6 secors.

	(-36, 7)  ≡ (0, 1) mod L
	(31, -6)  ≡ (1, 0) mod L
	(-5, 1)   ≡ (1, 1) mod L
	(26, -5)  ≡ (2, 1) mod L
	(-10, 2)  ≡ (2, 0) mod L
	(57, -11) ≡ (3, 1) mod L
	(21, -4)  ≡ (3, 0) mod L

Hendec:
Generator: √ϕ
L is spanned by 2 ~ (0, 2) and 3 ~ (6, -1)

	(-36, 25) ≡  (6, 0) mod L
	(49, -34) ≡  (1, 0) mod L
	(13, -9)  ≡ (-5, 0) mod L
	(26, -18) ≡  (2, 0) mod L
	(-10, 7)  ≡ (-4, 0) mod L
	(3, -2)   ≡  (3, 0) mod L

Semivalentine
Generator: ³√(ϕ/√2)
L is spanned by 2 ~ (0, 2) and 3/2 ~ (9, 0)

	(-54, 7) ≡  (0, 1) mod L
	(31, -4) ≡  (4, 0) mod L
	(-23, 3) ≡  (4, 1) mod L
	(62, -8) ≡ (-1, 0) mod L
	(8, -1)  ≡ (-1, 1) mod L
	(39, -5) ≡  (3, 1) mod L
	(-15, 2) ≡  (3, 0) mod L
	(70, -9) ≡ (-2, 1) mod L
	(16, -2) ≡ (-2, 0) mod L

The Sagittal forum

Notating phi temperaments (lemba, hendec, semimiracle, semivalentine and semimothra)

Notating phi temperaments (lemba, hendec, semimiracle, semivalentine and semimothra)

Re: Notating phi temperaments (lemba, hendec, semimiracle, semivalentine and semimothra)

Re: Notating phi temperaments (lemba, hendec, semimiracle, semivalentine and semimothra)