miracle temperament

Post Reply
User avatar
herman.miller
Posts: 37
Joined: Sun Sep 06, 2015 8:27 am

miracle temperament

Post by herman.miller »

I've made a chart for the proposed miracle notation. I had to come up with an accidental for +52 and there aren't many options, so I went with :~||(: [-9, 5, -1, 0, 1> based on the sum of :/|: [-4, 4, -1> and :/|\: [-5, 1, 0, 0, 1>.

Image
Attachments
miracle-10p-31p-11limit.png
(15.33 KiB) Not downloaded yet
User avatar
Dave Keenan
Site Admin
Posts: 2180
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: miracle temperament

Post by Dave Keenan »

Thanks for that, Herman. Am I right in assuming that the vertical dimension on these charts corresponds to generator count? It's clear that horizontal is pitch.

We seem to have different approaches to creating notations for linear temperaments that are based on 7 nominals in a chain of fifths. Can you describe your procedure? I don't understand why you feel the need for accidentals for -41 and -20 generators. And what's wrong with :||): for 52 generators, as the apotome complement of :|): ?

I note that you have an unwanted duplication in a previous post where you write:
A sequence of miracle generators starting from D up to A would look like this:
D E:!!/: E:|): F:/|\: F:/|\: G:!): G:||\: A
My procedure is to ask how many generators of the linear temperament are required to produce a fifth. Then I only need enough single-shaft accidentals to distinguish that many interleaved chains of fifths, with one chain notated ... B:b: F C G D A E B #: ... or ... B:\!!/: F C G D A E B F:/||\: ...

Because these single-shaft accidentals come in up/down pairs, and one chain doesn't need any, if there are n parallel chains of fifths, I only need Floor(n/2) such up/down pairs of single-shaft symbols (for the Evo notation). If using the Revo notation, I only need those multi-shaft symbols that represent combinations of these single-shaft symbols with sharps, flats and their doubles.

Since Miracle has 6 generators to the fifth, my set of accidentals for notating open Miracle tunings is identical to that for 72edo. And in practice, there is almost no reason to use more than 72 notes in a chain of miracle generators.

That's why George and I never explicitly gave a notation for Miracle, because we thought it obvious that it would be the same as the notation for 72edo.
User avatar
Dave Keenan
Site Admin
Posts: 2180
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: miracle temperament

Post by Dave Keenan »

cmloegcmluin wrote: Sat Feb 20, 2021 9:24 am Hey... shouldn't Sagittal offer a miracle temperament notation, since George himself invented it?
It does. It's the 72edo notation. In the last paragraph on page 18 of the XH article you will find:
George and Dave wrote:While the notation of linear or higher-dimensional temperaments has not been investigated by us in great detail, it has become evident that such a temperament may be readily notated using the notation for an EDO that closely approximates it. Meantone temperament may therefore be treated as if it were a subset of 31-EDO, while the Miracle temperament may be notated like 72-EDO.

cmloegcmluin wrote: Sat Feb 20, 2021 9:24 amI see this interval naming from Dave; perhaps the symbols there could be substituted for :/|:, :(|:, and :/|\: ?
Whadya mean "perhaps they could be substituted"? They are those symbols (except you have :|): back-to-front). They are just the short Sagitype for them. :)
User avatar
cmloegcmluin
Site Admin
Posts: 1700
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer (he/him/his)
Contact:

Re: miracle temperament

Post by cmloegcmluin »

That looks awesome Herman.

This topic seems to start sort of out of nowhere. That's because when I extracted it from the one about general linear temperament notation, which in turn was broken out from the porcupine thread, I couldn't change the order of the posts to add an introductory post. So this is how things actually started:
cmloegcmluin wrote:Hey... shouldn't Sagittal offer a miracle temperament notation, since George himself invented it? I see this interval naming from Dave; perhaps the symbols there could be substituted for :/|:, :(|:, and :/|\: ? I see that miracle is a member of the gamelismic clan, and these have been previously discussed, but I don't see a straightforward notation of miracle on that topic.
After this, Herman wrote:
herman.miller wrote: Sat Feb 20, 2021 12:06 pm All you need [to define Sagittal notations for regular temperaments] is the generator mapping, which shows how the primes can be approximated by the temperament.

[<1, 2, 3, 2], <0, -3, -5, 6]> porcupine
[<1, 1, 3, 3], <0, 6, -7, -2]> miracle

With that, you can take any Sagittal represented as a monzo (e.g. [-4, 4, -1> for :/|: 81/80) and do a matrix multiplication in a spreadsheet to see which interval of the temperament this symbol represents. For porcupine it's [+1, -7> or up one period and down 7 generators; for miracle it's [-3, +31>. The tricky part is deciding which Sagittals to use. Previously I've proposed one notation for miracle temperament using these accidentals:

:/|: [-4, 4, -1> (-3, +31)
:)|(: [7, -4, 0, 1, -1> (+4, -41)
:|): [6, -2, 0, -1> (+1, -10)
:/|\: [-5, 1, 0, 0, 1> (-2, +21)
:)||(: [-3, -1, 2> (+2, -20)
:||\: [-7, 3, 1> (-1, +11)
:/||\: [-11, 7> (-4, +42)

A sequence of miracle generators starting from D up to A would look like this:
D E:!!/: E:|): F:/|\: F:/|\: G:!): G:||\: A
So that's the "proposed notation" he's referring to in the first post here.
User avatar
cmloegcmluin
Site Admin
Posts: 1700
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer (he/him/his)
Contact:

Re: miracle temperament

Post by cmloegcmluin »

Dave Keenan wrote: Fri Mar 05, 2021 12:59 pm
cmloegcmluin wrote: Sat Feb 20, 2021 9:24 am Hey... shouldn't Sagittal offer a miracle temperament notation, since George himself invented it?
It does. It's the 72edo notation. In the last paragraph on page 18 of the XH article you will find:
George and Dave wrote:While the notation of linear or higher-dimensional temperaments has not been investigated by us in great detail, it has become evident that such a temperament may be readily notated using the notation for an EDO that closely approximates it. Meantone temperament may therefore be treated as if it were a subset of 31-EDO, while the Miracle temperament may be notated like 72-EDO.
Got it. Thank you.
cmloegcmluin wrote: Sat Feb 20, 2021 9:24 amI see this interval naming from Dave; perhaps the symbols there could be substituted for :/|:, :(|:, and :/|\: ?
Whadya mean "perhaps they could be substituted"? They are those symbols (except you have :|): back-to-front). They are just the short Sagitype for them. :)
So they are. My bad. Well, my double bad, rather.
User avatar
herman.miller
Posts: 37
Joined: Sun Sep 06, 2015 8:27 am

Re: miracle temperament

Post by herman.miller »

:||): for 52 is certainly a valid option, and I don't have any strong preferences. I like your reasoning that it's the apotome complement of :|): better than my suggestion of :~||(: as the sum of :/|: and :/|\: . As for the extra symbols beyond the 72-EDO notation, I like being able to name a pitch by reference to the nearest nominal (although I make exceptions for the case of :/||\: and :\!!/:). But it depends on the needs of the music. In some cases I prefer the extended symbols with more than two shafts.
User avatar
Dave Keenan
Site Admin
Posts: 2180
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: miracle temperament

Post by Dave Keenan »

herman.miller wrote: Sat Mar 06, 2021 10:06 am As for the extra symbols beyond the 72-EDO notation, I like being able to name a pitch by reference to the nearest nominal ...
Right. That's fair enough. It's similar to the way that we could notate 31edo using only sharps and flats and their doubles, but that would required that we accept "crossed nominals" e.g. F:x: being a higher pitch than G:b:. So we include a sagittal for one degree of 31edo so we're never forced to do that. We can call F:x: G:\!/: . Although we can still cross nominals if there's a reason, e.g. due to musical context.

Thanks for explaining.
Post Reply