User:Eufalesio/Ultimate: Difference between revisions

(6 intermediate revisions by the same user not shown)

Line 1:

is 41&53&217, with mapping {{Mapping|1 0 0 25 -33 -13|0 1 0 -14 23 12|0 0 1 0 0 -1}}. It's otherwise known by in the wiki as ''[[cassaschismic]]'' (~~technical info inside~~), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]]; but we will simply call it '''Ultimate'''. Our reasoning of this will become clear. Or at least, we expect you to understand why it's clear in our mind.

iis 41&53&217, with mapping {{Mapping|1 0 0 25 -33 -13|0 1 0 -14 23 12|0 0 1 0 0 -1}}. It's otherwise known by in the wiki as ''[[cassaschismic]]'' (tad less rambly), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]]; but we will simply call it '''Ultimate'''. Our reasoning of this will become clear. Or at least, we expect you to understand why it's clear in our mind.

Special thanks for [[Kite Giedraitis]] for feedback and edits.

Line 32:

|-

!0

|~~720~~/~~361~~

|351/176

|1/1

|'''1/1'''

|-

!1

Line 49:

!4

|24/19

|81/64

|19/15

|-

!5

|36/19

|~~243~~/~~128~~

|19/10

|-

!6

|64/45

|27/19

|~~729~~/~~512~~

|57/40

|-

!7

Line 84:

|-

!13

|~~243~~/~~160~~

|38/25

|32/21

|'''32/21'''

|-

!14

|~~729~~/~~640~~

|57/50

|'''8/7'''

|-

!15

|~~416~~/~~243~~

|77/45

|12/7

|-

!16

|~~104~~/81

|77/60

|9/7

|-

Line 157:

==== 12e ====

The coarsest tuning that makes sense. It can be written just with ~~sharps and flats~~, since the poma and the MC are tempered out in all its possible expressions. 12e because patent val tunes 11/8 as a tritone, not fourth. The cassandra mapping is based on 11/8 as a kind of fourth, not tritone. Either way, p11 is NOT there. Consider it an extremely coarse [[yazatha]] tuning. 24edo '''can't''' be used, as it uses a new mapping disjoint from the circle of fitfths and the poma is still tempered out.

The coarsest tuning that makes sense. It can be written just with apotomes, since the poma and the MC are tempered out in all its possible expressions. 12e because patent val tunes 11/8 as a tritone, not fourth. The cassandra mapping is based on 11/8 as a kind of fourth, not tritone. Either way, p11 is NOT there. Consider it an extremely coarse [[yazatha]] tuning. 24edo '''can't''' be used, as it uses a new mapping disjoint from the circle of fitfths and the poma is still tempered out.

=== Cassandra edos ===

They are the simplest of the bunch and the easiest to work with. They can be written with ~~sharps and flats~~, ↑/'''↓''' for the pomas reaching qualites of p5 and p7, and ⇑/'''⇓''' for doubled pomas reaching qualities of p11 and p13.

They are the simplest of the bunch and the easiest to work with. They can be written with apotomes, ↑/'''↓''' for the pomas reaching qualites of p5 and p7, and ⇑/'''⇓''' for doubled pomas reaching qualities of p11 and p13.

==== 41 ====

Line 169:

==== 94 ====

Best cassandra tuning. Since the chain takes much longer to close, ¡ and ! may be used to raise or lower by a ~~half-poma.~~ (~~Note that~~ a half-poma ~~implies~~ a ~~half-octave,~~ and thus an even-numbered edo). This tuning is optimal and technically usable in the Lumatone, but only as a subset, requiring more than one preset to reach within the [[Standard Lumatone mapping for Pythagorean]]. The cassandra layout can be used in a guitar with the [[skip fretting system 94 7 16]]. However, in a 6-string guitar there will be no other unisons.

Best cassandra tuning. Since the chain takes much longer to close, ¡ and ! may be used to raise or lower by a [[2835/2816|fwiwisma]] (a half-poma, as a result of tempering out kalisma and thus an even-numbered edo). This tuning is optimal and technically usable in the Lumatone, but only as a subset, requiring more than one preset to reach within the [[Standard Lumatone mapping for Pythagorean]]. The cassandra layout can be used in a guitar with the [[skip fretting system 94 7 16]]. However, in a 6-string guitar there will be no other unisons.

Note that 94edo is already quite fine for most real instruments, and though its step is very much discernible, human error can begin to slip pitches into the wrong edostep even in very skilled musicians.

=== Non-cassandra Ultimate ===

They are very fine and likely impossible to implement into real instruments with an Ultimate layout. They can be written with ~~sharps and flats~~, ↑/'''↓''' for the pomas, ⇑/'''⇓''' for doubled pomas, and the addition of ^/v for the MC, taken directly from the [[Kite's ups and downs notation|ups-and-downs notation.]] This is completely unfeasible to use with a Lumatone or with any acoustic instrument. Though, it can still be used in a DAW without much problem. Because Ultimate is rank-3, the layout is 3D and thus it is impossible to play on a flat surface, requiring some sort of eldritch holographic "keyspace".

They are very fine and likely impossible to implement into real instruments with an Ultimate layout. They can be written with apotomes, ↑/'''↓''' for the pomas, ⇑/'''⇓''' for doubled pomas, and the addition of ^/v for the MC, taken directly from the [[Kite's ups and downs notation|ups-and-downs notation.]] This is completely unfeasible to use with a Lumatone or with any acoustic instrument. Though, it can still be used in a DAW without much problem. Because Ultimate is rank-3, the layout is 3D and thus it is impossible to play on a flat surface, requiring some sort of eldritch holographic "keyspace".

==== 217 ====

Line 198:

94edo and 270edo have the key property of being even, tempering out the [[kalisma]] and allowing the poma to be halved. Using them this way is reminiscent of [[Gariwizmic]], a very similar subset of Ultimate, but with the MC found deep in the generator chain, not independent. This is useful for easier navigation within a DAW. It's possible to use Gariwizmic wholesale, though it only slightly improves 270edo in precision, Newt is a much better choice for accuracy's sake, though 94edo does ''not'' support it. Gariwizmic provides structure, not the tuning.

== Ultimate still not enough? ==

There's a reason why I deem Ultimate ultimate. You're supposed to end there and go no further, because Ultimate is right at the limit of practicality. If you're stubborn enough to ignore my warnings and venture into the land of impossible... I still know how you can continue following the same path of reasoning, and add extra pairs of accidentals that are fully retrocompatible.

=== Olympic ===

The best course of action for detempering Ultimate is to observe the garischisma, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma into a schisma (◿/◹) and a garischisma (◺/◸). Unlike in JI, where the schisma is around half the garischisma, here the schisma is ~1.6x LARGER, not smaller.

The simplest choice is to make ↑ 64/63; the poma is ◸↑ now, or if you want to be sleek, '''p'''.

Everything else is the same. v↑ for 81/80, ⇑ for 33/32, v⇑ for 1053/1024. It's something I don't think I'll ever see myself doing because this accuracy is enough to represent highly fine edos such as 494, 764, or 935edo, which is already too much for me.

Extensions to the yazalathana are bad: the strong ones keep either of 1540/1539 or 1729/1728, and a decent but weak one adding 5776/5775 needs you to split the septimal comma in halves, which is not ideal. Acknowledgements to [[Flora Canou]] for this insight. In short: Olympic kinda sucks: it is better, but not ''way'' better than Ultimate, which for a new dimension to worry about is not justifiable; unlike the jump from cassandra to Ultimate being a quantum leap.

=== Insanic ===

Olympic STILL not enough? Split your losses and use [[5767168/5767125|{S64/S65}]]. Now you observe the olympia and get a schismina accidental. An olympia is 3 of these schisminas. You could write this as dots above or below the accidentals but this is possibly getting a tad crowded. ↑ is 64/63 and 81/80 is v↑. ⇑̱ is 33/32 whilst ⇑ is 4096/3969. v̇⇑ is 1053/1024 whilst v⇑ is 36/35. {S64/S65} is at a level of precision comparable to 8539edo and much finer. The people at sagittal.org had already declared its own version of this notation to be of "Insane" precision, so if you need anything finer, you are ''beyond insane''.

Unlike Olympic, this one is undeniably '''WAY''' more accurate. Supposedly, It can be extended to the no-17 19-limit by doing {S64/S65, [[228096/228095|S96/S98]]} and p17 can be added by tempering out S324/S351. I'll leave the question of how to write 17/16 and 19/16 as an excercise left to the reader to figure out, because I know it's going to be extremely contrived. My notation is not going to serve you well here.

To recap: 12edo (apotomes), Cassandra (+pomas), Ultimate (+saruyomas), Olympic (+sasarumas), {S64/S65} (+schisminas).

== Nomenclature and notation ==

This notation can be easily spoken as well as written, adapting Kite's color notation and ups and downs into a nice collage. At least, the part of Ultimate. What's beyond Ultimate, not my problem.

=== Easy tables ===

These are all the accidentals you need to know to write in Ultimate, and even beyond it. For most cases, there is no need to go beyond two of anything, in the case of pomas however, you can end up using three or even more if you don't respell enharmonically (if possible), or go too far down the chain of fifths. One instance is writing 5/4 above a 16/11, which is 20/11. Above a C, this is '''⇓'''G - ^'''↓⇓'''B.

{| class="wikitable"

|+

! rowspan="2" |Main accidentals

! colspan="7" |Ultimate

! colspan="4" |Beyond Ultimate

|-

!Natural

! colspan="2" |Apotome

! colspan="2" |Poma/Ruma

! colspan="2" |Saruyoma

! colspan="2" |Sasaruma

! colspan="2" |Schismina

|-

| rowspan="2" |Symbols

| rowspan="2" |{{Sagittal|h}}

|{{Sagittal|#}}

|{{Sagittal|b}}

|↑

|'''↓'''

|^

|v

| rowspan="2" |◺

| rowspan="2" |◸

| rowspan="2" |Ȯ Ö Ō

| rowspan="2" |Ọ O̤ O̱

|-

|{{Sagittal|x}}

|{{Sagittal|bb}}

|⇑

|'''⇓'''

|M

|W

|-

| rowspan="2" |Spoken

| rowspan="2" |natural

|sharp

|flat

|po

|qu

|up

|down

| colspan="4" |Color notation, or some other JI phonetic coding

|-

|–

|pop(o)

|quq(u)

|dup

|dud

| colspan="2" |7-limit JI

| colspan="2" |11-limit JI

|}

The following accidentals are not part of the main bunch because they are either only part of rank-2s and edos or ligatures.

{| class="wikitable"

!Extra accidentals

! colspan="2" |Fwiwisma

! colspan="2" |Half-apotome

! colspan="2" |Non-gary poma*

|-

| rowspan="2" |Symbols

| rowspan="2" |¡

| rowspan="2" |!

|{{Sagittal|t}}

|{{Sagittal|d}}

|p

|q

|-

|{{Sagittal|t#}}

|{{Sagittal|db}}

|pp

|qq

|-

| rowspan="2" |Spoken

| rowspan="2" |halfpo

| rowspan="2" |halfqu

|halfsharp

|halfflat

|po

|qu

|-

|sesquisharp

|sesquiflat

|pop(o)

|quq(u)

|}

<nowiki>*</nowiki>Not necessary in Ultimate. This can prevent filling the page with garischismas. Logically, here the arrow is spoken ru/zo like in color notation, while the p/q is po/qu.

=== Syntax ===

For an accidental: (schismina) + (sasaruma) + [saruyoma] + [poma] + [apotome]

* v# = downsharp

* M'''⇓''' = dupquq

For intervals: [accidental] + [5L 7s 6|5 nominal]

* '''↓'''M2 = qumajor second

* ^'''↓'''M3 = upqumajor third

* v⇑m6 = downpopominor sixth

For pitches: [5L 7s 6|5 nominal] + [accidental] + [octave number]

* D4 = dee four

* E⇑b4 = e popoflat four

* F^'''↓'''#5 = ef upqusharp five

@@ Line 1: / Line 1: @@
-is 41&53&217, with mapping {{Mapping|1 0 0 25 -33 -13|0 1 0 -14 23 12|0 0 1 0 0 -1}}. It's otherwise known by in the wiki as ''[[cassaschismic]]'' (technical info inside), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]]; but we will simply call it '''Ultimate'''. Our reasoning of this will become clear. Or at least, we expect you to understand why it's clear in our mind.
+iis 41&53&217, with mapping {{Mapping|1 0 0 25 -33 -13|0 1 0 -14 23 12|0 0 1 0 0 -1}}. It's otherwise known by in the wiki as ''[[cassaschismic]]'' (tad less rambly), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]]; but we will simply call it '''Ultimate'''. Our reasoning of this will become clear. Or at least, we expect you to understand why it's clear in our mind.
 Special thanks for [[Kite Giedraitis]] for feedback and edits.
@@ Line 32: / Line 32: @@
 |-
 !0
-|720/361
+|351/176
-|1/1
+|'''1/1'''
 |-
 !1
@@ Line 49: / Line 49: @@
 !4
 |24/19
-|81/64
+|19/15
 |-
 !5
 |36/19
-|243/128
+|19/10
 |-
 !6
-|64/45
+|27/19
-|729/512
+|57/40
 |-
 !7
@@ Line 84: / Line 84: @@
 |-
 !13
-|243/160
+|38/25
-|32/21
+|'''32/21'''
 |-
 !14
-|729/640
+|57/50
 |'''8/7'''
 |-
 !15
-|416/243
+|77/45
 |12/7
 |-
 !16
-|104/81
+|77/60
 |9/7
 |-
@@ Line 157: / Line 157: @@
 ==== 12e ====
-The coarsest tuning that makes sense. It can be written just with sharps and flats, since the poma and the MC are tempered out in all its possible expressions. 12e because patent val tunes 11/8 as a tritone, not fourth. The cassandra mapping is based on 11/8 as a kind of fourth, not tritone. Either way, p11 is NOT there. Consider it an extremely coarse [[yazatha]] tuning. 24edo '''can't''' be used, as it uses a new mapping disjoint from the circle of fitfths and the poma is still tempered out.
+The coarsest tuning that makes sense. It can be written just with apotomes, since the poma and the MC are tempered out in all its possible expressions. 12e because patent val tunes 11/8 as a tritone, not fourth. The cassandra mapping is based on 11/8 as a kind of fourth, not tritone. Either way, p11 is NOT there. Consider it an extremely coarse [[yazatha]] tuning. 24edo '''can't''' be used, as it uses a new mapping disjoint from the circle of fitfths and the poma is still tempered out.
 === Cassandra edos ===
-They are the simplest of the bunch and the easiest to work with. They can be written with sharps and flats, ↑/'''↓''' for the pomas reaching qualites of p5 and p7, and ⇑/'''⇓''' for doubled pomas reaching qualities of p11 and p13.
+They are the simplest of the bunch and the easiest to work with. They can be written with apotomes, ↑/'''↓''' for the pomas reaching qualites of p5 and p7, and ⇑/'''⇓''' for doubled pomas reaching qualities of p11 and p13.
 ==== 41 ====
@@ Line 169: / Line 169: @@
 ==== 94 ====
-Best cassandra tuning. Since the chain takes much longer to close, ¡ and ! may be used to raise or lower by a half-poma. (Note that a half-poma implies a half-octave, and thus an even-numbered edo). This tuning is optimal and technically usable in the Lumatone, but only as a subset, requiring more than one preset to reach within the [[Standard Lumatone mapping for Pythagorean]]. The cassandra layout can be used in a guitar with the [[skip fretting system 94 7 16]]. However, in a 6-string guitar there will be no other unisons.
+Best cassandra tuning. Since the chain takes much longer to close, ¡ and ! may be used to raise or lower by a [[2835/2816|fwiwisma]] (a half-poma, as a result of tempering out kalisma and thus an even-numbered edo). This tuning is optimal and technically usable in the Lumatone, but only as a subset, requiring more than one preset to reach within the [[Standard Lumatone mapping for Pythagorean]]. The cassandra layout can be used in a guitar with the [[skip fretting system 94 7 16]]. However, in a 6-string guitar there will be no other unisons.
 Note that 94edo is already quite fine for most real instruments, and though its step is very much discernible, human error can begin to slip pitches into the wrong edostep even in very skilled musicians.
 === Non-cassandra Ultimate ===
-They are very fine and likely impossible to implement into real instruments with an Ultimate layout. They can be written with sharps and flats, ↑/'''↓''' for the pomas, ⇑/'''⇓''' for doubled pomas, and the addition of ^/v for the MC, taken directly from the [[Kite's ups and downs notation|ups-and-downs notation.]] This is completely unfeasible to use with a Lumatone or with any acoustic instrument. Though, it can still be used in a DAW without much problem. Because Ultimate is rank-3, the layout is 3D and thus it is impossible to play on a flat surface, requiring some sort of eldritch holographic "keyspace".
+They are very fine and likely impossible to implement into real instruments with an Ultimate layout. They can be written with apotomes, ↑/'''↓''' for the pomas, ⇑/'''⇓''' for doubled pomas, and the addition of ^/v for the MC, taken directly from the [[Kite's ups and downs notation|ups-and-downs notation.]] This is completely unfeasible to use with a Lumatone or with any acoustic instrument. Though, it can still be used in a DAW without much problem. Because Ultimate is rank-3, the layout is 3D and thus it is impossible to play on a flat surface, requiring some sort of eldritch holographic "keyspace".
 ==== 217 ====
@@ Line 198: / Line 198: @@
 edo and 270edo have the key property of being even, tempering out the [[kalisma]] and allowing the poma to be halved. Using them this way is reminiscent of [[Gariwizmic]], a very similar subset of Ultimate, but with the MC found deep in the generator chain, not independent. This is useful for easier navigation within a DAW. It's possible to use Gariwizmic wholesale, though it only slightly improves 270edo in precision, Newt is a much better choice for accuracy's sake, though 94edo does ''not'' support it. Gariwizmic provides structure, not the tuning.
+== Ultimate still not enough? ==
+There's a reason why I deem Ultimate ultimate. You're supposed to end there and go no further, because Ultimate is right at the limit of practicality. If you're stubborn enough to ignore my warnings and venture into the land of impossible... I still know how you can continue following the same path of reasoning, and add extra pairs of accidentals that are fully retrocompatible.
+=== Olympic ===
+The best course of action for detempering Ultimate is to observe the garischisma, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma into a schisma (◿/◹) and a garischisma (◺/◸). Unlike in JI, where the schisma is around half the garischisma, here the schisma is ~1.6x LARGER, not smaller.
+The simplest choice is to make ↑ 64/63; the poma is ◸↑ now, or if you want to be sleek, '''p'''.
+Everything else is the same. v↑ for 81/80, ⇑ for 33/32, v⇑ for 1053/1024. It's something I don't think I'll ever see myself doing because this accuracy is enough to represent highly fine edos such as 494, 764, or 935edo, which is already too much for me.
+Extensions to the yazalathana are bad: the strong ones keep either of 1540/1539 or 1729/1728, and a decent but weak one adding 5776/5775 needs you to split the septimal comma in halves, which is not ideal. Acknowledgements to [[Flora Canou]] for this insight. In short: Olympic kinda sucks: it is better, but not ''way'' better than Ultimate, which for a new dimension to worry about is not justifiable; unlike the jump from cassandra to Ultimate being a quantum leap.
+=== Insanic ===
+Olympic STILL not enough? Split your losses and use [[5767168/5767125|{S64/S65}]]. Now you observe the olympia and get a schismina accidental. An olympia is 3 of these schisminas. You could write this as dots above or below the accidentals but this is possibly getting a tad crowded. ↑ is 64/63 and 81/80 is v↑. ⇑̱ is 33/32 whilst ⇑ is 4096/3969. v̇⇑ is 1053/1024 whilst v⇑ is 36/35. {S64/S65} is at a level of precision comparable to 8539edo and much finer. The people at sagittal.org had already declared its own version of this notation to be of "Insane" precision, so if you need anything finer, you are ''beyond insane''.
+Unlike Olympic, this one is undeniably '''WAY''' more accurate. Supposedly, It can be extended to the no-17 19-limit by doing {S64/S65, [[228096/228095|S96/S98]]} and p17 can be added by tempering out S324/S351. I'll leave the question of how to write 17/16 and 19/16 as an excercise left to the reader to figure out, because I know it's going to be extremely contrived. My notation is not going to serve you well here.
+To recap: 12edo (apotomes), Cassandra (+pomas), Ultimate (+saruyomas), Olympic (+sasarumas), {S64/S65} (+schisminas).
+== Nomenclature and notation ==
+This notation can be easily spoken as well as written, adapting Kite's color notation and ups and downs into a nice collage. At least, the part of Ultimate. What's beyond Ultimate, not my problem.
+=== Easy tables ===
+These are all the accidentals you need to know to write in Ultimate, and even beyond it. For most cases, there is no need to go beyond two of anything, in the case of pomas however, you can end up using three or even more if you don't respell enharmonically (if possible), or go too far down the chain of fifths. One instance is writing 5/4 above a 16/11, which is 20/11. Above a C, this is '''⇓'''G - ^'''↓⇓'''B.
+{| class="wikitable"
+|+
+! rowspan="2" |Main accidentals
+! colspan="7" |Ultimate
+! colspan="4" |Beyond Ultimate
+|-
+!Natural
+! colspan="2" |Apotome
+! colspan="2" |Poma/Ruma
+! colspan="2" |Saruyoma
+! colspan="2" |Sasaruma
+! colspan="2" |Schismina
+|-
+| rowspan="2" |Symbols
+| rowspan="2" |{{Sagittal|h}}
+|{{Sagittal|#}}
+|{{Sagittal|b}}
+|↑
+|'''↓'''
+|^
+|v
+| rowspan="2" |◺
+| rowspan="2" |◸
+| rowspan="2" |Ȯ Ö Ō
+| rowspan="2" |Ọ O̤ O̱
+|-
+|{{Sagittal|x}}
+|{{Sagittal|bb}}
+|⇑
+|'''⇓'''
+|M
+|W
+|-
+| rowspan="2" |Spoken
+| rowspan="2" |natural
+|sharp
+|flat
+|po
+|qu
+|up
+|down
+| colspan="4" |Color notation, or some other JI phonetic coding
+|-
+|–
+|–
+|pop(o)
+|quq(u)
+|dup
+|dud
+| colspan="2" |7-limit JI
+| colspan="2" |11-limit JI
+|}
+The following accidentals are not part of the main bunch because they are either only part of rank-2s and edos or ligatures.
+{| class="wikitable"
+!Extra accidentals
+! colspan="2" |Fwiwisma
+! colspan="2" |Half-apotome
+! colspan="2" |Non-gary poma*
+|-
+| rowspan="2" |Symbols
+| rowspan="2" |¡
+| rowspan="2" |!
+|{{Sagittal|t}}
+|{{Sagittal|d}}
+|p
+|q
+|-
+|{{Sagittal|t#}}
+|{{Sagittal|db}}
+|pp
+|qq
+|-
+| rowspan="2" |Spoken
+| rowspan="2" |halfpo
+| rowspan="2" |halfqu
+|halfsharp
+|halfflat
+|po
+|qu
+|-
+|sesquisharp
+|sesquiflat
+|pop(o)
+|quq(u)
+|}
+<nowiki>*</nowiki>Not necessary in Ultimate. This can prevent filling the page with garischismas. Logically, here the arrow is spoken ru/zo like in color notation, while the p/q is po/qu.
+=== Syntax ===
+For an accidental: (schismina) + (sasaruma) + [saruyoma] + [poma] + [apotome]
+* v# = downsharp
+* M'''⇓''' = dupquq
+For intervals: [accidental] + [5L 7s 6|5 nominal]
+* '''↓'''M2 = qumajor second
+* ^'''↓'''M3 = upqumajor third
+* v⇑m6 = downpopominor sixth
+For pitches: [5L 7s 6|5 nominal] + [accidental] + [octave number]
+* D4 = dee four
+* E⇑b4 = e popoflat four
+* F^'''↓'''#5 = ef upqusharp five