User:Eufalesio/Ultimate: Difference between revisions

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== Quick definition ==

Ultimate can be easily defined in the 13-limit as nullifying the [[2080/2079|sinaisma]], [[minisma]], and [[eufalesma]]. This indirectly means that the [[Symbiotic comma|salozo]], [[Nexus comma|tribilo]], [[Wilschisma|sathoyo]], [[Olympia|salururu]], [[Garischisma|sasaru]], [[Argyria|lolotrizo]] commas, among an infinitude more, are all nullified too.

Ultimate can be easily defined in the 13-limit as nullifying the [[2080/2079|sinaisma]], [[minisma]], and [[eufalesma]]. This indirectly means that the [[Symbiotic comma|salozo]], [[Nexus comma|tribilo]], [[Wilschisma|sathoyo]], [[Olympia|salururu]], [[Garischisma|sasaru]], [[Argyria|lolotrizo]] commas, among an infinitude more, are all nullified too. In the 19-limit, it also tempers out the [[1225/1224|subizoyoma]] and [[1216/1215|sanoguma]].

Ultimate has the following notable equations:

Line 13:

* 2 pomas = [[33/32]] (salururuma tempered out)

* [[36/35]] = [[1053/1024]] (minisma tempered out)

* 35/34 = 36/35 (subizoyoma tempered out)

* [[32/27]] = [[11/10]] * [[14/13]] (sinaisma tempered out)

* [[5120/5103]] = [[513/512]] (sanoguma tempered out)

et cetera...

The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma" (from this point forward refered to as "MC"); a 3~5c interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13 is notated as P1 ^'''↓'''M3 P5 '''↓'''m7 ~~M9 ↑↑11 v↑↑m13.~~ Using pomas (pythagorean commas, [↑/'''↓''']) improves this notation for reasons that will be exposed later.

The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma" (from this point forward refered to as "MC"); a 3~5c interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13:15:17:19 octave reduced is notated as:

P1 – ^'''↓'''M3 – P5 – '''↓'''m7 – M2 – ⇑4 – v⇑m6 – ^'''↓'''M7 – ^^'''⇓⇓'''M2 – ^m3

Using pomas (pythagorean commas, [↑/'''↓''']) improves this notation for reasons that will be exposed later.

Some temperament properties of Ultimate and its subsets are exposed [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]].

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==== 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 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".

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 (qualities of p7), ⇑/'''⇓''' for doubled pomas (qualities of p11), and the addition of ^/v for the MC (for qualities of p5, p13, p17, p19 aside from pomas) 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 ====

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Line 209:

=== 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.

[[Olympic|This]]. 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, ↑̵.

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.

~~{{Note|Find good extensions to the 19-limit.}}~~

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. ~~v̈⇑~~ 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''.

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''. Or, to be more crude... '''batshit''' insane.

~~{{Note|Find good extensions~~ to the ~~whatever~~ limit.}}

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

Unlike Olympic, this one is undeniably '''WAY''' more accurate, at least 100 times more accurate. It can be extended to the whole 19-limit by adding S76/S77 and S2431 to the comma list, however, a tina is required to reach prime 17 and 19, which is close, but not equal to a third of a schismina. The tina is 10241/10240. 19/16 is /◿̄m3. 17/16 is \M'''⇓⇓̱'''M2.

Look how goddamn accurate this temp is! Look, I even made a technical temp data section!

=== Batshit - the temperament or simply call it Insanic if profanity is not allowed ===

[[Subgroup]]: 2.3.5.7.11.13.17.19

[[Comma list]]: [[5767168/5767125|S64/S65]], [[11413376/11413325|S76/S77]], [[633556/633555]]

[[Mapping]]:

{| class="right-all"

|[⟨

|1

|0

|5

|8

| -9

|0

|],

|-

|⟨

|0

|1

|0

| -2

|1

|],

|-

|⟨

|0

|1

|0

| -1

| -2

|1

|],

|-

|⟨

|0

|1

|0

|2

|],

|-

|⟨

|0

| -9

| -3

|8

| -8

|]]

|}

Mapping generators: ~2, ~3, ~5, ~7, ~608/385

[[Optimal tuning|Optimal tunings]]?

: [[CWE]]: ~2 = 1200.000 ¢, ~3/2 = 701.956 ¢, ~5/4 = 386.316 ¢, ~7/4 = 968.828 ¢, ~608/385 = 791.052¢

'''Optimal''' ET sequence: {{EDOs|53,94,270,???,3395,???,8539…???}} (I have no idea what follows.)

CWE errors: 0.000, 0.001, 0.002, 0.002, 0.001, 0.002, 0.001, 0.001 (the calculator isn't very helpful here...)

Badness (Sintel): 0.279

––––––––––––

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

== 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 ~~table~~ ===

=== 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" |~~Accidental type~~

! rowspan="2" |Main accidentals

! colspan="7" |Ultimate

! colspan="4" |Beyond Ultimate

! colspan="5" |Beyond Ultimate

|-

!Natural

! colspan="2" |Apotome

! colspan="2" |Poma/''Ruma''

! colspan="2" |Poma/Ruma

! colspan="2" |Saruyoma

! colspan="2" |Sasaruma

! colspan="2" |<nowiki>Layoma | Sasaruma</nowiki>

! colspan="2" |Schismina

!Tina

|-

| rowspan="2" |Symbols

| rowspan="2" |♮

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

|♯

|{{Sagittal|#}}

|♭

|{{Sagittal|b}}

|↑

|'''↓'''

|^

| rowspan="2" |^

|v

| rowspan="2" |v

| rowspan="2" |◺

| rowspan="2" |<nowiki>◿ | ◺</nowiki>

| rowspan="2" |◸

| rowspan="2" |<nowiki>◹ | ◸</nowiki>

| rowspan="2" |Ȯ Ö Ō

| rowspan="2" |Ọ O̤ O̱

|/

|-

|𝄪

|{{Sagittal|x}}

|𝄫

|{{Sagittal|bb}}

|⇑

|'''⇓'''

|M

|\

|W

|-

| rowspan="2" |Spoken

Line 260:

Line 347:

|up

|down

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

| colspan="5" |Color notation, or some other JI phonetic coding

|-

|–

Line 269:

Line 356:

|dud

| colspan="2" |7-limit JI

| colspan="2" |11-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" |Mercator comma*

! colspan="2" |Fwiwisma

! colspan="2" |Half-apotome

|-

| rowspan="2" |Symbols

| rowspan="2" |M

| rowspan="2" |W

| rowspan="2" |¡

| rowspan="2" |!

|{{Sagittal|t}}

|{{Sagittal|d}}

|-

|{{Sagittal|t#}}

|{{Sagittal|db}}

|-

| rowspan="2" |Spoken

| rowspan="2" |mo

| rowspan="2" |mu

| rowspan="2" |halfpo

| rowspan="2" |halfqu

|halfsharp

|halfflat

|-

|sesquisharp

|sesquiflat

|}

<nowiki>*</nowiki>Not strictly necessary, but it does simplify notation immensely, specially for ultra-long chains of fifths and notating p17, which requires at least four pomas. A mercator comma follows the following equation: ⇑⇑P1 = Mm2, ⇑⇑m2 = W'''↓'''M2; 17/16 is then ^^W↑m2.

=== 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]

* ~~For an accidental: (schismina) + (sasaruma) + [saruyoma] + [poma] + [apotome] v#~~ = ~~downsharp, M'''⇓''' = dupquq~~

* D4 = dee four

* ~~For intervals: [accidental] + [5L 7s 6|5 nominal] '''↓'''M2, ^'''↓'''M3, v⇑m6; qumajor second, upqumajor third, downpopominor sixth~~

* E⇑b4 = e popoflat four

* ~~For pitches: [5L 7s 6|5 nominal] + [accidental] + [octave number] Example: D4, E⇑b4,~~ F^'''↓'''#5~~; dee four, e popoflat four,~~ ef upqusharp five

* F^'''↓'''#5 = ef upqusharp five

@@ Line 4: / Line 4: @@
 == Quick definition ==
-Ultimate can be easily defined in the 13-limit as nullifying the [[2080/2079|sinaisma]], [[minisma]], and [[eufalesma]]. This indirectly means that the [[Symbiotic comma|salozo]], [[Nexus comma|tribilo]], [[Wilschisma|sathoyo]], [[Olympia|salururu]], [[Garischisma|sasaru]], [[Argyria|lolotrizo]] commas, among an infinitude more, are all nullified too.
+Ultimate can be easily defined in the 13-limit as nullifying the [[2080/2079|sinaisma]], [[minisma]], and [[eufalesma]]. This indirectly means that the [[Symbiotic comma|salozo]], [[Nexus comma|tribilo]], [[Wilschisma|sathoyo]], [[Olympia|salururu]], [[Garischisma|sasaru]], [[Argyria|lolotrizo]] commas, among an infinitude more, are all nullified too. In the 19-limit, it also tempers out the [[1225/1224|subizoyoma]] and [[1216/1215|sanoguma]].
 Ultimate has the following notable equations:
@@ Line 13: / Line 13: @@
 * 2 pomas = [[33/32]] (salururuma tempered out)
 * [[36/35]] = [[1053/1024]] (minisma tempered out)
+* 35/34 = 36/35 (subizoyoma tempered out)
 * [[32/27]] = [[11/10]] * [[14/13]] (sinaisma tempered out)
+* [[5120/5103]] = [[513/512]] (sanoguma tempered out)
 et cetera...
-The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma" (from this point forward refered to as "MC"); a 3~5c interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13 is notated as P1 ^'''↓'''M3 P5 '''↓'''m7 M9 ↑↑11 v↑↑m13. Using pomas (pythagorean commas, [↑/'''↓''']) improves this notation for reasons that will be exposed later.
+The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma" (from this point forward refered to as "MC"); a 3~5c interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13:15:17:19 octave reduced is notated as:
+P1 – ^'''↓'''M3 – P5 – '''↓'''m7 – M2 – ⇑4 – v⇑m6 – ^'''↓'''M7 – ^^'''⇓⇓'''M2 – ^m3
+Using pomas (pythagorean commas, [↑/'''↓''']) improves this notation for reasons that will be exposed later.
 Some temperament properties of Ultimate and its subsets are exposed [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]].
@@ Line 169: / Line 175: @@
 ==== 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 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".
+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 (qualities of p7), ⇑/'''⇓''' for doubled pomas (qualities of p11), and the addition of ^/v for the MC (for qualities of p5, p13, p17, p19 aside from pomas) 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 203: / Line 209: @@
 === 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.
+[[Olympic|This]]. 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, ↑̵.
+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.
-{{Note|Find good extensions to the 19-limit.}}
+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. v̈⇑ 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''.
+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''. Or, to be more crude... '''batshit''' insane.
-{{Note|Find good extensions to the whatever limit.}}
-To recap: 12edo (apotomes), Cassandra (+pomas), Ultimate (+saruyomas), Olympic (+sasarumas), {S64/S65} (+schisminas).
+Unlike Olympic, this one is undeniably '''WAY''' more accurate, at least 100 times more accurate. It can be extended to the whole 19-limit by adding S76/S77 and S2431 to the comma list, however, a tina is required to reach prime 17 and 19, which is close, but not equal to a third of a schismina. The tina is 10241/10240. 19/16 is /◿̄m3. 17/16 is \M'''⇓⇓̱'''M2.
+Look how goddamn accurate this temp is! Look, I even made a technical temp data section!
+=== Batshit - the temperament <small><small><small><small><small>or simply call it</small></small></small></small></small> Insanic<small><small><small><small><small> if profanity is not allowed</small></small></small></small></small> ===
+[[Subgroup]]: 2.3.5.7.11.13.17.19
+[[Comma list]]: [[5767168/5767125|S64/S65]], [[11413376/11413325|S76/S77]], [[633556/633555]]
+[[Mapping]]:
+{| class="right-all"
+|[⟨
+|1
+|0
+|0
+|0
+|5
+|8
+| -9
+|0
+|],
+|-
+|⟨
+|0
+|1
+|0
+|0
+|0
+|0
+| -2
+|1
+|],
+|-
+|⟨
+|0
+|0
+|1
+|0
+|0
+| -1
+| -2
+|1
+|],
+|-
+|⟨
+|0
+|0
+|0
+|1
+|1
+|0
+|0
+|2
+|],
+|-
+|⟨
+|0
+|0
+|0
+|0
+| -9
+| -3
+|8
+| -8
+|]]
+|}
+Mapping generators: ~2, ~3, ~5, ~7, ~608/385
+[[Optimal tuning|Optimal tunings]]?
+: [[CWE]]: ~2 = 1200.000 ¢, ~3/2 = 701.956 ¢, ~5/4 = 386.316 ¢, ~7/4 = 968.828 ¢, ~608/385 = 791.052¢
+'''Optimal''' ET sequence: {{EDOs|53,94,270,???,3395,???,8539…???}} (I have no idea what follows.)
+CWE errors: 0.000, 0.001, 0.002, 0.002, 0.001, 0.002, 0.001, 0.001 (the calculator isn't very helpful here...)
+Badness (Sintel): 0.279
+––––––––––––
+To recap: 12edo (apotomes), Cassandra (+pomas), Ultimate (+saruyomas), Olympic (+sasarumas), {S64/S65} (+schisminas + tinas).
 == 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 table ===
+=== 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" |Accidental type
+! rowspan="2" |Main accidentals
 ! colspan="7" |Ultimate
-! colspan="4" |Beyond Ultimate
+! colspan="5" |Beyond Ultimate
 |-
 !Natural
 ! colspan="2" |Apotome
-! colspan="2" |Poma/''Ruma''
+! colspan="2" |Poma/Ruma
 ! colspan="2" |Saruyoma
-! colspan="2" |Sasaruma
+! colspan="2" |<nowiki>Layoma | Sasaruma</nowiki>
 ! colspan="2" |Schismina
+!Tina
 |-
 | rowspan="2" |Symbols
-| rowspan="2" |♮
+| rowspan="2" |{{Sagittal|h}}
-|♯
+|{{Sagittal|#}}
-|♭
+|{{Sagittal|b}}
 |↑
 |'''↓'''
-|^
+| rowspan="2" |^
-|v
+| rowspan="2" |v
-| rowspan="2" |◺
+| rowspan="2" |<nowiki>◿ | ◺</nowiki>
-| rowspan="2" |◸
+| rowspan="2" |<nowiki>◹ | ◸</nowiki>
 | rowspan="2" |Ȯ Ö Ō
 | rowspan="2" |Ọ O̤ O̱
+|/
 |-
-|𝄪
+|{{Sagittal|x}}
-|𝄫
+|{{Sagittal|bb}}
 |⇑
 |'''⇓'''
-|M
+|\
-|W
 |-
 | rowspan="2" |Spoken
@@ Line 260: / Line 347: @@
 |up
 |down
-| colspan="4" |Color notation, or some other JI phonetic coding
+| colspan="5" |Color notation, or some other JI phonetic coding
 |-
 |–
@@ Line 269: / Line 356: @@
 |dud
 | colspan="2" |7-limit JI
-| colspan="2" |11-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" |Mercator comma*
+! colspan="2" |Fwiwisma
+! colspan="2" |Half-apotome
+|-
+| rowspan="2" |Symbols
+| rowspan="2" |M
+| rowspan="2" |W
+| rowspan="2" |¡
+| rowspan="2" |!
+|{{Sagittal|t}}
+|{{Sagittal|d}}
+|-
+|{{Sagittal|t#}}
+|{{Sagittal|db}}
+|-
+| rowspan="2" |Spoken
+| rowspan="2" |mo
+| rowspan="2" |mu
+| rowspan="2" |halfpo
+| rowspan="2" |halfqu
+|halfsharp
+|halfflat
+|-
+|sesquisharp
+|sesquiflat
+|}
+<nowiki>*</nowiki>Not strictly necessary, but it does simplify notation immensely, specially for ultra-long chains of fifths and notating p17, which requires at least four pomas. A mercator comma follows the following equation: ⇑⇑P1 = Mm2, ⇑⇑m2 = W'''↓'''M2; 17/16 is then ^^W↑m2.
 === 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]
-* For an accidental: (schismina) + (sasaruma) + [saruyoma] + [poma] + [apotome] v# = downsharp, M'''⇓''' = dupquq
+* D4 = dee four
-* For intervals: [accidental] + [5L 7s 6|5 nominal] '''↓'''M2, ^'''↓'''M3, v⇑m6; qumajor second, upqumajor third, downpopominor sixth
+* E⇑b4 = e popoflat four
-* For pitches: [5L 7s 6|5 nominal] + [accidental] + [octave number] Example: D4, E⇑b4, F^'''↓'''#5; dee four, e popoflat four, ef upqusharp five
+* F^'''↓'''#5 = ef upqusharp five