User:Eufalesio/Ultimate: Difference between revisions

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The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma" (hereafter noted as MC); a ~4{{C}} 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

P1 – ^'''↓'''M3 – P5 – '''↓'''m7 – M2 – ⇈4 – v⇈m6 – ^'''↓'''M7 – ^^⇊⇊M2 / 4^m2* – ^m3

Using pomas (pythagorean commas, [↑/'''↓''']) improves [[User:Eufalesio/Ultimate#Nomenclature and notation|the notation]] for reasons that will be exposed later.

<nowiki>*</nowiki>See [[#17/16 conundrum]].

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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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 good structure, not necessarily the best tuning.

== 17/16 conundrum ==

The yazalathana is trivial, easy, very well tuned and well behaved in Ultimate, cassandra, and 12edo. Filling the 17 gap is harder.

Prime 17, while obvious in 12edo, is complex in cassandra and not nice in Ultimate. In Ultimate there are two valid extensions to reach 17/16, of which "septinoellismic" - so called because it tempers out 1225/1224 - is deemed to be the one with lowest badness. Sure. That doesn't mean it's the best. It's not ideal because 17/16 detempers from a major second (^^⇊⇊M2), which means that it is very far away in a keyboard.

The other one, called "septisperasmic" - so called because it tempers out 2500/2499 - does detemper from a minor second (4^m2), however, it requires minicommas and thus this cannot be used in cassandra as m2 is wayy too flat - though it can be used in 12edo. It is a more accurate and intuitive mapping with less distance, but it now requires the third generator to have a larger span. Not that septinoellismic didn't do that, but it was still a mapping that could be found within the 21-odd-limit (most span is 26/25 and 25/13, -/+3 minicommas).

== Beyond Ultimate - the ULTRAOLYMPIC sequence ==

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=== Olympic - Metaolympic ===

[[Olympic|This]]. The best course of action for detempering Ultimate is to observe the sasaruma, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma in halves with no meaningful notation changes~~. Unlike in JI, where the layoma is around half the sasaruma, here~~ the ~~layoma is ~1.6x LARGER, not smaller. A heavily inflated [[10241/10240|tina (~10241/10240)]] is one~~ of ~~these sasaruma halves; it and distinct layomas are required to reach primes 17 and 19~~.

[[Olympic|This]]. The best course of action for detempering Ultimate is to observe the sasaruma, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma in halves with basically no meaningful notation changes until higher primes come into play or you modulate far enough along the chain of fifths.

~~Olympic decouples 64/63 from~~ the ~~chain of fifths; 64/63~~ is ~~now its own thing~~, ~~and~~ the ~~poma~~ is \|↑. (\| is a sasaruma ~~down)~~

Unlike in JI, where the layoma is around half the sasaruma, here the layoma is ~1.6x LARGER, not smaller. A heavily inflated [[10241/10240|tina (~10241/10240)]] is one of these sasaruma halves; it and distinct layomas are required to reach primes 17 and 19.

{{Note|Olympic and beyond require detempering from septinoellismic - the least-badness mapping, for good or for worse.}}

Everything else is the same:

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=== 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 with saquarumas, or go too far down the chain of fifths.

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 more if you don't respell enharmonically with saquarumas, or go too far down the chain of fifths, which can lead to things being technically correct but pragmatically confusing.

One instance is ~~writing 5~~/4 ~~above~~ a 16/11, which is ~~20/11. Above~~ a C, ~~this~~ is '''⇊'''G - ^'''↓'''⇊B.

One instance is the famous 14/11 - often deemed a major third. It isn't here. From C, it is '''↓'''⇊F. This is because 7/4 is a kind of minor seventh, 11/8 is a fourth. Minor seventh-fourth = fourth. Subtract the commas independently: '''↓ - (⇈) = ↓'''⇊. Similar thing occurs with 11/10, which is a v↑'''⇈'''m2, 14/13 which is a ^'''↓'''⇊M2. It all makes complete and logical sense from the perspective of the temperament,

But it feels wrong. That's why we have saquarumas. 14/13 sounds minor-second-y. 14/11 sounds major-third-y. With saquarumas you can spell 11/10 as a major second ('''↓'''ΓM2), 14/11 as a major third (↑LM3).

A phonetic coding proposal and the alterations for written text are:

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|}

<nowiki>*</nowiki>Not strictly necessary, but it does simplify notation immensely. The saquaruma equals the mercator comma in Ultimate (also being tempered out in 53edo), but not beyond it. It is useful for ultra-long chains of fifths and notating p17, which requires at least four rumas. It follows the following equation: ⇈⇈P1 = Lm2, ⇈⇈m2 = L'''↓'''M2; 17/16 is then ^^L↑m2 in Ultimate; or 𐕣|\^L↑̱m2 in Batshittin.

Note that calling ^^L↑m2

It is also unnecessary in 12edo and 41edo, since it equals a flat and a poma up respectively in those systems, and it is a way to skip the bad 19-limit mappings of 12eg and 41g.

@@ Line 21: / Line 21: @@
 The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma" (hereafter noted as MC); a ~4{{C}} 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
+P1 – ^'''↓'''M3 – P5 – '''↓'''m7 – M2 – ⇈4 – v⇈m6 – ^'''↓'''M7 – ^^⇊⇊M2 / 4^m2* – ^m3
 Using pomas (pythagorean commas, [↑/'''↓''']) improves [[User:Eufalesio/Ultimate#Nomenclature and notation|the notation]] for reasons that will be exposed later.
+<nowiki>*</nowiki>See [[#17/16 conundrum]].
 Some temperament properties of Ultimate and its subsets are exposed [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]].
@@ Line 224: / Line 226: @@
 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 good structure, not necessarily the best tuning.
+== 17/16 conundrum ==
+The yazalathana is trivial, easy, very well tuned and well behaved in Ultimate, cassandra, and 12edo. Filling the 17 gap is harder.
+Prime 17, while obvious in 12edo, is complex in cassandra and not nice in Ultimate. In Ultimate there are two valid extensions to reach 17/16, of which "septinoellismic" - so called because it tempers out 1225/1224 - is deemed to be the one with lowest badness. Sure. That doesn't mean it's the best. It's not ideal because 17/16 detempers from a major second (^^⇊⇊M2), which means that it is very far away in a keyboard.
+The other one, called "septisperasmic" - so called because it tempers out 2500/2499 - does detemper from a minor second (4^m2), however, it requires minicommas and thus this cannot be used in cassandra as m2 is wayy too flat - though it can be used in 12edo. It is a more accurate and intuitive mapping with less distance, but it now requires the third generator to have a larger span. Not that septinoellismic didn't do that, but it was still a mapping that could be found within the 21-odd-limit (most span is 26/25 and 25/13, -/+3 minicommas).
 == Beyond Ultimate - the ULTRAOLYMPIC sequence ==
@@ Line 229: / Line 238: @@
 === Olympic - Metaolympic ===
-[[Olympic|This]]. The best course of action for detempering Ultimate is to observe the sasaruma, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma in halves with no meaningful notation changes. Unlike in JI, where the layoma is around half the sasaruma, here the layoma is ~1.6x LARGER, not smaller. A heavily inflated [[10241/10240|tina (~10241/10240)]] is one of these sasaruma halves; it and distinct layomas are required to reach primes 17 and 19.
+[[Olympic|This]]. The best course of action for detempering Ultimate is to observe the sasaruma, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma in halves with basically no meaningful notation changes until higher primes come into play or you modulate far enough along the chain of fifths.
-Olympic decouples 64/63 from the chain of fifths; 64/63 is now its own thing, and the poma is \|↑. (\| is a sasaruma down)
+Unlike in JI, where the layoma is around half the sasaruma, here the layoma is ~1.6x LARGER, not smaller. A heavily inflated [[10241/10240|tina (~10241/10240)]] is one of these sasaruma halves; it and distinct layomas are required to reach primes 17 and 19.
+{{Note|Olympic and beyond require detempering from septinoellismic - the least-badness mapping, for good or for worse.}}
 Everything else is the same:
@@ Line 323: / Line 334: @@
 === 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 with saquarumas, or go too far down the chain of fifths.
+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 more if you don't respell enharmonically with saquarumas, or go too far down the chain of fifths, which can lead to things being technically correct but pragmatically confusing.
-One instance is writing 5/4 above a 16/11, which is 20/11. Above a C, this is '''⇊'''G - ^'''↓'''⇊B.
+One instance is the famous 14/11 - often deemed a major third. It isn't here. From C, it is '''↓'''⇊F. This is because 7/4 is a kind of minor seventh, 11/8 is a fourth. Minor seventh-fourth = fourth. Subtract the commas independently: '''↓ - (⇈) = ↓'''⇊.  Similar thing occurs with 11/10, which is a v↑'''⇈'''m2, 14/13 which is a ^'''↓'''⇊M2. It all makes complete and logical sense from the perspective of the temperament,
+But it feels wrong. That's why we have saquarumas. 14/13 sounds minor-second-y. 14/11 sounds major-third-y. With saquarumas you can spell 11/10 as a major second ('''↓'''ΓM2), 14/11 as a major third (↑LM3).
 A phonetic coding proposal and the alterations for written text are:
@@ Line 436: / Line 449: @@
 |}
 <nowiki>*</nowiki>Not strictly necessary, but it does simplify notation immensely. The saquaruma equals the mercator comma in Ultimate (also being tempered out in 53edo), but not beyond it. It is useful for ultra-long chains of fifths and notating p17, which requires at least four rumas. It follows the following equation: ⇈⇈P1 = Lm2, ⇈⇈m2 = L'''↓'''M2; 17/16 is then ^^L↑m2 in Ultimate; or 𐕣|\^L↑̱m2 in Batshittin.
+Note that calling ^^L↑m2
 It is also unnecessary in 12edo and 41edo, since it equals a flat and a poma up respectively in those systems, and it is a way to skip the bad 19-limit mappings of 12eg and 41g.