5L 3s: Difference between revisions

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More extreme oneirotonic temperaments include:

* [[Chromatic pairs#Tridec|Tridec]] (a 2.3.7/5.11/5.13/5 subgroup temperament that approximates 5:7:11:13:15), when the generator is between 453.~~33c~~ (17\45) and 457.~~14c~~ (8\21). These have near-equal L/s ratios of 6/5 to 3/2.

* [[Chromatic pairs#Tridec|Tridec]] (a 2.3.7/5.11/5.13/5 subgroup temperament that approximates 5:7:11:13:15), when the generator is between 453.33¢ (17\45) and 457.14¢ (8\21). These have near-equal L/s ratios of 6/5 to 3/2.

* [[Hemifamity_temperaments#Buzzard|Buzzard]], when the generator is between 471.42¢ (11\28) and 480¢ (2\5). While this is a harmonically accurate temperament, with 4 generators reaching [[3/2]] and -3 generators [[7/4]], it is relatively weak melodically, as the optimum size of the small steps is around 20-25 cents, making it difficult to distinguish from equal pentatonic.

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EDOs that support A-Team include [[13edo]], [[18edo]], and [[31edo]].

* 13edo has characteristically small major mosseconds of about 185c. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best [[11/8]] out of all A-team tunings.

* 18edo can be used for a large L/s ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic), or for its nearly pure 9/8 and 7/6. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.

* 18edo can be used for a large L/s ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.

* 31edo is very close to the 2.9.5.21 POTE tuning, and can be used to make the major mos3rd a near-just 5/4.

* [[44edo]] (generator 17\44 = 463.~~64c~~), [[57edo]] (generator 22\57 = 463.~~16c~~), and [[70edo]] (generator 27\70 = 462.~~857c~~) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.

* [[44edo]] (generator 17\44 = 463.64¢), [[57edo]] (generator 22\57 = 463.16¢), and [[70edo]] (generator 27\70 = 462.857¢) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.

The sizes of the generator, large step and small step of oneirotonic are as follows in various A-Team tunings.

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=== Petrtri (13&21) ===

Petrtri tunings (with generator between 8\21 and 5\13) have less extreme L-to-s ratios than A-Team tunings, between 3/2 and 2/1. The 8\21-to-5\13 range of oneirotonic tunings remains relatively unexplored. In these tunings,

* the large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about ~~92c~~ to ~~114c~~.

* the large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92¢ to 114¢.

* The major mosthird (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (~~342c~~) to 4\13 (~~369c~~), and the temperament interprets it as both [[11/9]] and [[16/13]].

* The major mosthird (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342¢) to 4\13 (369¢), and the temperament interprets it as both [[11/9]] and [[16/13]].

The three major edos in this range, [[13edo]], [[21edo]] and [[34edo]], all nominally support petrtri, but [[34edo]] is close to optimal for the temperament, with a generator only .~~33c~~ flat of the optimal ([[POTE]]) petrtri generator of 459.1502c. Close-to-optimal petrtri tunings such as 34edo may be particularly useful for the Sarnathian mode, as Sarnathian in these tunings uniquely approximates four over-2 harmonics plausibly, namely 17/16, 5/4, 11/8, and 13/8.

The three major edos in this range, [[13edo]], [[21edo]] and [[34edo]], all nominally support petrtri, but [[34edo]] is close to optimal for the temperament, with a generator only 0.33¢ flat of the optimal ([[POTE]]) petrtri generator of 459.1502c. Close-to-optimal petrtri tunings such as 34edo may be particularly useful for the Sarnathian mode, as Sarnathian in these tunings uniquely approximates four over-2 harmonics plausibly, namely 17/16, 5/4, 11/8, and 13/8.

The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings.

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=== Tridec (29&37) ===

In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a [[4/3]] perfect fourth. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] Based on the JI interpretations of the [[29edo]] and [[37edo]] tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a [[Non-over-2 temperament|non-over-2 temperament]] that approximates the chord 5:7:11:13:15. The optimal generator is 455.~~2178c~~, which is very close to 29edo's 11\29 (455.~~17c~~), but we could accept any generator between 17\45 (453.~~33c~~) and 8\21 (457.~~14c~~), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.

In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a [[4/3]] perfect fourth. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] Based on the JI interpretations of the [[29edo]] and [[37edo]] tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a [[Non-over-2 temperament|non-over-2 temperament]] that approximates the chord 5:7:11:13:15. The optimal generator is 455.2178¢, which is very close to 29edo's 11\29 (455.17¢), but we could accept any generator between 17\45 (453.33¢) and 8\21 (457.14¢), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.

Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]].

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A-Team oneirotonic may be a particularly good place to bring to bear [[primodality]]'s high harmonic series chords, as A-Team temperament doesn't yield many low-complexity chords.

18edo may be a better basis for a style of oneirotonic primodality using comma sharp and comma flat fifths than 13edo (in particular diesis sharp and diesis flat fifths; diesis is a category with a central region of 32 to ~~40c~~). In 18edo both the major fifth (+31.4c) and the minor fifth (-35.3c) are about a diesis off from a just perfect fifth. In 13edo only the major fifth is a diesis sharp, and it is +36.5c off from just; so there's less wiggle room for a [[neji]] if you want every major fifth to be at most a diesis sharp).

18edo may be a better basis for a style of oneirotonic primodality using comma sharp and comma flat fifths than 13edo (in particular diesis sharp and diesis flat fifths; diesis is a category with a central region of 32 to 40¢). In 18edo both the major fifth (+31.4¢) and the minor fifth (-35.3¢) are about a diesis off from a just perfect fifth. In 13edo only the major fifth is a diesis sharp, and it is +36.5¢ off from just; so there's less wiggle room for a [[neji]] if you want every major fifth to be at most a diesis sharp).

31nejis and 34nejis (though 34edo is not an A-Team tuning) also provide opportunities to use dieses directly, since 1\31 (38.~~71c~~) and 1\34 (35.~~29c~~) are both dieses.

31nejis and 34nejis (though 34edo is not an A-Team tuning) also provide opportunities to use dieses directly, since 1\31 (38.71¢) and 1\34 (35.29¢) are both dieses.

=== Primodal chords ===

Some relatively low-complexity oneirotonic-inspired primodal chords. They are grouped by [[prime family]].

@@ Line 13: / Line 13: @@
 More extreme oneirotonic temperaments include:
-* [[Chromatic pairs#Tridec|Tridec]] (a 2.3.7/5.11/5.13/5 subgroup temperament that approximates 5:7:11:13:15), when the generator is between 453.33c (17\45) and 457.14c (8\21). These have near-equal L/s ratios of 6/5 to 3/2.
+* [[Chromatic pairs#Tridec|Tridec]] (a 2.3.7/5.11/5.13/5 subgroup temperament that approximates 5:7:11:13:15), when the generator is between 453.33¢ (17\45) and 457.14¢ (8\21). These have near-equal L/s ratios of 6/5 to 3/2.
 * [[Hemifamity_temperaments#Buzzard|Buzzard]], when the generator is between 471.42¢ (11\28) and 480¢ (2\5). While this is a harmonically accurate temperament, with 4 generators reaching [[3/2]] and -3 generators [[7/4]], it is relatively weak melodically, as the optimum size of the small steps is around 20-25 cents, making it difficult to distinguish from equal pentatonic.
@@ Line 434: / Line 434: @@
 EDOs that support A-Team include [[13edo]], [[18edo]], and [[31edo]].
 * 13edo has characteristically small major mosseconds of about 185c. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best [[11/8]] out of all A-team tunings.
-* 18edo can be used for a large L/s ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic), or for its nearly pure 9/8 and 7/6. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
+* 18edo can be used for a large L/s ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
 * 31edo is very close to the 2.9.5.21 POTE tuning, and can be used to make the major mos3rd a near-just 5/4.
-* [[44edo]] (generator 17\44 = 463.64c), [[57edo]] (generator 22\57 = 463.16c), and [[70edo]] (generator 27\70 = 462.857c) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.
+* [[44edo]] (generator 17\44 = 463.64¢), [[57edo]] (generator 22\57 = 463.16¢), and [[70edo]] (generator 27\70 = 462.857¢) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.
 The sizes of the generator, large step and small step of oneirotonic are as follows in various A-Team tunings.
@@ Line 474: / Line 474: @@
 === Petrtri (13&21) ===
 Petrtri tunings (with generator between 8\21 and 5\13) have less extreme L-to-s ratios than A-Team tunings, between 3/2 and 2/1. The 8\21-to-5\13 range of oneirotonic tunings remains relatively unexplored. In these tunings,
-* the large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92c to 114c.
+* the large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92¢ to 114¢.
-* The major mosthird (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342c) to 4\13 (369c), and the temperament interprets it as both [[11/9]] and [[16/13]].
+* The major mosthird (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342¢) to 4\13 (369¢), and the temperament interprets it as both [[11/9]] and [[16/13]].
-The three major edos in this range, [[13edo]], [[21edo]] and [[34edo]], all nominally support petrtri, but [[34edo]] is close to optimal for the temperament, with a generator only .33c flat of the optimal ([[POTE]]) petrtri generator of 459.1502c. Close-to-optimal petrtri tunings such as 34edo may be particularly useful for the Sarnathian mode, as Sarnathian in these tunings uniquely approximates four over-2 harmonics plausibly, namely 17/16, 5/4, 11/8, and 13/8.
+The three major edos in this range, [[13edo]], [[21edo]] and [[34edo]], all nominally support petrtri, but [[34edo]] is close to optimal for the temperament, with a generator only 0.33¢ flat of the optimal ([[POTE]]) petrtri generator of 459.1502c. Close-to-optimal petrtri tunings such as 34edo may be particularly useful for the Sarnathian mode, as Sarnathian in these tunings uniquely approximates four over-2 harmonics plausibly, namely 17/16, 5/4, 11/8, and 13/8.
 The sizes of the generator, large step and small step of oneirotonic are as follows in various petrtri tunings.
@@ Line 512: / Line 512: @@
 === Tridec (29&37) ===
-In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a [[4/3]] perfect fourth. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] Based on the JI interpretations of the [[29edo]] and [[37edo]] tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a [[Non-over-2 temperament|non-over-2 temperament]] that approximates the chord 5:7:11:13:15. The optimal generator is 455.2178c, which is very close to 29edo's 11\29 (455.17c), but we could accept any generator between 17\45 (453.33c) and 8\21 (457.14c), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.
+In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a [[4/3]] perfect fourth. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] Based on the JI interpretations of the [[29edo]] and [[37edo]] tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a [[Non-over-2 temperament|non-over-2 temperament]] that approximates the chord 5:7:11:13:15. The optimal generator is 455.2178¢, which is very close to 29edo's 11\29 (455.17¢), but we could accept any generator between 17\45 (453.33¢) and 8\21 (457.14¢), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.
 Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]].
@@ Line 875: / Line 875: @@
 A-Team oneirotonic may be a particularly good place to bring to bear [[primodality]]'s high harmonic series chords, as A-Team temperament doesn't yield many low-complexity chords.
-edo may be a better basis for a style of oneirotonic primodality using comma sharp and comma flat fifths than 13edo (in particular diesis sharp and diesis flat fifths; diesis is a category with a central region of 32 to 40c). In 18edo both the major fifth (+31.4c) and the minor fifth (-35.3c) are about a diesis off from a just perfect fifth. In 13edo only the major fifth is a diesis sharp, and it is +36.5c off from just; so there's less wiggle room for a [[neji]] if you want every major fifth to be at most a diesis sharp).
+edo may be a better basis for a style of oneirotonic primodality using comma sharp and comma flat fifths than 13edo (in particular diesis sharp and diesis flat fifths; diesis is a category with a central region of 32 to 40¢). In 18edo both the major fifth (+31.4¢) and the minor fifth (-35.3¢) are about a diesis off from a just perfect fifth. In 13edo only the major fifth is a diesis sharp, and it is +36.5¢ off from just; so there's less wiggle room for a [[neji]] if you want every major fifth to be at most a diesis sharp).
-nejis and 34nejis (though 34edo is not an A-Team tuning) also provide opportunities to use dieses directly, since 1\31 (38.71c) and 1\34 (35.29c) are both dieses.
+nejis and 34nejis (though 34edo is not an A-Team tuning) also provide opportunities to use dieses directly, since 1\31 (38.71¢) and 1\34 (35.29¢) are both dieses.
 === Primodal chords ===
 Some relatively low-complexity oneirotonic-inspired primodal chords. They are grouped by [[prime family]].