Sensi: Difference between revisions

Line 6:

| Title = Sensi

| Subgroups = 2.3.5.7, 2.3.5.7.13

| Comma basis = [[245/243]], [[126/125]] (7-limit); <br> [[91/90]], [[126/125]], [[~~245~~/~~243~~]] (2.3.5.7.13)

| Comma basis = [[245/243]], [[126/125]] (7-limit); <br> [[91/90]], [[126/125]], [[169/168]] (2.3.5.7.13)

| Edo join 1 = 19 | Edo join 2 = 27

| Generator = 9/7 | Generator tuning = 443.3 | Optimization method = CWE

Line 15:

| Odd limit 2 = (2.3.5.7.13) 21 | Mistuning 2 = 11.1c | Complexity 2 = 27

}}

'''Sensi''' is a [[rank-2 temperament|rank-2]] [[regular temperament]] that is [[generator|generated]] by an extremely sharp major third of between 442 and 445{{cent}}, which is taken in the [[7-limit]] to represent a sharpened [[9/7]]. The most important equivalence in sensi (i.e. [[tempering out]] the comma [[245/243]]) is known as ''sensamagic'', by which two of these thirds stack to a major sixth which approximates [[5/3]]. Sensi then makes the additional tempering of [[126/125]], through which three of these major sixths approximate [[7/6]], two octaves up. The [[6/1|6th harmonic]] is therefore split into seven. Furthermore, since the supermajor third is tempered so sharply, it makes sense to have it represent both 9/7 and [[13/10]], which results in [[91/90]] being tempered out in the 2.3.5.7.13 [[subgroup]].

'''Sensi''' is a [[rank-2 temperament|rank-2]] [[regular temperament]] that is [[generator|generated]] by an extremely sharp major third of between 442 and 445{{cent}}, which is taken in the [[7-limit]] to represent a sharpened [[9/7]]. The most important equivalence in sensi (i.e. [[tempering out]] the comma [[245/243]]) is known as ''sensamagic'', by which two of these thirds stack to a major sixth which approximates [[5/3]]. Sensi then makes the additional tempering of [[126/125]], through which three of these major sixths approximate [[7/6]], two octaves up. The [[6/1|6th harmonic]] is therefore split into seven, and [[5/4]] is divided into three parts, each identified with [[15/14]]. Furthermore, since the supermajor third is tempered so sharply, it makes sense to have it represent both 9/7 and [[13/10]]; the 15/14 interval then also represents 14/13 and 13/12, which results in [[169/168]] and [[196/195]] being tempered out in the 2.3.5.7.13 [[subgroup]] in addition to their product [[91/90]].

The structure whereby 5/3 is split into two supermajor thirds is obviously xenharmonic as this cannot occur in [[12edo]]. But particularly, as the simplest [[EDO]]s with similar structures are [[8edo]] and [[11edo]] (whence the 8-note ([[3L 5s]], checkertonic) and 11-note ([[8L 3s]], flanatonic) [[MOS scale]]s), sensi has a very xenmelodic character compared to many other ways of organizing the 7-limit (such as [[superpyth]], which is based on the familiar [[chain of fifths]], and even [[porcupine]], which is fundamentally heptatonic).

@@ Line 6: / Line 6: @@
 | Title = Sensi
 | Subgroups = 2.3.5.7, 2.3.5.7.13
-| Comma basis = [[245/243]], [[126/125]] (7-limit); <br> [[91/90]], [[126/125]], [[245/243]] (2.3.5.7.13)
+| Comma basis = [[245/243]], [[126/125]] (7-limit); <br> [[91/90]], [[126/125]], [[169/168]] (2.3.5.7.13)
 | Edo join 1 = 19 | Edo join 2 = 27
 | Generator = 9/7 | Generator tuning = 443.3 | Optimization method = CWE
@@ Line 15: / Line 15: @@
 | Odd limit 2 = (2.3.5.7.13) 21 | Mistuning 2 = 11.1c | Complexity 2 = 27
 }}
-'''Sensi''' is a [[rank-2 temperament|rank-2]] [[regular temperament]] that is [[generator|generated]] by an extremely sharp major third of between 442 and 445{{cent}}, which is taken in the [[7-limit]] to represent a sharpened [[9/7]]. The most important equivalence in sensi (i.e. [[tempering out]] the comma [[245/243]]) is known as ''sensamagic'', by which two of these thirds stack to a major sixth which approximates [[5/3]]. Sensi then makes the additional tempering of [[126/125]], through which three of these major sixths approximate [[7/6]], two octaves up. The [[6/1|6th harmonic]] is therefore split into seven. Furthermore, since the supermajor third is tempered so sharply, it makes sense to have it represent both 9/7 and [[13/10]], which results in [[91/90]] being tempered out in the 2.3.5.7.13 [[subgroup]].
+'''Sensi''' is a [[rank-2 temperament|rank-2]] [[regular temperament]] that is [[generator|generated]] by an extremely sharp major third of between 442 and 445{{cent}}, which is taken in the [[7-limit]] to represent a sharpened [[9/7]]. The most important equivalence in sensi (i.e. [[tempering out]] the comma [[245/243]]) is known as ''sensamagic'', by which two of these thirds stack to a major sixth which approximates [[5/3]]. Sensi then makes the additional tempering of [[126/125]], through which three of these major sixths approximate [[7/6]], two octaves up. The [[6/1|6th harmonic]] is therefore split into seven, and [[5/4]] is divided into three parts, each identified with [[15/14]]. Furthermore, since the supermajor third is tempered so sharply, it makes sense to have it represent both 9/7 and [[13/10]]; the 15/14 interval then also represents 14/13 and 13/12, which results in [[169/168]] and [[196/195]] being tempered out in the 2.3.5.7.13 [[subgroup]] in addition to their product [[91/90]].
 The structure whereby 5/3 is split into two supermajor thirds is obviously xenharmonic as this cannot occur in [[12edo]]. But particularly, as the simplest [[EDO]]s with similar structures are [[8edo]] and [[11edo]] (whence the 8-note ([[3L 5s]], checkertonic) and 11-note ([[8L 3s]], flanatonic) [[MOS scale]]s), sensi has a very xenmelodic character compared to many other ways of organizing the 7-limit (such as [[superpyth]], which is based on the familiar [[chain of fifths]], and even [[porcupine]], which is fundamentally heptatonic).