Superpyth: Difference between revisions

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'''Superpyth''', a member of the [[archytas clan]], has 4/3 as a generator, and the Archytas comma [[64/63]] is [[tempering out|tempered out]], so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for [[meantone]] and [[12edo]], with the understanding that, for example, A♯ is sharper than B♭ (in contrast to meantone where A♯ is flatter than B♭, or 12edo where they are identical). An interesting coincidence is that the [[Wikipedia: Plastic number|plastic number]] has a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.

'''Superpyth''', a member of the [[archytas clan]] (and thus a version of [[Archytas_clan#Archy|archy]] with a mapping for prime 5), has 4/3 as a generator, and the Archytas comma [[64/63]] is [[tempering out|tempered out]], so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for [[meantone]] and [[12edo]], with the understanding that, for example, A♯ is sharper than B♭ (in contrast to meantone where A♯ is flatter than B♭, or 12edo where they are identical). An interesting coincidence is that the [[Wikipedia: Plastic number|plastic number]] has a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.

Such a temperament without the 5th harmonic is also called '''archy'''. If the 5th harmonic is used at all, it is mapped to -9 generators through tempering out [[245/243]], so C-D♯ is 5/4. So superpyth is the "opposite" of septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex.

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-'''Superpyth''', a member of the [[archytas clan]], has 4/3 as a generator, and the Archytas comma [[64/63]] is [[tempering out|tempered out]], so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for [[meantone]] and [[12edo]], with the understanding that, for example, A♯ is sharper than B♭ (in contrast to meantone where A♯ is flatter than B♭, or 12edo where they are identical). An interesting coincidence is that the [[Wikipedia: Plastic number|plastic number]] has a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.
+'''Superpyth''', a member of the [[archytas clan]] (and thus a version of [[Archytas_clan#Archy|archy]] with a mapping for prime 5), has 4/3 as a generator, and the Archytas comma [[64/63]] is [[tempering out|tempered out]], so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for [[meantone]] and [[12edo]], with the understanding that, for example, A♯ is sharper than B♭ (in contrast to meantone where A♯ is flatter than B♭, or 12edo where they are identical). An interesting coincidence is that the [[Wikipedia: Plastic number|plastic number]] has a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.
 Such a temperament without the 5th harmonic is also called '''archy'''. If the 5th harmonic is used at all, it is mapped to -9 generators through tempering out [[245/243]], so C-D♯ is 5/4. So superpyth is the "opposite" of septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex.