Gene Ward Smith - Revision history

BudjarnLambeth: /* See also */ add link to orphaned page

2025-06-06T01:07:39Z

ArrowHead294 at 13:09, 10 April 2025

2025-04-10T13:09:05Z

← Older revision		Revision as of 13:09, 10 April 2025
Line 1:		Line 1:
	'''Gene Ward Smith''' (1947–2021) was an American mathematician, music theorist, and composer.		'''Gene Ward Smith''' (1947–2021) was an American mathematician, music theorist, and composer.

	In mathematics, he worked in the areas of ~~[[Wikipedia:Galois theory~~\|Galois theory]] and ~~[[Wikipedia:Monstrous moonshine~~\|Moonshine theory]].		In mathematics, he worked in the areas of {{w\|Galois theory}} and {{w\|Moonshine theory}}.

	In music theory, he introduced ~~[[Wikipedia:Exterior_algebra~~\|wedge ~~products]]~~ as a way of classifying [[~~Regular_Temperaments\|~~regular ~~temperaments~~]]. In this system, a temperament is specified by means of a [[~~Wedgies_and_Multivals~~\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal temperaments\|equal divisions of the octave]] and the [[the Riemann zeta function and tuning\|Riemann zeta function]].<ref>Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]</ref><ref>OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.</ref><ref>OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.</ref> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].		In music theory, he introduced {{w\|wedge product}}s as a way of classifying [[regular temperament]]s. In this system, a temperament is specified by means of a [[Wedgies and multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal temperaments\|equal divisions of the octave]] and the [[the Riemann zeta function and tuning\|Riemann zeta function]].<ref>Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]</ref><ref>OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.</ref><ref>OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.</ref> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].

	In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a ~~[[Wikipedia:crystal oscillator~~\|crystal oscillator]]. The device in turn was controlled by ~~[[Wikipedia:HP 9800 series desktop computers~~\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara]].		In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a {{w\|crystal oscillator}}. The device in turn was controlled by {{w\|HP 9800 series desktop computers}}, initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara]].

	Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note [[highschool scales]], led to interesting ([[Omnitetrachordality\|omnitetrachordal]]) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.		Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note [[highschool scales]], led to interesting ([[Omnitetrachordality\|omnitetrachordal]]) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.

	Gene was among the first to consider extending the ~~[[Wikipedia:Tonnetz~~\|Tonnetz]] of Hugo Riemann beyond the 5-limit and hence into higher dimensional ~~[[Wikipedia:~~Lattice (group)\|lattices]]. In three dimensions, the hexagonal lattice of [[5-limit]] harmony extends to a lattice of type A3 ~ D3. He is also the first to write music in a number of exotic intonation systems.		Gene was among the first to consider extending the {{w\|Tonnetz}} of Hugo Riemann beyond the 5-limit and hence into higher dimensional {{w\|Lattice (group)\|lattices}}. In three dimensions, the hexagonal lattice of [[5-limit]] harmony extends to a lattice of type {{nowrap\|A3 ~ D3}}. He is also the first to write music in a number of exotic intonation systems.

	== Historical interest ==		== Historical interest ==

ArrowHead294 at 13:07, 10 April 2025

2025-04-10T13:07:19Z

← Older revision		Revision as of 13:07, 10 April 2025
Line 3:		Line 3:
	In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].		In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].

	In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[~~Equal_Temperaments~~\|equal divisions of the octave]] and the [[~~The_Riemann_Zeta_Function_and_Tuning~~\|Riemann zeta function]].<ref>Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]</ref><ref>OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.</ref><ref>OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.</ref> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].		In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal temperaments\|equal divisions of the octave]] and the [[the Riemann zeta function and tuning\|Riemann zeta function]].<ref>Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]</ref><ref>OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.</ref><ref>OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.</ref> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].

	In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara]].		In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara]].

Fredg999: Cleanup, misc. edits, categories, sort key

2022-04-02T16:33:31Z

Cleanup, misc. edits, categories, sort key

← Older revision		Revision as of 16:33, 2 April 2022
Line 1:		Line 1:
	'''Gene Ward Smith''' (~~b. 1947, d. 2021~~) was a mathematician, music theorist, and composer.		'''Gene Ward Smith''' (1947–2021) was an American mathematician, music theorist, and composer.

	In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].		In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].

	In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].<~~/sup~~>[~~[#cite_note~~-~~1\|[1]]~~]</~~sup~~><~~/sup~~>~~[[#cite_note-2~~\|~~[2]]]~~</~~sup~~><~~/sup~~>~~[[#cite_note-3~~\|~~[3]]]~~</~~sup~~> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].		In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].<ref>Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]</ref><ref>OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.</ref><ref>OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.</ref> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].

	In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[~~pajara\|~~pajara]].		In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara]].

	Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note [[~~Highschool_scales\|~~highschool scales]], led to interesting (omnitetrachordal) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.		Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note [[highschool scales]], led to interesting ([[Omnitetrachordality\|omnitetrachordal]]) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.

	Gene was among the first to consider extending the [[Wikipedia:Tonnetz\|Tonnetz]] of Hugo Riemann beyond the 5-limit and hence into higher dimensional [[Wikipedia:Lattice (group)\|lattices]]. In three dimensions, the hexagonal lattice of [[~~Harmonic_Limit\|~~5-limit ~~harmony~~]] extends to a lattice of type A3 ~ D3. He is also the first to write music in a number of exotic intonation systems~~. See [[Microtonal_Music_by_Gene_Ward_Smith\|Microtonal Music by Gene Ward Smith]]~~.		Gene was among the first to consider extending the [[Wikipedia:Tonnetz\|Tonnetz]] of Hugo Riemann beyond the 5-limit and hence into higher dimensional [[Wikipedia:Lattice (group)\|lattices]]. In three dimensions, the hexagonal lattice of [[5-limit]] harmony extends to a lattice of type A3 ~ D3. He is also the first to write music in a number of exotic intonation systems.

	=Historical interest=		== Historical interest ==
	[[USENET_post_from_1990_by_Gene_Smith_on_homomorphisms_and_kernels\|Usenet post from 1990 by Gene Smith on homomorphisms and kernels]]		* [[USENET_post_from_1990_by_Gene_Smith_on_homomorphisms_and_kernels\|Usenet post from 1990 by Gene Smith on homomorphisms and kernels]]
			* [[USENET_post_from_1995_by_Gene_Smith_on_homomorphisms_and_kernels\|Usenet post from 1995 by Gene Smith on homomorphisms and kernels]]

	[[~~USENET_post_from_1995_by_Gene_Smith_on_homomorphisms_and_kernels\|Usenet post from 1995~~ by Gene Smith ~~on homomorphisms and kernels~~]]		== See also ==
	~~-----~~		* [[Microtonal music by Gene Ward Smith]].
	~~# [[#cite_ref-1\|^]] Rusin, Dave. "Why 12 tones per octave?" [http:~~/~~/www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]~~
	~~# [[#cite_ref-2\|^]] OEIS. Increasingly large peaks of the Riemann zeta function on the critical line:~~ {{~~OEIS\|A117536~~}}.		== References ==
	# [[~~#cite_ref-3\|^~~]] ~~OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.~~		<references/>

			{{DEFAULTSORT:Smith, Gene Ward}}
			[[Category:People]]
	[[Category:Composers]]		[[Category:Composers]]
	[[Category:~~mathematician]]~~		[[Category:Mathematicians]]
	~~[[Category:people~~]]
	[[Category:Theorists]]		[[Category:Theorists]]

Fredg999 category edits: Moving from Category:Theorist to Category:Theorists using Cat-a-lot

2022-02-23T00:03:29Z

Moving from Category:Theorist to Category:Theorists using Cat-a-lot

← Older revision		Revision as of 00:03, 23 February 2022
Line 22:		Line 22:
	[[Category:mathematician]]		[[Category:mathematician]]
	[[Category:people]]		[[Category:people]]
	[[Category:~~theorist~~]]		[[Category:Theorists]]

Fredg999 category edits: Moving from Category:Composer to Category:Composers using Cat-a-lot

2022-02-22T23:57:04Z

Moving from Category:Composer to Category:Composers using Cat-a-lot

← Older revision		Revision as of 23:57, 22 February 2022
Line 19:		Line 19:
	# [[#cite_ref-2\|^]] OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.		# [[#cite_ref-2\|^]] OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.
	# [[#cite_ref-3\|^]] OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.		# [[#cite_ref-3\|^]] OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.
	[[Category:~~composer~~]]		[[Category:Composers]]
	[[Category:mathematician]]		[[Category:mathematician]]
	[[Category:people]]		[[Category:people]]
	[[Category:theorist]]		[[Category:theorist]]

Inthar: internal wiki link

2021-06-23T17:51:18Z

internal wiki link

← Older revision		Revision as of 17:51, 23 June 2021
Line 3:		Line 3:
	In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].		In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].

	In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [~~http://x31eq.com/paradigm.html~~ regular mapping paradigm].		In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].

	In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara\|pajara]].		In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara\|pajara]].

Aura: Realized I missed something in editing this page to reflect Ward's death earlier this year.

2021-03-10T05:15:06Z

Realized I missed something in editing this page to reflect Ward's death earlier this year.

← Older revision		Revision as of 05:15, 10 March 2021
Line 3:		Line 3:
	In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].		In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].

	In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He ~~has~~ long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [http://x31eq.com/paradigm.html regular mapping paradigm].		In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [http://x31eq.com/paradigm.html regular mapping paradigm].

	In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara\|pajara]].		In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara\|pajara]].

Aura: Updated in part to reflect the news of his death from COVID, which I just found out about on Discord

2021-03-09T18:32:27Z

Updated in part to reflect the news of his death from COVID, which I just found out about on Discord

← Older revision		Revision as of 18:32, 9 March 2021
Line 1:		Line 1:
	'''Gene Ward Smith''' (b. 1947) is a mathematician, music theorist, and composer.		'''Gene Ward Smith''' (b. 1947, d. 2021) was a mathematician, music theorist, and composer.

	In mathematics, he ~~has~~ worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].		In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].

	In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He has long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [http://x31eq.com/paradigm.html regular mapping paradigm].		In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He has long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [http://x31eq.com/paradigm.html regular mapping paradigm].

Oak Blood Three: Cleanup wikipedia & OEIS links

2019-05-22T20:09:17Z

Cleanup wikipedia & OEIS links

← Older revision		Revision as of 20:09, 22 May 2019
Line 1:		Line 1:
	'''Gene Ward Smith''' (b. 1947) is a mathematician, music theorist, and composer.		'''Gene Ward Smith''' (b. 1947) is a mathematician, music theorist, and composer.

	In mathematics, he has worked in the areas of [~~http~~:~~//en.wikipedia.org/wiki/Galois_theory~~ Galois theory] and [~~http~~:~~//en.wikipedia.org/wiki/Monstrous_moonshine~~ Moonshine theory].		In mathematics, he has worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].

	In music theory, he introduced [~~http~~:~~//en.wikipedia.org/wiki/~~Exterior_algebra wedge products] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [~~http~~:~~//en.wikipedia.org/wiki/~~Grassmannian Grassmannian]. He has long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [http://x31eq.com/paradigm.html regular mapping paradigm].		In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He has long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [http://x31eq.com/paradigm.html regular mapping paradigm].

	In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [~~http~~:~~//en.wikipedia.org/wiki/Crystal_oscillator~~ crystal oscillator]. The device in turn was controlled by [~~http~~:~~//en.wikipedia.org/wiki/HP_9800_series_desktop_computers~~ HP 9800 series desktop computers], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara\|pajara]].		In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara\|pajara]].

	Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note [[Highschool_scales\|highschool scales]], led to interesting (omnitetrachordal) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.		Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note [[Highschool_scales\|highschool scales]], led to interesting (omnitetrachordal) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.

	Gene was among the first to consider extending the [~~http~~:~~//en.wikipedia.org/wiki/~~Tonnetz Tonnetz] of Hugo Riemann beyond the 5-limit and hence into higher dimensional [~~http~~:~~//en.wikipedia.org/wiki/Lattice_%28group%29~~ lattices]. In three dimensions, the hexagonal lattice of [[Harmonic_Limit\|5-limit harmony]] extends to a lattice of type A3 ~ D3. He is also the first to write music in a number of exotic intonation systems. See [[Microtonal_Music_by_Gene_Ward_Smith\|Microtonal Music by Gene Ward Smith]].		Gene was among the first to consider extending the [[Wikipedia:Tonnetz\|Tonnetz]] of Hugo Riemann beyond the 5-limit and hence into higher dimensional [[Wikipedia:Lattice (group)\|lattices]]. In three dimensions, the hexagonal lattice of [[Harmonic_Limit\|5-limit harmony]] extends to a lattice of type A3 ~ D3. He is also the first to write music in a number of exotic intonation systems. See [[Microtonal_Music_by_Gene_Ward_Smith\|Microtonal Music by Gene Ward Smith]].

	=Historical interest=		=Historical interest=
Line 16:		Line 16:
	[[USENET_post_from_1995_by_Gene_Smith_on_homomorphisms_and_kernels\|Usenet post from 1995 by Gene Smith on homomorphisms and kernels]]		[[USENET_post_from_1995_by_Gene_Smith_on_homomorphisms_and_kernels\|Usenet post from 1995 by Gene Smith on homomorphisms and kernels]]
	-----		-----
	~~<ol><li>~~[[#cite_ref-1\|^]] Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]~~</li><li>~~[[#cite_ref-2\|^]] OEIS. Increasingly large peaks of the Riemann zeta function on the critical line ~~[http~~:~~//oeis.org/~~A117536 ~~http://oeis~~.~~org/A117536]</li><li>~~[[#cite_ref-3\|^]] OEIS. Increasingly large integrals of the Z function between zeros ~~[http~~:~~//oeis.org/~~A117538 ~~http://oeis~~.~~org/A117538]</li></ol>~~ [[Category:composer]]		# [[#cite_ref-1\|^]] Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]
			# [[#cite_ref-2\|^]] OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.
			# [[#cite_ref-3\|^]] OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.
			[[Category:composer]]
	[[Category:mathematician]]		[[Category:mathematician]]
	[[Category:people]]		[[Category:people]]
	[[Category:theorist]]		[[Category:theorist]]

← Older revision		Revision as of 13:09, 10 April 2025
Line 1:		Line 1:
	'''Gene Ward Smith''' (1947–2021) was an American mathematician, music theorist, and composer.		'''Gene Ward Smith''' (1947–2021) was an American mathematician, music theorist, and composer.

	In mathematics, he worked in the areas of ~~[[Wikipedia:Galois theory~~\|Galois theory]] and ~~[[Wikipedia:Monstrous moonshine~~\|Moonshine theory]].		In mathematics, he worked in the areas of {{w\|Galois theory}} and {{w\|Moonshine theory}}.

	In music theory, he introduced ~~[[Wikipedia:Exterior_algebra~~\|wedge ~~products]]~~ as a way of classifying [[~~Regular_Temperaments\|~~regular ~~temperaments~~]]. In this system, a temperament is specified by means of a [[~~Wedgies_and_Multivals~~\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal temperaments\|equal divisions of the octave]] and the [[the Riemann zeta function and tuning\|Riemann zeta function]].<ref>Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]</ref><ref>OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.</ref><ref>OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.</ref> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].		In music theory, he introduced {{w\|wedge product}}s as a way of classifying [[regular temperament]]s. In this system, a temperament is specified by means of a [[Wedgies and multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal temperaments\|equal divisions of the octave]] and the [[the Riemann zeta function and tuning\|Riemann zeta function]].<ref>Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]</ref><ref>OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.</ref><ref>OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.</ref> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].

	In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a ~~[[Wikipedia:crystal oscillator~~\|crystal oscillator]]. The device in turn was controlled by ~~[[Wikipedia:HP 9800 series desktop computers~~\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara]].		In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a {{w\|crystal oscillator}}. The device in turn was controlled by {{w\|HP 9800 series desktop computers}}, initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara]].

	Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note [[highschool scales]], led to interesting ([[Omnitetrachordality\|omnitetrachordal]]) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.		Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note [[highschool scales]], led to interesting ([[Omnitetrachordality\|omnitetrachordal]]) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.

	Gene was among the first to consider extending the ~~[[Wikipedia:Tonnetz~~\|Tonnetz]] of Hugo Riemann beyond the 5-limit and hence into higher dimensional ~~[[Wikipedia:~~Lattice (group)\|lattices]]. In three dimensions, the hexagonal lattice of [[5-limit]] harmony extends to a lattice of type A3 ~ D3. He is also the first to write music in a number of exotic intonation systems.		Gene was among the first to consider extending the {{w\|Tonnetz}} of Hugo Riemann beyond the 5-limit and hence into higher dimensional {{w\|Lattice (group)\|lattices}}. In three dimensions, the hexagonal lattice of [[5-limit]] harmony extends to a lattice of type {{nowrap\|A3 ~ D3}}. He is also the first to write music in a number of exotic intonation systems.

	== Historical interest ==		== Historical interest ==

← Older revision		Revision as of 13:07, 10 April 2025
Line 3:		Line 3:
	In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].		In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].

	In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[~~Equal_Temperaments~~\|equal divisions of the octave]] and the [[~~The_Riemann_Zeta_Function_and_Tuning~~\|Riemann zeta function]].<ref>Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]</ref><ref>OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.</ref><ref>OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.</ref> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].		In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal temperaments\|equal divisions of the octave]] and the [[the Riemann zeta function and tuning\|Riemann zeta function]].<ref>Rusin, Dave. "Why 12 tones per octave?" [http://www.math.niu.edu/~rusin/uses-math/music/12 http://www.math.niu.edu/~rusin/uses-math/music/12]</ref><ref>OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: {{OEIS\|A117536}}.</ref><ref>OEIS. Increasingly large integrals of the Z function between zeros: {{OEIS\|A117538}}.</ref> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].

	In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara]].		In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara]].

← Older revision		Revision as of 17:51, 23 June 2021
Line 3:		Line 3:
	In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].		In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].

	In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [~~http://x31eq.com/paradigm.html~~ regular mapping paradigm].		In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[Regular temperament\|regular mapping paradigm]].

	In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara\|pajara]].		In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara\|pajara]].

← Older revision		Revision as of 05:15, 10 March 2021
Line 3:		Line 3:
	In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].		In mathematics, he worked in the areas of [[Wikipedia:Galois theory\|Galois theory]] and [[Wikipedia:Monstrous moonshine\|Moonshine theory]].

	In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He ~~has~~ long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [http://x31eq.com/paradigm.html regular mapping paradigm].		In music theory, he introduced [[Wikipedia:Exterior_algebra\|wedge products]] as a way of classifying [[Regular_Temperaments\|regular temperaments]]. In this system, a temperament is specified by means of a [[Wedgies_and_Multivals\|wedgie]], which may technically be identified as a point on a [[Wikipedia:Grassmannian\|Grassmannian]]. He had long drawn attention to the relationship between [[Equal_Temperaments\|equal divisions of the octave]] and the [[The_Riemann_Zeta_Function_and_Tuning\|Riemann zeta function]].</sup>[[#cite_note-1\|[1]]]</sup></sup>[[#cite_note-2\|[2]]]</sup></sup>[[#cite_note-3\|[3]]]</sup> He [http://www.webcitation.org/67ZUSajSK early on] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [http://x31eq.com/paradigm.html regular mapping paradigm].

	In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara\|pajara]].		In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a [[Wikipedia:crystal oscillator\|crystal oscillator]]. The device in turn was controlled by [[Wikipedia:HP 9800 series desktop computers\|HP 9800 series desktop computers]], initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and [[pajara\|pajara]].

← Older revision		Revision as of 01:07, 6 June 2025
Line 16:		Line 16:

	== See also ==		== See also ==
	* [[Microtonal music by Gene Ward Smith]].		* [[Microtonal music by Gene Ward Smith]]
			* [[Hypergenesis58]] (a scale described by Gene Ward Smith)

	== References ==		== References ==