Prime EDO - Revision history

Fredg999: Merged

2022-10-09T19:47:39Z

Merged

← Older revision		Revision as of 19:47, 9 October 2022
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	~~A '''prime edo''' is an [[edo]] with a [[prime number]] of different pitches per [[octave]].~~		#REDIRECT [[Prime equal division]]

	~~== Prime numbers in edos ==~~
	~~Whether or not a number ''n'' is prime has important consequences for the properties of the corresponding ''n''-[[edo]], especially for lower values of ''n''. In these instances:~~

	* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo]])
	* Excepting the scale comprising all notes of the edo, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone scale in 12edo)
	* There are no [[Wikipedia: Modes of limited transposition\|modes of limited transposition]], such as used by the composer Olivier Messiaen
	* There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)
	* Making a chain of any interval of the ''n''-edo, one can reach every tone in ''n'' steps. (For composite edos, this works with intervals that are co-prime to ''n'', for example, 5 degrees of 12edo)

	~~For these or similar reasons, some musicians do not like prime edos (e.g. the makers of [http://www.armodue.com/risorse.htm Armodue]) and others love them.~~

	Primality may be desirable if you want, for example, a whole tone scale that is ''not'' absolutely uniform. In this case you might like [[19edo]] (with whole tone scale 3 3 3 3 3 4, [[mos scale]] of type [[1L 5s]]) or [[17edo]] (with whole tone scale 3 3 3 3 3 2, MOS scale of type [[5L 1s]]). In general, making a chain of any interval of a prime ''n''-edo, thus treating the interval as the generator of a mos scale, one can reach every tone in ''n'' steps. For composite edos, this will only work with intervals that are co-prime to the edo, for example 5 degrees of [[12edo]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo]]).

	A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo. Since 12 is 2 × 2 × 3, it contains [[2edo]], 3edo, [[4edo]] and [[6edo]]. All edos with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12edo, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo]], will work just as well for this purpose.

	If you like a certain edo for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[EDT\|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt]] is a non-prime system very similar to 17edo, while [[19edt\|19edt (Stopper tuning)]] is a prime system very similar to the ubiquitous 12edo. (See [[EDT #~~EDT-EDO correspondence\|edt-edo correspondence]] for more of these.) Anyway, for every prime edo system there is a non-prime [[ed4]] system with identical step sizes.~~

	~~The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.~~

	~~The opposite of a prime EDO is a [[highly melodic EDO]].~~

	~~== The first 46 prime edos ==~~
	Multiples of an edo, including multiples of a prime edo, can inherit properties from that edo, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime edo is always the least complex edo divisible by that prime, and these are listed below:

	~~[[2edo\|2]], [[3edo\|3]], [[5edo\|5]], [[7edo\|7]], [[11edo\|11]], [[13edo\|13]], [[17edo\|17]], [[19edo\|19]], <br>~~
	~~[[23edo\|23]], [[29edo\|29]], [[31edo\|31]], [[37edo\|37]], [[41edo\|41]], [[43edo\|43]], [[47edo\|47]], [[53edo\|53]], <br>~~
	~~[[59edo\|59]], [[61edo\|61]], [[67edo\|67]], [[71edo\|71]], [[73edo\|73]], [[79edo\|79]], [[83edo\|83]], [[89edo\|89]], <br>~~
	~~[[97edo\|97]], [[101edo\|101]], [[103edo\|103]], [[107edo\|107]], [[109edo\|109]], [[113edo\|113]], [[127edo\|127]], [[131edo\|131]], <br>~~
	~~[[137edo\|137]], [[139edo\|139]], [[149edo\|149]], [[151edo\|151]], [[157edo\|157]], [[163edo\|163]], [[167edo\|167]], [[173edo\|173]], <br>~~
	~~[[179edo\|179]], [[181edo\|181]], [[191edo\|191]], [[193edo\|193]], [[197edo\|197]], [[199edo\|199]].~~

	~~[[Category:EDO theory pages]]~~
	~~[[Category:Prime EDO\| ]] <!-- main article -->~~
	[[~~Category:~~Prime]]

Eliora: /* Prime numbers in edos */

2022-05-04T11:53:52Z

Prime numbers in edos

← Older revision		Revision as of 11:53, 4 May 2022
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	The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.		The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.

			The opposite of a prime EDO is a [[highly melodic EDO]].

	== The first 46 prime edos ==		== The first 46 prime edos ==

FloraC: Re-style (per discussion of convention)

2022-04-02T15:58:49Z

Re-style (per discussion of convention)

← Older revision		Revision as of 15:58, 2 April 2022
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	A '''prime ~~EDO~~''' is an [[~~EDO~~]] with a [[prime number]] of different pitches per [[octave]].		A '''prime edo''' is an [[edo]] with a [[prime number]] of different pitches per [[octave]].

	== Prime numbers in ~~EDOs~~ ==		== Prime numbers in edos ==
			Whether or not a number ''n'' is prime has important consequences for the properties of the corresponding ''n''-[[edo]], especially for lower values of ''n''. In these instances:

	~~Whether or not a number ''n'' is prime has important consequences for the properties of the corresponding ''n''-[[EDO]], especially for lower values of ''n''. In these instances:~~		* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo]])
			* Excepting the scale comprising all notes of the edo, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone scale in 12edo)
	* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo~~\|12-EDO~~]])
	* Excepting the scale comprising all notes of the ~~EDO~~, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone scale in ~~12-EDO~~)
	* There are no [[Wikipedia: Modes of limited transposition\|modes of limited transposition]], such as used by the composer Olivier Messiaen		* There are no [[Wikipedia: Modes of limited transposition\|modes of limited transposition]], such as used by the composer Olivier Messiaen
	* There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)		* There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)
	* Making a chain of any interval of the ''n''-~~EDO~~, one can reach every tone in ''n'' steps. (For composite ~~EDOs~~, this works with intervals that are co-prime to ''n'', for example, 5 degrees of ~~12-EDO~~)		* Making a chain of any interval of the ''n''-edo, one can reach every tone in ''n'' steps. (For composite edos, this works with intervals that are co-prime to ''n'', for example, 5 degrees of 12edo)

	For these or similar reasons, some musicians do not like prime ~~EDOs~~ (e.g. the makers of [http://www.armodue.com/risorse.htm Armodue]) and others love them.		For these or similar reasons, some musicians do not like prime edos (e.g. the makers of [http://www.armodue.com/risorse.htm Armodue]) and others love them.

	Primality may be desirable if you want, for example, a whole tone scale that is ''not'' absolutely uniform. In this case you might like [[~~19-EDO~~]] (with whole tone scale 3 3 3 3 3 4, [[~~MOS~~ scale]] of type [[1L 5s]]) or [[17edo~~\|17-EDO~~]] (with whole tone scale 3 3 3 3 3 2, MOS scale of type [[5L 1s]]). In general, making a chain of any interval of a prime ''n''-~~EDO~~, thus treating the interval as the generator of a ~~MOS~~ scale, one can reach every tone in ''n'' steps. For composite ~~EDOs~~, this will only work with intervals that are co-prime to the ~~EDO~~, for example 5 degrees of [[12edo~~\|12-EDO~~]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo~~\|3-EDO~~]]).		Primality may be desirable if you want, for example, a whole tone scale that is ''not'' absolutely uniform. In this case you might like [[19edo]] (with whole tone scale 3 3 3 3 3 4, [[mos scale]] of type [[1L 5s]]) or [[17edo]] (with whole tone scale 3 3 3 3 3 2, MOS scale of type [[5L 1s]]). In general, making a chain of any interval of a prime ''n''-edo, thus treating the interval as the generator of a mos scale, one can reach every tone in ''n'' steps. For composite edos, this will only work with intervals that are co-prime to the edo, for example 5 degrees of [[12edo]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo]]).

	A prime ~~EDO~~ is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in ~~12-EDO~~. Since 12 is 2 × 2 × 3, it contains [[2edo~~\|2-EDO~~]], ~~3-EDO~~, [[4edo~~\|4-EDO~~]] and [[6edo~~\|6-EDO~~]]. All ~~EDOs~~ with a 2, 3, 4, or 6 in their factorization will share at least one interval with ~~12-EDO~~, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo~~\|35-EDO~~]], will work just as well for this purpose.		A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo. Since 12 is 2 × 2 × 3, it contains [[2edo]], 3edo, [[4edo]] and [[6edo]]. All edos with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12edo, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo]], will work just as well for this purpose.

	If you like a certain ~~EDO~~ for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[EDT\|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt~~\|27-EDT~~]] is a non-prime system very similar to ~~17-EDO~~, while [[19edt\|~~19-EDT~~ (Stopper tuning)]] is a prime system very similar to the ubiquitous ~~12-EDO~~. (See [[EDT #EDT-EDO correspondence\|~~EDT~~-~~EDO~~ correspondence]] for more of these.) Anyway, for every prime ~~EDO~~ system there is a non-prime [[ed4~~\|ED4~~]] system with identical step sizes.		If you like a certain edo for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[EDT\|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt]] is a non-prime system very similar to 17edo, while [[19edt\|19edt (Stopper tuning)]] is a prime system very similar to the ubiquitous 12edo. (See [[EDT #EDT-EDO correspondence\|edt-edo correspondence]] for more of these.) Anyway, for every prime edo system there is a non-prime [[ed4]] system with identical step sizes.

	The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.		The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.

	== The first 46 prime ~~EDOs~~ ==		== The first 46 prime edos ==
	Multiples of an ~~EDO~~, including multiples of a prime ~~EDO~~, can inherit properties from that ~~EDO~~, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime ~~EDO~~ is always the least complex ~~EDO~~ divisible by that prime, and these are listed below:		Multiples of an edo, including multiples of a prime edo, can inherit properties from that edo, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime edo is always the least complex edo divisible by that prime, and these are listed below:

	[[2edo\|2]], [[3edo\|3]], [[5edo\|5]], [[7edo\|7]], [[11edo\|11]], [[13edo\|13]], [[17edo\|17]], [[19edo\|19]], <br>		[[2edo\|2]], [[3edo\|3]], [[5edo\|5]], [[7edo\|7]], [[11edo\|11]], [[13edo\|13]], [[17edo\|17]], [[19edo\|19]], <br>

Ketsuban: /* Prime numbers in EDOs */

2022-02-25T23:28:26Z

Prime numbers in EDOs

← Older revision		Revision as of 23:28, 25 February 2022
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	* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo\|12-EDO]])		* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo\|12-EDO]])
	* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone scale in 12-EDO)		* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone scale in 12-EDO)
	* There are no [[Wikipedia: Modes of limited transposition\|modes of limited ~~transpostion~~]], such as used by the composer Olivier Messiaen		* There are no [[Wikipedia: Modes of limited transposition\|modes of limited transposition]], such as used by the composer Olivier Messiaen
	* There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)		* There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)
	* Making a chain of any interval of the ''n''-EDO, one can reach every tone in ''n'' steps. (For composite EDOs, this works with intervals that are co-prime to ''n'', for example, 5 degrees of 12-EDO)		* Making a chain of any interval of the ''n''-EDO, one can reach every tone in ''n'' steps. (For composite EDOs, this works with intervals that are co-prime to ''n'', for example, 5 degrees of 12-EDO)

FloraC: Styling +1 (and it's funny how the prime edos were grouped by seven)

2021-07-04T12:53:55Z

Styling +1 (and it's funny how the prime edos were grouped by seven)

← Older revision		Revision as of 12:53, 4 July 2021
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	* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo\|12-EDO]])		* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo\|12-EDO]])
	* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone scale in ~~12EDO~~)		* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone scale in 12-EDO)
	* There are no [[Wikipedia: Modes of limited transposition\|modes of limited transpostion]], such as used by the composer Olivier Messiaen		* There are no [[Wikipedia: Modes of limited transposition\|modes of limited transpostion]], such as used by the composer Olivier Messiaen
	* There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)		* There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)
Line 15:		Line 15:
	Primality may be desirable if you want, for example, a whole tone scale that is ''not'' absolutely uniform. In this case you might like [[19-EDO]] (with whole tone scale 3 3 3 3 3 4, [[MOS scale]] of type [[1L 5s]]) or [[17edo\|17-EDO]] (with whole tone scale 3 3 3 3 3 2, MOS scale of type [[5L 1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a MOS scale, one can reach every tone in ''n'' steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12edo\|12-EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo\|3-EDO]]).		Primality may be desirable if you want, for example, a whole tone scale that is ''not'' absolutely uniform. In this case you might like [[19-EDO]] (with whole tone scale 3 3 3 3 3 4, [[MOS scale]] of type [[1L 5s]]) or [[17edo\|17-EDO]] (with whole tone scale 3 3 3 3 3 2, MOS scale of type [[5L 1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a MOS scale, one can reach every tone in ''n'' steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12edo\|12-EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo\|3-EDO]]).

	A prime ~~edo~~ is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12-EDO. Since 12 is 2 × 2 × 3, it contains [[2edo\|2-EDO]], 3-EDO, [[4edo\|4-EDO]] and [[6edo\|6-EDO]]. All EDOs with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12-EDO, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo\|35-EDO]], will work just as well for this purpose.		A prime EDO is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12-EDO. Since 12 is 2 × 2 × 3, it contains [[2edo\|2-EDO]], 3-EDO, [[4edo\|4-EDO]] and [[6edo\|6-EDO]]. All EDOs with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12-EDO, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo\|35-EDO]], will work just as well for this purpose.

	If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[EDT\|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt\|27-EDT]] is a non-prime system very similar to 17-EDO, while [[19edt\|19-EDT (Stopper tuning)]] is a prime system very similar to the ubiquitous 12-EDO. (See [[EDT #EDT-EDO correspondence\|EDT-EDO correspondence]] for more of these.) Anyway, for every prime EDO system there is a non-prime [[ed4\|ED4]] system with identical step sizes.		If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[EDT\|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt\|27-EDT]] is a non-prime system very similar to 17-EDO, while [[19edt\|19-EDT (Stopper tuning)]] is a prime system very similar to the ubiquitous 12-EDO. (See [[EDT #EDT-EDO correspondence\|EDT-EDO correspondence]] for more of these.) Anyway, for every prime EDO system there is a non-prime [[ed4\|ED4]] system with identical step sizes.
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	The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.		The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.

	== The first 46 ~~Prime~~ EDOs ==		== The first 46 prime EDOs ==
	Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below:		Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below:

	[[2edo\|2]], [[3edo\|3]], [[5edo\|5]], [[7edo\|7]], [[11edo\|11]], [[13edo\|13]], [[17edo\|17]], ~~<br/>~~		[[2edo\|2]], [[3edo\|3]], [[5edo\|5]], [[7edo\|7]], [[11edo\|11]], [[13edo\|13]], [[17edo\|17]], [[19edo\|19]], <br>
	[[19edo\|19]], [[23edo\|23]], [[29edo\|29]], [[31edo\|31]], [[37edo\|37]], [[41edo\|41]], [[43edo\|43]], ~~<br/>~~		[[23edo\|23]], [[29edo\|29]], [[31edo\|31]], [[37edo\|37]], [[41edo\|41]], [[43edo\|43]], [[47edo\|47]], [[53edo\|53]], <br>
	[[47edo\|47]], [[53edo\|53]], [[59edo\|59]], [[61edo\|61]], [[67edo\|67]], [[71edo\|71]], [[73edo\|73]], ~~<br/>~~		[[59edo\|59]], [[61edo\|61]], [[67edo\|67]], [[71edo\|71]], [[73edo\|73]], [[79edo\|79]], [[83edo\|83]], [[89edo\|89]], <br>
	[[79edo\|79]], [[83edo\|83]], [[89edo\|89]], [[97edo\|97]], [[101edo\|101]], [[103edo\|103]], [[107edo\|107]], ~~<br/>~~		[[97edo\|97]], [[101edo\|101]], [[103edo\|103]], [[107edo\|107]], [[109edo\|109]], [[113edo\|113]], [[127edo\|127]], [[131edo\|131]], <br>
	[[109edo\|109]], [[113edo\|113]], [[127edo\|127]], [[131edo\|131]], [[137edo\|137]], [[139edo\|139]], [[149edo\|149]], ~~<br/>~~		[[137edo\|137]], [[139edo\|139]], [[149edo\|149]], [[151edo\|151]], [[157edo\|157]], [[163edo\|163]], [[167edo\|167]], [[173edo\|173]], <br>
	[[151edo\|151]], [[157edo\|157]], [[163edo\|163]], [[167edo\|167]], [[173edo\|173]], [[179edo\|179]], [[181edo\|181]], ~~<br/>~~		[[179edo\|179]], [[181edo\|181]], [[191edo\|191]], [[193edo\|193]], [[197edo\|197]], [[199edo\|199]].
	[[191edo\|191]], [[193edo\|193]], [[197edo\|197]], [[199edo\|199]].

	[[Category:EDO theory pages]]		[[Category:EDO theory pages]]
	[[Category:Prime EDO\| ]] <!-- main article -->		[[Category:Prime EDO\| ]] <!-- main article -->
	[[Category:Prime]]		[[Category:Prime]]

FloraC: Internalize wikipedia links and unify the style (-EDO for now)

2021-07-04T11:39:32Z

Internalize wikipedia links and unify the style (-EDO for now)

FloraC: Copy from the prime number page

2021-07-04T11:26:37Z

Copy from the prime number page

← Older revision		Revision as of 11:26, 4 July 2021
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	''~~todo: improve definition rearrange contents of pages~~ [[prime number]] ~~and~~ [[prime EDO]]''		A '''prime edo''' is an [[EDO]] with a [[prime number]] of different pitches per [[octave]].

			== Prime numbers in EDOs ==

			Whether or not a number ''n'' is prime has important consequences for the properties of the corresponding ''n''-[[EDO]], especially for lower values of ''n''. In these instances:

	A '''prime ~~edo~~''' is an [[EDO]] with a [[prime ~~number~~]] of ~~different pitches per~~ [[~~octave~~]].		* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo\|12EDO]])
			* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the wholetone scale in 12EDO)
			* There are no [http://en.wikipedia.org/wiki/Modes_of_limited_transposition modes of limited transpostion], such as used by the composer Olivier Messiaen
			* There is no support for rank-two temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)
			* Making a chain of any interval of the ''n''-EDO, one can reach every tone in ''n'' steps. (For composite EDOs, this works with intervals that are co-prime to ''n'', for example, 5 degrees of 12EDO)

			For these or similar reasons, some musicians do not like the prime EDOs (e.g. the makers of [http://www.armodue.com/risorse.htm Armodue]) and others love them.

			Primality may be desirable if you want, for example, a wholetone scale that is ''not'' absolutely uniform. In this case you might like [[19edo]] (with whole tone scale 3 3 3 3 3 4, MOS scale of type [[1L 5s\|1L+5s]]) or [[17edo\|17EDO]] (with whole tone scale 3 3 3 3 3 2, MOS Scale of type [[5L 1s\|5L+1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a [[MOSScales\|Moment of Symmetry]] scale, one can reach every tone in ''n'' steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12edo\|12EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo\|3EDO]]).

			A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo. Since 12 is 223, it contains [[2edo\|2EDO]], [[3edo\|3EDO]], [[4edo\|4EDO]] and [[6edo\|6EDO]]. All edos with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12edo, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo\|35EDO]], will work just as well for this purpose.

	~~== Examples ==~~		If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[Edt\|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt]] is a non-prime system very similar to [[17edo]], while [[19ED3\|19edt (Stopper tuning)]] is a prime system very similar to the ubiquitous [[12edo]]. (See [[Edt#EDT-EDO correspondence\|EDT-EDO correspondence]] for more of these.) Anyway, for every prime EDO system there is a non-prime [[Ed4\|ED4]] system with identical step sizes.

	~~macrotonal~~		The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.
	* [[2edo]]
	* [[3edo]]
	* [[5edo]]
	* [[7edo]]
	* [[11edo]]

	~~mesotonal~~		[TODO: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here.]
	* [~~[13edo]]~~
	* [[17edo]]
	* [[19edo]]
	* [[23edo]]

	~~microtonal~~		== The first 46 Prime EDOs ==
			Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below:

	* [[29edo]]		[[2edo\|2]], [[3edo\|3]], [[5edo\|5]], [[7edo\|7]], [[11edo\|11]], [[13edo\|13]], [[17edo\|17]], <br/>
	* [[31edo]]		[[19edo\|19]], [[23edo\|23]], [[29edo\|29]], [[31edo\|31]], [[37edo\|37]], [[41edo\|41]], [[43edo\|43]], <br/>
	* [[41edo]]		[[47edo\|47]], [[53edo\|53]], [[59edo\|59]], [[61edo\|61]], [[67edo\|67]], [[71edo\|71]], [[73edo\|73]], <br/>
	* [[43edo]]		[[79edo\|79]], [[83edo\|83]], [[89edo\|89]], [[97edo\|97]], [[101edo\|101]], [[103edo\|103]], [[107edo\|107]], <br/>
	* [[53edo]]		[[109edo\|109]], [[113edo\|113]], [[127edo\|127]], [[131edo\|131]], [[137edo\|137]], [[139edo\|139]], [[149edo\|149]], <br/>
			[[151edo\|151]], [[157edo\|157]], [[163edo\|163]], [[167edo\|167]], [[173edo\|173]], [[179edo\|179]], [[181edo\|181]], <br/>
			[[191edo\|191]], [[193edo\|193]], [[197edo\|197]], [[199edo\|199]].

			[[Category:EDO theory pages]]
	[[Category:Prime EDO\| ]] <!-- main article -->		[[Category:Prime EDO\| ]] <!-- main article -->
			[[Category:Prime]]

Xenwolf: typo fixed

2018-11-02T09:55:38Z

typo fixed

← Older revision		Revision as of 09:55, 2 November 2018
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	* [[53edo]]		* [[53edo]]

	[[~~Categorie~~:Prime EDO\| ]] <!-- main article -->		[[Category:Prime EDO\| ]] <!-- main article -->

Xenwolf: simplified page categorized

2018-11-02T08:51:11Z

simplified page categorized

← Older revision		Revision as of 08:51, 2 November 2018
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	''todo: improve definition rearrange contents of pages [[~~prime_number\|~~prime number]] and [[~~prime_edo\|~~prime ~~edo~~]]''		''todo: improve definition rearrange contents of pages [[prime number]] and [[prime EDO]]''

	A '''prime edo''' is an [[EDO~~\|edo~~]] with a [[~~prime_number\|~~prime number]] of different pitches per [[~~Octave\|~~octave]].		A '''prime edo''' is an [[EDO]] with a [[prime number]] of different pitches per [[octave]].

	== Examples ==		== Examples ==

	macrotonal		macrotonal
			* [[2edo]]
			* [[3edo]]
			* [[5edo]]
			* [[7edo]]
			* [[11edo]]

			mesotonal
			* [[13edo]]
			* [[17edo]]
			* [[19edo]]
			* [[23edo]]

	~~<ul><li>[[2edo\|2edo]]</li><li>[[3edo\|3edo]]</li><li>[[5edo\|5edo]]</li><li>[[7edo\|7edo]]</li><li>[[11edo\|11edo]]</li></ul>mesotonal~~		microtonal

	~~<ul><li>~~[[~~13edo\|13edo~~]]~~</li><li>~~[[~~17edo\|17edo~~]]~~</li><li>~~[[~~19edo\|19edo~~]]~~</li><li>~~[[~~23edo\|23edo~~]]~~</li></ul>microtonal~~		* [[29edo]]
			* [[31edo]]
			* [[41edo]]
			* [[43edo]]
			* [[53edo]]

	~~<ul><li>[[29edo\|29edo]]</li><li>[[31edo\|31edo]]</li><li>~~[[~~41edo~~\|~~41edo~~]]<~~/li><li>[[43edo\|43edo]]</li><li>[[53edo\|53edo]]</li></ul~~>		[[Categorie:Prime EDO\| ]] <!-- main article -->

Xenwolf: Xenwolf moved page Prime edos to Prime EDO: better title (abreviation in capitals, singular case)

2018-11-02T08:41:14Z

Xenwolf moved page Prime edos to Prime EDO: better title (abreviation in capitals, singular case)

← Older revision	Revision as of 08:41, 2 November 2018
(No difference)

← Older revision		Revision as of 12:53, 4 July 2021
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	* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo\|12-EDO]])		* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo\|12-EDO]])
	* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone scale in ~~12EDO~~)		* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone scale in 12-EDO)
	* There are no [[Wikipedia: Modes of limited transposition\|modes of limited transpostion]], such as used by the composer Olivier Messiaen		* There are no [[Wikipedia: Modes of limited transposition\|modes of limited transpostion]], such as used by the composer Olivier Messiaen
	* There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)		* There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)
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	Primality may be desirable if you want, for example, a whole tone scale that is ''not'' absolutely uniform. In this case you might like [[19-EDO]] (with whole tone scale 3 3 3 3 3 4, [[MOS scale]] of type [[1L 5s]]) or [[17edo\|17-EDO]] (with whole tone scale 3 3 3 3 3 2, MOS scale of type [[5L 1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a MOS scale, one can reach every tone in ''n'' steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12edo\|12-EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo\|3-EDO]]).		Primality may be desirable if you want, for example, a whole tone scale that is ''not'' absolutely uniform. In this case you might like [[19-EDO]] (with whole tone scale 3 3 3 3 3 4, [[MOS scale]] of type [[1L 5s]]) or [[17edo\|17-EDO]] (with whole tone scale 3 3 3 3 3 2, MOS scale of type [[5L 1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a MOS scale, one can reach every tone in ''n'' steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12edo\|12-EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo\|3-EDO]]).

	A prime ~~edo~~ is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12-EDO. Since 12 is 2 × 2 × 3, it contains [[2edo\|2-EDO]], 3-EDO, [[4edo\|4-EDO]] and [[6edo\|6-EDO]]. All EDOs with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12-EDO, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo\|35-EDO]], will work just as well for this purpose.		A prime EDO is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12-EDO. Since 12 is 2 × 2 × 3, it contains [[2edo\|2-EDO]], 3-EDO, [[4edo\|4-EDO]] and [[6edo\|6-EDO]]. All EDOs with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12-EDO, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo\|35-EDO]], will work just as well for this purpose.

	If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[EDT\|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt\|27-EDT]] is a non-prime system very similar to 17-EDO, while [[19edt\|19-EDT (Stopper tuning)]] is a prime system very similar to the ubiquitous 12-EDO. (See [[EDT #EDT-EDO correspondence\|EDT-EDO correspondence]] for more of these.) Anyway, for every prime EDO system there is a non-prime [[ed4\|ED4]] system with identical step sizes.		If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[EDT\|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt\|27-EDT]] is a non-prime system very similar to 17-EDO, while [[19edt\|19-EDT (Stopper tuning)]] is a prime system very similar to the ubiquitous 12-EDO. (See [[EDT #EDT-EDO correspondence\|EDT-EDO correspondence]] for more of these.) Anyway, for every prime EDO system there is a non-prime [[ed4\|ED4]] system with identical step sizes.
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	The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.		The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.

	== The first 46 ~~Prime~~ EDOs ==		== The first 46 prime EDOs ==
	Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below:		Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below:

	[[2edo\|2]], [[3edo\|3]], [[5edo\|5]], [[7edo\|7]], [[11edo\|11]], [[13edo\|13]], [[17edo\|17]], ~~<br/>~~		[[2edo\|2]], [[3edo\|3]], [[5edo\|5]], [[7edo\|7]], [[11edo\|11]], [[13edo\|13]], [[17edo\|17]], [[19edo\|19]], <br>
	[[19edo\|19]], [[23edo\|23]], [[29edo\|29]], [[31edo\|31]], [[37edo\|37]], [[41edo\|41]], [[43edo\|43]], ~~<br/>~~		[[23edo\|23]], [[29edo\|29]], [[31edo\|31]], [[37edo\|37]], [[41edo\|41]], [[43edo\|43]], [[47edo\|47]], [[53edo\|53]], <br>
	[[47edo\|47]], [[53edo\|53]], [[59edo\|59]], [[61edo\|61]], [[67edo\|67]], [[71edo\|71]], [[73edo\|73]], ~~<br/>~~		[[59edo\|59]], [[61edo\|61]], [[67edo\|67]], [[71edo\|71]], [[73edo\|73]], [[79edo\|79]], [[83edo\|83]], [[89edo\|89]], <br>
	[[79edo\|79]], [[83edo\|83]], [[89edo\|89]], [[97edo\|97]], [[101edo\|101]], [[103edo\|103]], [[107edo\|107]], ~~<br/>~~		[[97edo\|97]], [[101edo\|101]], [[103edo\|103]], [[107edo\|107]], [[109edo\|109]], [[113edo\|113]], [[127edo\|127]], [[131edo\|131]], <br>
	[[109edo\|109]], [[113edo\|113]], [[127edo\|127]], [[131edo\|131]], [[137edo\|137]], [[139edo\|139]], [[149edo\|149]], ~~<br/>~~		[[137edo\|137]], [[139edo\|139]], [[149edo\|149]], [[151edo\|151]], [[157edo\|157]], [[163edo\|163]], [[167edo\|167]], [[173edo\|173]], <br>
	[[151edo\|151]], [[157edo\|157]], [[163edo\|163]], [[167edo\|167]], [[173edo\|173]], [[179edo\|179]], [[181edo\|181]], ~~<br/>~~		[[179edo\|179]], [[181edo\|181]], [[191edo\|191]], [[193edo\|193]], [[197edo\|197]], [[199edo\|199]].
	[[191edo\|191]], [[193edo\|193]], [[197edo\|197]], [[199edo\|199]].

	[[Category:EDO theory pages]]		[[Category:EDO theory pages]]
	[[Category:Prime EDO\| ]] <!-- main article -->		[[Category:Prime EDO\| ]] <!-- main article -->
	[[Category:Prime]]		[[Category:Prime]]

← Older revision		Revision as of 11:39, 4 July 2021
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	A '''prime ~~edo~~''' is an [[EDO]] with a [[prime number]] of different pitches per [[octave]].		A '''prime EDO''' is an [[EDO]] with a [[prime number]] of different pitches per [[octave]].

	== Prime numbers in EDOs ==		== Prime numbers in EDOs ==
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	Whether or not a number ''n'' is prime has important consequences for the properties of the corresponding ''n''-[[EDO]], especially for lower values of ''n''. In these instances:		Whether or not a number ''n'' is prime has important consequences for the properties of the corresponding ''n''-[[EDO]], especially for lower values of ''n''. In these instances:

	* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo\|~~12EDO~~]])		* There is ''no fully symmetric chord'' (such as the diminished seventh chord in [[12edo\|12-EDO]])
	* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the ~~wholetone~~ scale in 12EDO)		* Excepting the scale comprising all notes of the EDO, there is ''no absolutely uniform, octave-repeating scale'' (such as the whole tone scale in 12EDO)
	* There are no [~~http~~:~~//en.wikipedia.org/wiki/Modes_of_limited_transposition~~ modes of limited transpostion], such as used by the composer Olivier Messiaen		* There are no [[Wikipedia: Modes of limited transposition\|modes of limited transpostion]], such as used by the composer Olivier Messiaen
	* There is no support for rank-~~two~~ temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)		* There is no support for rank-2 temperaments whose period is a fraction of the octave (all such temperaments are ''linear'' temperaments)
	* Making a chain of any interval of the ''n''-EDO, one can reach every tone in ''n'' steps. (For composite EDOs, this works with intervals that are co-prime to ''n'', for example, 5 degrees of ~~12EDO~~)		* Making a chain of any interval of the ''n''-EDO, one can reach every tone in ''n'' steps. (For composite EDOs, this works with intervals that are co-prime to ''n'', for example, 5 degrees of 12-EDO)

	For these or similar reasons, some musicians do not like ~~the~~ prime EDOs (e.g. the makers of [http://www.armodue.com/risorse.htm Armodue]) and others love them.		For these or similar reasons, some musicians do not like prime EDOs (e.g. the makers of [http://www.armodue.com/risorse.htm Armodue]) and others love them.

	Primality may be desirable if you want, for example, a ~~wholetone~~ scale that is ''not'' absolutely uniform. In this case you might like [[~~19edo~~]] (with whole tone scale 3 3 3 3 3 4, MOS scale of type [[1L ~~5s\|1L+~~5s]]) or [[17edo\|~~17EDO~~]] (with whole tone scale 3 3 3 3 3 2, MOS ~~Scale~~ of type [[5L ~~1s\|5L+~~1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a ~~[[MOSScales\|Moment of Symmetry]]~~ scale, one can reach every tone in ''n'' steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12edo\|~~12EDO~~]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo\|~~3EDO~~]]).		Primality may be desirable if you want, for example, a whole tone scale that is ''not'' absolutely uniform. In this case you might like [[19-EDO]] (with whole tone scale 3 3 3 3 3 4, [[MOS scale]] of type [[1L 5s]]) or [[17edo\|17-EDO]] (with whole tone scale 3 3 3 3 3 2, MOS scale of type [[5L 1s]]). In general, making a chain of any interval of a prime ''n''-EDO, thus treating the interval as the generator of a MOS scale, one can reach every tone in ''n'' steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12edo\|12-EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo\|3-EDO]]).

	A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in ~~12edo~~. Since 12 is 223, it contains [[2edo\|~~2EDO~~]], ~~[[3edo\|3EDO]]~~, [[4edo\|~~4EDO~~]] and [[6edo\|~~6EDO~~]]. All ~~edos~~ with a 2, 3, 4, or 6 in their factorization will share at least one interval with ~~12edo~~, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo\|~~35EDO~~]], will work just as well for this purpose.		A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12-EDO. Since 12 is 2 × 2 × 3, it contains [[2edo\|2-EDO]], 3-EDO, [[4edo\|4-EDO]] and [[6edo\|6-EDO]]. All EDOs with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12-EDO, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo\|35-EDO]], will work just as well for this purpose.

	If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[~~Edt~~\|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt]] is a non-prime system very similar to ~~[[17edo]]~~, while [[~~19ED3~~\|~~19edt~~ (Stopper tuning)]] is a prime system very similar to the ubiquitous ~~[[12edo]]~~. (See [[~~Edt~~#EDT-EDO correspondence\|EDT-EDO correspondence]] for more of these.) Anyway, for every prime EDO system there is a non-prime [[~~Ed4~~\|ED4]] system with identical step sizes.		If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[EDT\|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt\|27-EDT]] is a non-prime system very similar to 17-EDO, while [[19edt\|19-EDT (Stopper tuning)]] is a prime system very similar to the ubiquitous 12-EDO. (See [[EDT #EDT-EDO correspondence\|EDT-EDO correspondence]] for more of these.) Anyway, for every prime EDO system there is a non-prime [[ed4\|ED4]] system with identical step sizes.

	The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.		The larger ''n'' is, the less these points matter, since the difference between an ''absolutely'' uniform scale and an approximated, ''nearly'' uniform scale eventually become inaudible.

	~~[TODO: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here.]~~

	== The first 46 Prime EDOs ==		== The first 46 Prime EDOs ==