Huygens vs meanpop: Difference between revisions

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'''Undecimal meantone''' (also known as '''huygens''') and '''meanpop''', both discussed at [[meantone family]], are two different temperaments in the 11-limit. This page compares and contrasts them in detail.

'''Undecimal meantone''' (also known as '''huygens''') and '''meanpop''', both discussed at [[meantone family]], are two different temperaments in the [[11-limit]]. This page compares and contrasts them in detail.

Extending meantone from the 5-limit to the 7-limit, there is one obvious mapping (for standard meantone tunings) which ~~doesn't~~ split the fifth that is not too complex and adds hardly any additional error (so we~~'re~~ not talking about dominant temperament here). This is called "7-limit meantone" or "septimal meantone" and is an amazingly efficient (and beautiful) temperament. But extending it from the 7-limit to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: huygens (12&~~amp;~~31) and meanpop (19&~~amp;~~31).

Extending meantone from the [[5-limit]] to the [[7-limit]], there is one obvious mapping (for standard meantone tunings) which does not split the fifth that is not too complex and adds hardly any additional error (so we are not talking about [[dominant (temperament)|dominant]] here). This is called ''7-limit meantone'' or ''septimal meantone'' and is an amazingly efficient and beautiful temperament. But extending it from the 7-limit to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: huygens (12 & 31) and meanpop (19 & 31).

In 11-limit huygens, 11/8 is represented by the doubly augmented third, for example ~~C-Ex (where "x" represents the standard double sharp symbol, equivalent in meaning to "##")~~. This is 18 fifths along the ~~circle~~ of fifths; Ex is 18 fifths up from C. Huygens is tuned best sharp of 31edo, around 697 cents.

In 11-limit huygens, 11/8 is represented by the doubly augmented third, for example C–E𝄪. This is 18 fifths along the [[chain of fifths]]; E𝄪 is 18 fifths up from C. Huygens is tuned best sharp of 31edo, around 697 cents.

In meanpop, 11/8 is represented by the doubly diminished fifth, for example ~~C-Gbb~~. This is in the opposite direction along the circle of fifths - 13 fifths down. Meanpop is tuned best flat of 31edo, around 696 cents.

In meanpop, 11/8 is represented by the doubly diminished fifth, for example C–G𝄫. This is in the opposite direction along the circle of fifths – 13 fifths down. Meanpop is tuned best flat of 31edo, around 696 cents.

In 13-limit, ~~they extend~~ by ~~the~~ 105/104 ~~comma. Alternatively huygens extends~~ into grosstone ~~by 144/143~~.

In the [[13-limit]], meanpop extends by [[105/104]], whereas meantone forks into fokkertone, grosstone, and meridetone.

Can huygens and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is [[31edo]]. In 31edo the circle of fifths closes perfectly after 31 fifths, so Ex and ~~Gbb~~ are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, setting a minor third equal to four chromatic semitones. Expressed in tempered fifths and octave-reduced, this interval is the 31-comma [-49 ~~31⟩~~, which is the 3-limit comma tempered out in 31edo.) This makes everything much simpler and results in 121/120 and 243/242 being tempered out, so that 12/11~11/10 is a "neutral second" (exactly half of a minor third), and 11/9 is a "neutral third" (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either huygens or meanpop.

Can huygens and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is [[31edo]]. In 31edo the circle of fifths closes perfectly after 31 fifths, so E𝄪 and G𝄫 are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, setting a minor third equal to four chromatic semitones. Expressed in tempered fifths and octave-reduced, this interval is the [[31-comma]] {{monzo| -49 31 }}, which is the 3-limit comma tempered out in 31edo.) This makes everything much simpler and results in [[121/120]] and [[243/242]] being tempered out, so that 12/11~11/10 is a true neutral second (exactly half of a minor third), and 11/9 is a true neutral third (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either huygens or meanpop.

== Interval chain ==

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 {{Breadcrumb|Meantone}}
-{{Wikipedia| Septimal meantone temperament#11-limit meantone }}
+{{Wikipedia|Septimal meantone temperament #11-limit meantone}}
-'''Undecimal meantone''' (also known as '''huygens''') and '''meanpop''', both discussed at [[meantone family]], are two different temperaments in the 11-limit. This page compares and contrasts them in detail.
+'''Undecimal meantone''' (also known as '''huygens''') and '''meanpop''', both discussed at [[meantone family]], are two different temperaments in the [[11-limit]]. This page compares and contrasts them in detail.
-Extending meantone from the 5-limit to the 7-limit, there is one obvious mapping (for standard meantone tunings) which doesn't split the fifth that is not too complex and adds hardly any additional error (so we're not talking about dominant temperament here). This is called "7-limit meantone" or "septimal meantone" and is an amazingly efficient (and beautiful) temperament. But extending it from the 7-limit to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: huygens (12&amp;31) and meanpop (19&amp;31).
+Extending meantone from the [[5-limit]] to the [[7-limit]], there is one obvious mapping (for standard meantone tunings) which does not split the fifth that is not too complex and adds hardly any additional error (so we are not talking about [[dominant (temperament)|dominant]] here). This is called ''7-limit meantone'' or ''septimal meantone'' and is an amazingly efficient and beautiful temperament. But extending it from the 7-limit to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: huygens (12 & 31) and meanpop (19 & 31).
-In 11-limit huygens, 11/8 is represented by the doubly augmented third, for example C-Ex (where "x" represents the standard double sharp symbol, equivalent in meaning to "##"). This is 18 fifths along the circle of fifths; Ex is 18 fifths up from C. Huygens is tuned best sharp of 31edo, around 697 cents.
+In 11-limit huygens, 11/8 is represented by the doubly augmented third, for example C–E𝄪. This is 18 fifths along the [[chain of fifths]]; E𝄪 is 18 fifths up from C. Huygens is tuned best sharp of 31edo, around 697 cents.
-In meanpop, 11/8 is represented by the doubly diminished fifth, for example C-Gbb. This is in the opposite direction along the circle of fifths - 13 fifths down. Meanpop is tuned best flat of 31edo, around 696 cents.
+In meanpop, 11/8 is represented by the doubly diminished fifth, for example C–G𝄫. This is in the opposite direction along the circle of fifths – 13 fifths down. Meanpop is tuned best flat of 31edo, around 696 cents.
-In 13-limit, they extend by the 105/104 comma. Alternatively huygens extends into grosstone by 144/143.
+In the [[13-limit]], meanpop extends by [[105/104]], whereas meantone forks into fokkertone, grosstone, and meridetone.
-Can huygens and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is [[31edo]]. In 31edo the circle of fifths closes perfectly after 31 fifths, so Ex and Gbb are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, setting a minor third equal to four chromatic semitones. Expressed in tempered fifths and octave-reduced, this interval is the 31-comma [-49 31⟩, which is the 3-limit comma tempered out in 31edo.) This makes everything much simpler and results in 121/120 and 243/242 being tempered out, so that 12/11~11/10 is a "neutral second" (exactly half of a minor third), and 11/9 is a "neutral third" (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either huygens or meanpop.
+Can huygens and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is [[31edo]]. In 31edo the circle of fifths closes perfectly after 31 fifths, so E𝄪 and G𝄫 are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, setting a minor third equal to four chromatic semitones. Expressed in tempered fifths and octave-reduced, this interval is the [[31-comma]] {{monzo| -49 31 }}, which is the 3-limit comma tempered out in 31edo.) This makes everything much simpler and results in [[121/120]] and [[243/242]] being tempered out, so that 12/11~11/10 is a true neutral second (exactly half of a minor third), and 11/9 is a true neutral third (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either huygens or meanpop.
 == Interval chain ==