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.

Extending meantone from the 5-limit to the 7-limit, there is one obvious mapping 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: ~~11-limit meantone~~ (12&31) and meanpop (19&31).

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&31) and meanpop (19&31).

In 11-limit ~~meantone~~, 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.

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 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.

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

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

Can ~~meantone~~ 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, if that makes any sense to you.) 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 ~~meantone~~ 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 Ex and Gbb are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, if that makes any sense to you.) 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.

{| class="wikitable center-3 center-5"

! rowspan="2" | JI Interval

! colspan="2" | ~~Meantone~~ mapping

! colspan="2" | Huygens mapping

! colspan="2" | Meanpop mapping

|-

@@ Line 5: / Line 5: @@
 '''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 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: 11-limit meantone (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 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).
-In 11-limit meantone, 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.
+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 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.
+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 13-limit, they extend by the 105/104 comma. Alternatively meantone extends into grosstone by 144/143.
+In 13-limit, they extend by the 105/104 comma. Alternatively huygens extends into grosstone by 144/143.
-Can meantone 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, if that makes any sense to you.) 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 meantone 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 Ex and Gbb are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, if that makes any sense to you.) 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.
 {| class="wikitable center-3 center-5"
 ! rowspan="2" | JI Interval
-! colspan="2" | Meantone mapping
+! colspan="2" | Huygens mapping
 ! colspan="2" | Meanpop mapping
 |-