441/440: Difference between revisions

Interval information
Ratio	441/440
Factorization	2-3 × 32 × 5-1 × 72 × 11-1
Monzo	[-3 2 -1 2 -1⟩
Size in cents	3.930158¢
Name	werckisma
Color name	1uzzg2, luzozogu 2nd,; Luzozogu comma
FJS name	[math]\displaystyle{ \text{m2}^{7,7}_{5,11} }[/math]
Special properties	square superparticular,; reduced
Tenney norm (log2 nd)	17.566
Weil norm (log2 max(n, d))	17.5693
Wilson norm (sopfr(nd))	42
Comma size	small
S-expression	S21
	Open this interval in xen-calc

Revision as of 11:22, 30 November 2025

441/440, the werckisma, also known as Werckmeister's undecimal septenarian schisma, is a small superparticular 11-limit comma with a size of roughly 3.93 cents. It is the difference between 21/20 and 22/21, and between 80/63 ( = (10/9)⋅(8/7)) and 14/11, as well as the amount by which a stack of two 21/16 septimal subfourths exceeds a single 55/32 supermajor sixth. It also arises as the following comma differences: (49/48)/(55/54), (56/55)/(64/63), (81/80)/(99/98), (126/125)/(176/175), and (243/242)/(540/539).

Tempering it out splits 11/10 into two even halves, equates 21/16 with 55/42 to create some interesting-sounding triads, and makes werckismic chords possible. See Rank-4 temperament #Werckismic (441/440) for some technical data. See Werckismic temperaments for a collection of rank-3 temperaments where it is tempered out.

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-'''441/440''', also known as '''Werckmeister's undecimal septenarian schisma''', or more simply, the '''werckisma''', is a [[small comma|small]] [[superparticular]] [[11-limit]] comma with a size of roughly 3.93 [[cent]]s. It is the difference between [[21/20]] and [[22/21]], and between [[80/63]] ( = ([[10/9]])([[8/7]])) and [[14/11]], as well as the amount by which a stack of two [[21/16]] septimal subfourths exceeds a single [[55/32]] supermajor sixth. It also arises as the following comma differences: ([[49/48]])/([[55/54]]), ([[56/55]])([[64/63]]), ([[81/80]])/([[99/98]]), ([[126/125]])/([[176/175]]), and ([[243/242]])/([[540/539]]).
+'''441/440''', the '''werckisma''', also known as ''Werckmeister's undecimal septenarian schisma'', is a [[small comma|small]] [[superparticular]] [[11-limit]] [[comma]] with a size of roughly 3.93 [[cent]]s. It is the difference between [[21/20]] and [[22/21]], and between [[80/63]] ( = ([[10/9]])⋅([[8/7]])) and [[14/11]], as well as the amount by which a stack of two [[21/16]] septimal subfourths exceeds a single [[55/32]] supermajor sixth. It also arises as the following comma differences: ([[49/48]])/([[55/54]]), ([[56/55]])/([[64/63]]), ([[81/80]])/([[99/98]]), ([[126/125]])/([[176/175]]), and ([[243/242]])/([[540/539]]).
 == Temperaments ==