159edo: Difference between revisions

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=== Mappings ===

A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[Overtone series|harmonics]] with [[53edo]]. However, compared to 53edo, the patent vals differ on the mappings for 7, 11 and 17 – in fact, this EDO has a very accurate 11 and an only slightly less accurate 17. Although 159edo is [[consistent]] up to the 17-odd-limit, it proves to be inconsistent in the 19-odd-limit, with the 19th harmonic having multiple reasonable mappings. Furthermore, even though 159edo is [[contorted]] in the 5-limit, there is a perfect match between the [[direct mapping]] and the more complicated traditional mapping for an [[octave-reduced]] stack of fifty-three tempered [[3/2]] perfect fifths – a complete [[circle of fifths]] for this EDO – as [[Mercator's comma]], which this EDO tempers out, is less than half the size of a single step in this EDO. However, for intervals such as [[49/32]] and [[128/125]], these two mappings don't match.

A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[Overtone series|harmonics]] with [[53edo]]. However, compared to 53edo, the patent vals differ on the mappings for 7, 11 and 17 – in fact, this EDO has a very accurate 11 and an only slightly less accurate 17. Although 159edo is [[consistent]] up to the 17-odd-limit, it proves to be inconsistent in the 19-odd-limit, with the 19th harmonic having multiple reasonable mappings. Furthermore, even though 159edo is [[contorted]] in the 5-limit, there is a perfect match between the [[direct mapping]] and the more complicated traditional mapping for an [[octave-reduced]] stack of fifty-three tempered [[3/2]] perfect fifths – a complete [[circle of fifths]] for this EDO – as [[Mercator's comma]], which this EDO tempers out, is less than half the size of a single step in this EDO. This means that 159edo demonstrates 3-to-2 [[telicity]] – in fact, 159edo is the largest EDO divisible by 53 for which this is the case. However, for intervals such as [[49/32]] and [[128/125]], these two mappings don't match. While the [[patent val]] supports supports [[Mercator family#Cartography|cartography]] temperament, which is among the best 13-limit temperaments in the [[Mercator family]], the 159d and 159e mapping support others in that family.

=== Commas ===

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In the 11-limit, it tempers out not only [[385/384]], [[441/440]], [[3025/3024]], [[4000/3993]], [[4375/4356]], and [[6250/6237]], but both [[Nexus comma|1771561/1769472]] and [[Quartisma|117440512/117406179]], which in turn means that [[Symbiotic comma|19712/19683]] is tempered out as well.

In the 13-limit it tempers out [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], [[6656/6655]], [[cantonisma|10985/10976]] and 13720/13689~~. This means, among other things, that 159edo supports 13-limit [[Mercator family#Cartography|cartography]] temperament~~.

In the 13-limit it tempers out [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], [[6656/6655]], [[cantonisma|10985/10976]] and 13720/13689.

In the 17-limit tempers out [[273/272]], 375/374, 595/594, 715/714, [[936/935]], [[1089/1088]], [[1701/1700]], 8624/8619, [[11271/11264]] and 15379/15300.

@@ Line 13: / Line 13: @@
 === Mappings ===
-A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[Overtone series|harmonics]] with [[53edo]]. However, compared to 53edo, the patent vals differ on the mappings for 7, 11 and 17 – in fact, this EDO has a very accurate 11 and an only slightly less accurate 17.  Although 159edo is [[consistent]] up to the 17-odd-limit, it proves to be inconsistent in the 19-odd-limit, with the 19th harmonic having multiple reasonable mappings.  Furthermore, even though 159edo is [[contorted]] in the 5-limit, there is a perfect match between the [[direct mapping]] and the more complicated traditional mapping for an [[octave-reduced]] stack of fifty-three tempered [[3/2]] perfect fifths –  a complete [[circle of fifths]] for this EDO – as [[Mercator's comma]], which this EDO tempers out, is less than half the size of a single step in this EDO.  However, for intervals such as [[49/32]] and [[128/125]], these two mappings don't match.
+A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[Overtone series|harmonics]] with [[53edo]]. However, compared to 53edo, the patent vals differ on the mappings for 7, 11 and 17 – in fact, this EDO has a very accurate 11 and an only slightly less accurate 17.  Although 159edo is [[consistent]] up to the 17-odd-limit, it proves to be inconsistent in the 19-odd-limit, with the 19th harmonic having multiple reasonable mappings.  Furthermore, even though 159edo is [[contorted]] in the 5-limit, there is a perfect match between the [[direct mapping]] and the more complicated traditional mapping for an [[octave-reduced]] stack of fifty-three tempered [[3/2]] perfect fifths – a complete [[circle of fifths]] for this EDO – as [[Mercator's comma]], which this EDO tempers out, is less than half the size of a single step in this EDO.  This means that 159edo demonstrates 3-to-2 [[telicity]] – in fact, 159edo is the largest EDO divisible by 53 for which this is the case.  However, for intervals such as [[49/32]] and [[128/125]], these two mappings don't match.  While the [[patent val]] supports supports [[Mercator family#Cartography|cartography]] temperament, which is among the best 13-limit temperaments in the [[Mercator family]], the 159d and 159e mapping support others in that family.
 === Commas ===
@@ Line 24: / Line 24: @@
 In the 11-limit, it tempers out not only [[385/384]], [[441/440]], [[3025/3024]], [[4000/3993]], [[4375/4356]], and [[6250/6237]], but both [[Nexus comma|1771561/1769472]] and [[Quartisma|117440512/117406179]], which in turn means that [[Symbiotic comma|19712/19683]] is tempered out as well.
-In the 13-limit it tempers out [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], [[6656/6655]], [[cantonisma|10985/10976]] and 13720/13689.  This means, among other things, that 159edo supports 13-limit [[Mercator family#Cartography|cartography]] temperament.
+In the 13-limit it tempers out [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], [[6656/6655]], [[cantonisma|10985/10976]] and 13720/13689.
 In the 17-limit tempers out [[273/272]], 375/374, 595/594, 715/714, [[936/935]], [[1089/1088]], [[1701/1700]], 8624/8619, [[11271/11264]] and 15379/15300.

159edo: Difference between revisions

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