Father–3 equivalence continuum: Difference between revisions

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The '''chromatic-diatonic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[25/24|chromatic semitones (25/24)]] with [[16/15|diatonic semitones (16/15)]].
The '''father–3 equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[16/15|classical diatonic semitones (16/15)]] with the [[32/27|Pythagorean minor third (32/27)]].


All temperaments in the continuum satisfy (25/24)<sup>''n''</sup> ~ 16/15. Varying ''n'' results in different temperaments listed in the table below. It converges to [[dicot]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.58097..., and temperaments having ''n'' near this value tend to be the most accurate ones.  
Note that because 3et is a record equal temperament in the [[2.5 subgroup]], the continuum can be conceptualized as the [[Father–3 equivalence continuum/Godtone's approach|''augmented–dicot equivalence continuum'']], which Godtone argues is easier to understand, with characteristic 2.5-subgroup [[comma]] [[128/125]] as the interval with a single factor of 3 is [[25/24]].


{| class="wikitable center-1 center-2"
All temperaments in the continuum satisfy {{nowrap|(16/15)<sup>''n''</sup> ~ 32/27}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[father]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 2.63252…, and temperaments having ''n'' near this value tend to be the most accurate ones.
|+ Temperaments in the continuum
 
32/27 is the characteristic 3-limit comma tempered out in 3edo. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|3]] in the generator chain.
 
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Very low accuracy temperaments #Alteraugment|Alteraugment]]
| [[32/27]]
| {{Monzo| 5 -3 }}
|-
| 1
| [[Very low accuracy temperaments #Antonian|Antonian]]
| [[10/9]]
| {{Monzo| 1 -2 1 }}
|-
| 2
| [[Dicot]]
| [[25/24]]
| {{Monzo| -3 -1 2 }}
|-
| 3
| [[Augmented (temperament)|Augmented]]
| [[128/125]]
| {{Monzo| 7 0 -3 }}
|-
| 4
| [[Smate]]
| [[2048/1875]]
| {{Monzo| 11 -1 -4 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Father]]
| [[16/15]]
| {{Monzo| 4 -1 -1 }}
|}
We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''antonian–3 equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.61255…
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments with integer ''m''
|-
! rowspan="2" | ''m''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Very low accuracy temperaments #Alteraugment|Alteraugment]]
| [[32/27]]
| {{Monzo| 5 -3 }}
|-
| 1
| [[Father]]
| [[16/15]]
| {{Monzo| 4 -1 -1 }}
|-
| 2
| [[Dicot]]
| [[25/24]]
| {{Monzo| -3 -1 2 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Very low accuracy temperaments #Antonian|Antonian]]
| [[10/9]]
| {{Monzo| 1 -2 1 }}
|}
{| class="wikitable"
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m''
|-
! ''n'' !! ''m''!! Temperament || Comma
|-
| 7/3 = 2.{{overline|3}} || 7/4 = 1.75 || [[Wesley]] || {{monzo| 13 2 -7 }}
|-
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| 10 1 -5 }}
|-
| 29/11 = 2.{{overline|63}} || 29/18 = 1.6{{overline|1}} || [[Squarschmidt]] || {{monzo| 61 4 -29 }}
|-
| 8/3 = 2.{{overline|6}} || 8/5 = 1.6 || [[Würschmidt]] || {{monzo| 17 1 -8 }}
|-
| 19/7 = 2.{{overline|714285}} || 19/12 = 1.58{{overline|3}} || [[#Isnes|Isnes]] || {{monzo| 41 2 -19 }}
|-
| 11/4 = 2.75 || 11/7 = 1.{{overline|571428}} || [[Magus]] || {{monzo| 24 1 -11 }}
|}
Because 3et is a record equal temperament in the 2.5 subgroup, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is 128/125, and the interval with a single factor of 3 is 25/24. As such, Godtone has conceptualized this continuum as ''augmented–dicot equivalence continuum''. See [[{{PAGENAME}}/Godtone's approach]].
Others prefer conceptualizing this continuum in terms of {{nowrap| ''k'' {{=}} {{sfrac|1|''n'' − 2}} }} such that temperaments satisfy {{nowrap|(25/24)<sup>''k''</sup> {{=}} 16/15}}. This gives rise to the name ''chromatic–diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097…
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments with integer ''k''
|-
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 14: Line 124:
|-
|-
| -1
| -1
| [[Yo]]
| [[Very low accuracy temperaments #Antonian|Antonian]]
| [[10/9]]
| [[10/9]]
| {{monzo|1 -2 1}}
| {{Monzo| 1 -2 1 }}
|-
|-
| 0
| 0
| [[Father]]
| [[Father]]
| [[16/15]]
| [[16/15]]
| {{monzo|4 -1 -1}}
| {{Monzo| 4 -1 -1 }}
|-
|-
| 1
| 1
| [[Augmented]]
| [[Augmented (temperament)|Augmented]]
| [[128/125]]
| [[128/125]]
| {{monzo|7 0 -3}}
| {{Monzo| 7 0 -3 }}
|-
|-
| 2
| 2
| [[Magic]]
| [[Magic]]
| [[3125/3072]]
| [[3125/3072]]
| {{monzo|10 1 -5}}
| {{Monzo| 10 1 -5 }}
|-
|-
| 3
| 3
| [[Wesley family|Wesley]]
| [[Wesley]]
| 78125/73728
| 78125/73728
| {{monzo| 13 2 -7}}
| {{monzo| 13 2 -7 }}
|-
|-
| 4
| 4
| 3 & 33c
| 3 & 33c
| 1953125/1769472
| 1953125/1769472
| {{monzo| 16 3 -9 }}
| {{Monzo| 16 3 -9 }}
|-
|-
| …
| …
Line 51: Line 161:
| [[Dicot]]
| [[Dicot]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2}}
| {{Monzo| -3 -1 2 }}
|}
|}


Examples of temperaments with fractional values of ''n'':
== 3 & 33c ==
* [[Very low accuracy temperaments#Alteraugment|Alteraugment]] (''n'' = -0.5)
This low-accuracy high-complexity temperament corresponds to {{nowrap| ''n'' {{=}} 9/4 }} and {{nowrap| ''m'' {{=}} 9/5 }}.
* [[Symbolic]] (''n'' = 0.5)
* [[Würschmidt]] (''n'' = 1.5)
* Isnes (''n'' = 1.6)
* [[Sensamagic clan#Magus|Magus]] (''n'' = 5/3 = 1.{{Overline|6}})


== 3 & 33c ==
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 1953125/1769472
 
{{Mapping|legend=1| 3 2 6 | 0 3 1 }}
: mapping generators: ~125/96, ~5/4


Comma list: {{monzo| 16 3 -9 }}
[[Optimal tuning]]s:
* [[WE]]: ~125/96 = 401.2633{{c}}, ~5/4 = 367.0585{{c}} (~25/24 = 34.2047{{c}})
: [[error map]]: {{val| +3.790 +1.747 -11.676 }}
* [[CWE]]: ~125/96 = 400.0000{{c}}, ~5/4 = 366.8103{{c}} (~25/24 = 33.1897{{c}})
: error map: {{val| 0.000 -1.524 -19.503 }}


POTE generator: 34.0971 cents
{{Optimal ET sequence|legend=1| 3, …, 33c, 36c, 69cc }}


Mapping: [{{val| 3 5 7 }}, {{val| 0 -3 -1 }}]
[[Badness]] (Sintel): 16.0


{{Val list|legend=1| 3, 6, 9b, 33c }}
== Isnes ==
Isnes is so called because the generator is half of a [[5/2]] major tenth, in a similar way that [[sensi]] has a generator of half a [[5/3]] major sixth. This corresponds to {{nowrap|''n'' {{=}} 19/7 }} and {{nowrap| ''m'' {{=}} 19/12 }}.


[http://x31eq.com/cgi-bin/rt.cgi?ets=3_33c&limit=5 The temperament finder - 5-limit 3 & 33c]
[[Subgroup]]: 2.3.5


== Symbolic ==
[[Comma list]]: {{monzo| 41 2 -19 }}


Comma list: [[2048/1875]]
{{Mapping|legend=1| 1 -11 1 | 0 19 2 }}
: mapping generators: ~2, ~3145728/1953125


POTE generator: ~5/4 = 420.8548 cents
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2782{{c}}, ~3145728/1953125 = 794.4174{{c}}
: [[error map]]: {{val| -0.722 -0.090 +1.799 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3145728/1953125 = 794.8728{{c}}
: error map: {{val| 0.000 +0.628 +3.432 }}


Mapping: [{{val| 1 3 2 }}, {{val| 0 -4 1 }}]
{{Optimal ET sequence|legend=1| 3, 71b, 74, 77, 157, 548ccc }}


{{Val list|legend=1| 3, 6, 11, 14, 17c }}
[[Badness]] (Sintel): 30.4


[http://x31eq.com/cgi-bin/rt.cgi?ets=3_14p&limit=5 The temperament finder - 5-limit symbolic]
== Squarschmidt (5-limit) ==
: ''For extensions, see [[Hemimage temperaments #Squarschmidt]].''


== Isnes ==
A generator for the squarschmidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents.


So called because the generator is half of a [[8/5]] minor sixth, in a similar way that [[sensi]] has a generator of half a [[5/3]].
[[Subgroup]]: 2.3.5


Comma list: {{Monzo|41 2 -19}}
[[Comma list]]: {{monzo| 61 4 -29 }}


POTE generator: 405.1047 cents
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
: mapping generators: ~2, ~98304/78125


Mapping: [{{val| 1 8 3 }}, {{val| 0 -19 -2 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9653{{c}}, ~98304/78125 = 396.6094{{c}}
: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98304/78125 = 396.6201{{c}}
: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}


{{Val list|legend=1| 3, 74, 77, 80, 83, 154, 157, 160 }}
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947, 9588cc, 10535cc, 11482ccc }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=3_77&limit=5 The temperament finder - 5-limit 3 & 77]
[[Badness]] (Sintel): 5.12


[[Category:3edo]]
[[Category:3edo]]
[[Category:Theory]]
[[Category:Temperament]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Latest revision as of 07:43, 11 June 2026

The father–3 equivalence continuum is a continuum of 5-limit temperaments which equate a number of classical diatonic semitones (16/15) with the Pythagorean minor third (32/27).

Note that because 3et is a record equal temperament in the 2.5 subgroup, the continuum can be conceptualized as the augmented–dicot equivalence continuum, which Godtone argues is easier to understand, with characteristic 2.5-subgroup comma 128/125 as the interval with a single factor of 3 is 25/24.

All temperaments in the continuum satisfy (16/15)n ~ 32/27. Varying n results in different temperaments listed in the table below. It converges to father as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 3edo due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of n is approximately 2.63252…, and temperaments having n near this value tend to be the most accurate ones.

32/27 is the characteristic 3-limit comma tempered out in 3edo. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of 3 in the generator chain.

Temperaments with integer n
n Temperament Comma
Ratio Monzo
0 Alteraugment 32/27 [5 -3⟩
1 Antonian 10/9 [1 -2 1⟩
2 Dicot 25/24 [-3 -1 2⟩
3 Augmented 128/125 [7 0 -3⟩
4 Smate 2048/1875 [11 -1 -4⟩
… … … …
∞ Father 16/15 [4 -1 -1⟩

We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the antonian–3 equivalence continuum, which is essentially the same thing. The just value of m is 1.61255…

Temperaments with integer m
m Temperament Comma
Ratio Monzo
0 Alteraugment 32/27 [5 -3⟩
1 Father 16/15 [4 -1 -1⟩
2 Dicot 25/24 [-3 -1 2⟩
… … … …
∞ Antonian 10/9 [1 -2 1⟩
Temperaments with fractional n and m
n m Temperament Comma
7/3 = 2.3 7/4 = 1.75 Wesley [13 2 -7⟩
5/2 = 2.5 5/3 = 1.6 Magic [10 1 -5⟩
29/11 = 2.63 29/18 = 1.61 Squarschmidt [61 4 -29⟩
8/3 = 2.6 8/5 = 1.6 Würschmidt [17 1 -8⟩
19/7 = 2.714285 19/12 = 1.583 Isnes [41 2 -19⟩
11/4 = 2.75 11/7 = 1.571428 Magus [24 1 -11⟩

Because 3et is a record equal temperament in the 2.5 subgroup, there is another way to conceptualize this continuum. The characteristic 2.5-subgroup comma is 128/125, and the interval with a single factor of 3 is 25/24. As such, Godtone has conceptualized this continuum as augmented–dicot equivalence continuum. See Father–3 equivalence continuum/Godtone's approach.

Others prefer conceptualizing this continuum in terms of k = ⁠1/n − 2⁠ such that temperaments satisfy (25/24)k = 16/15. This gives rise to the name chromatic–diatonic equivalence continuum, where both chromatic and diatonic refer to the classical versions of semitones. The just value of k is approximately 1.58097…

Temperaments with integer k
k Temperament Comma
Ratio Monzo
-1 Antonian 10/9 [1 -2 1⟩
0 Father 16/15 [4 -1 -1⟩
1 Augmented 128/125 [7 0 -3⟩
2 Magic 3125/3072 [10 1 -5⟩
3 Wesley 78125/73728 [13 2 -7⟩
4 3 & 33c 1953125/1769472 [16 3 -9⟩
… … … …
∞ Dicot 25/24 [-3 -1 2⟩

3 & 33c

This low-accuracy high-complexity temperament corresponds to n = 9/4 and m = 9/5.

Subgroup: 2.3.5

Comma list: 1953125/1769472

Mapping: [⟨3 2 6], ⟨0 3 1]]

mapping generators: ~125/96, ~5/4

Optimal tunings:

  • WE: ~125/96 = 401.2633 ¢, ~5/4 = 367.0585 ¢ (~25/24 = 34.2047 ¢)
error map: ⟨+3.790 +1.747 -11.676]
  • CWE: ~125/96 = 400.0000 ¢, ~5/4 = 366.8103 ¢ (~25/24 = 33.1897 ¢)
error map: ⟨0.000 -1.524 -19.503]

Optimal ET sequence: 3, …, 33c, 36c, 69cc

Badness (Sintel): 16.0

Isnes

Isnes is so called because the generator is half of a 5/2 major tenth, in a similar way that sensi has a generator of half a 5/3 major sixth. This corresponds to n = 19/7 and m = 19/12.

Subgroup: 2.3.5

Comma list: [41 2 -19⟩

Mapping: [⟨1 -11 1], ⟨0 19 2]]

mapping generators: ~2, ~3145728/1953125

Optimal tunings:

  • WE: ~2 = 1199.2782 ¢, ~3145728/1953125 = 794.4174 ¢
error map: ⟨-0.722 -0.090 +1.799]
  • CWE: ~2 = 1200.0000 ¢, ~3145728/1953125 = 794.8728 ¢
error map: ⟨0.000 +0.628 +3.432]

Optimal ET sequence: 3, 71b, 74, 77, 157, 548ccc

Badness (Sintel): 30.4

Squarschmidt (5-limit)

For extensions, see Hemimage temperaments #Squarschmidt.

A generator for the squarschmidt temperament is the fourth root of 5/2, (5/2)1/4, tuned around 396.6 cents.

Subgroup: 2.3.5

Comma list: [61 4 -29⟩

Mapping: [⟨1 -8 1], ⟨0 29 4]]

mapping generators: ~2, ~98304/78125

Optimal tunings:

  • WE: ~2 = 1199.9653 ¢, ~98304/78125 = 396.6094 ¢
error map: ⟨-0.099 +0.543 +0.029 -0.719]
  • CWE: ~2 = 1200.0000 ¢, ~98304/78125 = 396.6201 ¢
error map: ⟨0.000 +0.653 +0.253 -0.552]

Optimal ET sequence: 118, 593, 711, 829, 947, 9588cc, 10535cc, 11482ccc

Badness (Sintel): 5.12

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