Dicot: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
mNo edit summary
"It is also the first fully prototypical dicot temperament" only makes sense due to the hyperlink
 
(5 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{Infobox regtemp
{{Infobox regtemp
| Title = Dicot
| Title = Dicot
| Subgroups = 2.3.5
| Subgroups = 2.3.5, 2.3.5.11
| Comma basis = [[25/24]]
| Comma basis = [[25/24]] (2.3.5);<br>[[25/24]], [[45/44]] (2.3.5.11)
| Edo join 1 = 3 | Edo join 2 = 7
| Edo join 1 = 7 | Edo join 2 = 10
| Mapping = 1; 2 1
| Mapping = 1; 2 1 5
| Generators = 6/5 | Generators tuning = 351.1 | Optimization method = CWE
| Generators = 6/5 | Generators tuning = 351.1 | Optimization method = CWE
| MOS scales = [[3L 1s]], [[3L 4s]], [[7L 3s]]
| MOS scales = [[3L 1s]], [[3L 4s]], [[7L 3s]]
| Pergen = (P8, P5/2)
| Pergen = (P8, P5/2)
| Odd limit 1 = 5 | Mistuning 1 = 35.3 | Complexity 1 = 3
| Odd limit 1 = 5 | Mistuning 1 = 35.3 | Complexity 1 = 3
| Odd limit 2 = 5-limit 9 | Mistuning 2 = 35.3 | Complexity 2 = 7
| Odd limit 2 = 2.3.5.11 15 | Mistuning 2 = 35.3 | Complexity 2 = 7
}}
}}
{{About|the regular temperament|the ploidacot signature|Ploidacot/Dicot}}
{{About|the regular temperament|the ploidacot signature|Ploidacot/Dicot}}


'''Dicot''' is an [[exotemperament]] that [[tempering out|tempers out]] [[25/24]]. It is also the first fully prototypical [[ploidacot/Dicot|dicot]] temperament. It tempers [[6/5]] and [[5/4]] into the same [[neutral third]] interval, which, when the fifth is tuned pure, is [[sqrt(3/2)]]. It is useful to represent the structure of [[5-limit]] harmonies without fully representing them in its greater accuracy, with [[mos scale]]s [[3L 4s]] and [[7L 3s]].  
'''Dicot''' is an [[exotemperament]] that [[tempering out|tempers out]] [[25/24]]. This means it tempers [[6/5]] and [[5/4]] into the same [[neutral third]] interval, which, when the fifth is tuned pure, is [[sqrt(3/2)]]. It is useful to represent the structure of [[5-limit]] harmonies without fully representing them in its greater accuracy, with [[mos scale]]s [[3L 4s]] and [[7L 3s]].  


It can be extended by tempering out [[15/14]] and [[36/35]] in the [[7-limit]], though this could turn the [[3L 4s]] [[mos]] into a [[4L 3s]] [[mos]]. This makes [[7/6]] and [[9/7]] equated to the neutral third, viewing [[6:7:9]] as a tertian chord.  
It can be extended by tempering out [[15/14]] and [[36/35]] in the [[7-limit]], called ''[[mujannabic]]'', though this could turn the [[3L 4s]] [[mos]] into a [[4L 3s]] [[mos]]. This makes [[7/6]] and [[9/7]] equated to the neutral third, viewing [[6:7:9]] as a tertian chord.  


Another notable extension of dicot is [[decimal]], which splits the octave in two for [[7/5]][[~]][[10/7]] by tempering out [[50/49]], and equates [[7/6]] and [[8/7]] to the tritone complement of 5/4~6/5, neutralizing the 6:7:8 chord as well. This represents the structure of 7-limit harmonies in a way that is not based on tertian harmony and a heptatonic system, but rather a decatonic one.
Another notable extension of dicot is [[decimal]], which splits the octave in two for [[7/5]][[~]][[10/7]] by tempering out [[50/49]], and equates [[7/6]] and [[8/7]] to the tritone complement of 5/4~6/5, neutralizing the [[6:7:8]] chord as well. This represents the structure of 7-limit harmonies in a way that is not based on tertian harmony and a heptatonic system, but rather a decatonic one.


For technical data, see [[Dicot family #Dicot]].  
For technical data, see [[Dicot family #Dicot]].  
Line 42: Line 42:
== Tunings ==
== Tunings ==
=== Norm-based tunings ===
=== Norm-based tunings ===
{{Todo|complete section|inline=1}}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~5/4 = 354.664{{C}}
| CWE: ~5/4 = 351.086{{C}}
| POTE: ~5/4 = 348.594{{C}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.11-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~5/4 = 352.287{{C}}
| CWE: ~5/4 = 348.954{{C}}
| POTE: ~5/4 = 346.734{{C}}
|}


=== Tuning spectrum ===
=== Tuning spectrum ===
Line 51: Line 80:
| '''[[4edo|1\4]]''' ||  || '''300.000''' || '''Lower bound of 5-odd-limit diamond monotone'''
| '''[[4edo|1\4]]''' ||  || '''300.000''' || '''Lower bound of 5-odd-limit diamond monotone'''
|-
|-
|  || 5/3 || 315.641 || Full comma
|  || [[5/3]] || 315.641 || Full comma
|-
|-
|  || 9/5 || 339.199 || 2/3-comma
| [[11edo|3\11]] || || 327.273 || 11c val
|-
|  || [[9/5]] || 339.199 || 2/3-comma
|-
|-
| '''[[7edo|2\7]]''' ||  || '''342.857''' || '''Lower bound of 5-limit 9-odd-limit diamond monotone'''
| '''[[7edo|2\7]]''' ||  || '''342.857''' || '''Lower bound of 5-limit 9-odd-limit diamond monotone'''
|-
|-
|  || 27/20 || 343.910 || 3/5-comma
|  || [[27/20]] || 343.910 || 3/5-comma
|-
|-
| [[24edo|7\24]] ||  || 350.000 || 24c val
| [[24edo|7\24]] ||  || 350.000 || 24c val
|-
|-
|  || 3/2 || 350.978 || 1/2-comma
|  || [[3/2]] || 350.978 || 1/2-comma
|-
|-
| [[17edo|5\17]] ||  || 352.941 ||  
| [[17edo|5\17]] ||  || 352.941 ||  
|-
|-
|  || 45/32 || 358.045 || 2/5-comma
|  || [[45/32]] || 358.045 || 2/5-comma
|-
|-
| [[10edo|3\10]] ||  || 360.000 ||  
| [[10edo|3\10]] ||  || 360.000 ||  
|-
|-
|  || 15/8 || 362.756 || 1/3-comma
|  || [[15/8]] || 362.756 || 1/3-comma
|-
| [[13edo|4\13]] || || 369.231 ||
|-
|-
|  || 5/4 || 386.314 || Untempered tuning
|  || [[5/4]] || 386.314 || Untempered tuning
|-
|-
| '''[[3edo|1\3]]''' ||  || '''400.000''' || '''Upper bound of 5-odd-limit, <br>and 5-limit 9-odd-limit diamond monotone'''
| '''[[3edo|1\3]]''' ||  || '''400.000''' || '''Upper bound of 5-odd-limit, <br>and 5-limit 9-odd-limit diamond monotone'''
Line 77: Line 110:
<nowiki/>* Besides the octave
<nowiki/>* Besides the octave


[[Category:Dicot| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Exotemperaments]]
[[Category:Exotemperaments]]
[[Category:Dicot family]]
[[Category:Dicot family]]
[[Category:Mint temperaments]]
[[Category:Keemic temperaments]]

Latest revision as of 10:27, 28 June 2026

Dicot
Subgroups 2.3.5, 2.3.5.11
Comma basis 25/24 (2.3.5);
25/24, 45/44 (2.3.5.11)
Reduced mapping ⟨1; 2 1 5]
ET join 7 & 10
Generators (CWE) ~6/5 = 351.1 ¢
MOS scales 3L 1s, 3L 4s, 7L 3s
Ploidacot dicot
Pergen (P8, P5/2)
Minimax error 5-odd-limit: 35.3 ¢;
2.3.5.11 15-odd-limit: 35.3 ¢
Target scale size 5-odd-limit: 3 notes;
2.3.5.11 15-odd-limit: 7 notes
This page is about the regular temperament. For the ploidacot signature, see Ploidacot/Dicot.

Dicot is an exotemperament that tempers out 25/24. This means it tempers 6/5 and 5/4 into the same neutral third interval, which, when the fifth is tuned pure, is sqrt(3/2). It is useful to represent the structure of 5-limit harmonies without fully representing them in its greater accuracy, with mos scales 3L 4s and 7L 3s.

It can be extended by tempering out 15/14 and 36/35 in the 7-limit, called mujannabic, though this could turn the 3L 4s mos into a 4L 3s mos. This makes 7/6 and 9/7 equated to the neutral third, viewing 6:7:9 as a tertian chord.

Another notable extension of dicot is decimal, which splits the octave in two for 7/5~10/7 by tempering out 50/49, and equates 7/6 and 8/7 to the tritone complement of 5/4~6/5, neutralizing the 6:7:8 chord as well. This represents the structure of 7-limit harmonies in a way that is not based on tertian harmony and a heptatonic system, but rather a decatonic one.

For technical data, see Dicot family #Dicot.

Interval chain

In the following table, odd harmonics 1–9 are labeled in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 351.1 5/4, 6/5
2 702.2 3/2
3 1053.3 9/5, 15/8
4 204.3 9/8

* In 5-limit CWE tuning

Tunings

Norm-based tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/4 = 354.664 ¢ CWE: ~5/4 = 351.086 ¢ POTE: ~5/4 = 348.594 ¢
2.3.5.11-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/4 = 352.287 ¢ CWE: ~5/4 = 348.954 ¢ POTE: ~5/4 = 346.734 ¢

Tuning spectrum

Edo
generator		 Eigenmonzo
(Unchanged-interval)*		 Generator (¢) Comments
1\4 300.000 Lower bound of 5-odd-limit diamond monotone
5/3 315.641 Full comma
3\11 327.273 11c val
9/5 339.199 2/3-comma
2\7 342.857 Lower bound of 5-limit 9-odd-limit diamond monotone
27/20 343.910 3/5-comma
7\24 350.000 24c val
3/2 350.978 1/2-comma
5\17 352.941
45/32 358.045 2/5-comma
3\10 360.000
15/8 362.756 1/3-comma
4\13 369.231
5/4 386.314 Untempered tuning
1\3 400.000 Upper bound of 5-odd-limit,
and 5-limit 9-odd-limit diamond monotone

* Besides the octave

Retrieved from "https://en.xen.wiki/index.php?title=Dicot&oldid=232876"