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'''Dubhe''' is a [[temperament]] in the 3.5.7.17 [[subgroup]] generated by a sharp [[~]][[17/15]] (or equivalently a flat ~[[45/17]]). As a weak extension of [[BPS]], it [[tempering out|tempers out]] the sensamagic comma, [[245/243]], while splitting the BPS generator of ~[[9/7]] in two (by tempering out [[2025/2023]]) so that a stack of two generators represents 9/7, and a stack of four generators represents [[5/3]] in addition to 81/49, which generates a [[MOS scale]] of [[8L 1s (3/1-equivalent)|8L 1s]] against the tritave. The "canonical" tuning for the generator is [[26edt|3\26]]edt, representing the double [[Bohlen-Pierce]] scale, but a range of other tunings are valid, including [[17edt|2\17]]edt and [[43edt|5\43]]edt.
'''Dubhe''' is a [[temperament]] in the 3.5.7.17 [[subgroup]] generated by a sharp [[~]][[17/15]] (or equivalently a flat ~[[45/17]]). As a weak extension of [[BPS]], it [[tempering out|tempers out]] the sensamagic comma, [[245/243]], while splitting the BPS generator of ~[[9/7]] in two (by tempering out [[2025/2023]]) so that a stack of two generators represents 9/7, and a stack of four generators represents [[5/3]] in addition to 81/49, which generates [[MOS scale]]s of [[8L 1s (3/1-equivalent)|8L 1s]] and [[9L 8s (3/1-equivalent)|9L 8s]] against the tritave. The "canonical" tuning for the generator is [[26edt|3\26]]edt, representing the double [[Bohlen-Pierce]] scale, but a range of other tunings are valid, including [[17edt|2\17]]edt and [[43edt|5\43]]edt.


For technical data, see ''[[No-twos subgroup temperaments#Dubhe]]''.
For technical data, see ''[[No-twos subgroup temperaments#Dubhe]]''.


== Interval chains ==
== Interval chain ==
In the below, tritave-reduced harmonics below 243 are indicated in '''bold'''. All ratios are considered to be in the 287-[[odd limit#Nonoctave equaves|throdd limit]].
In the below, tritave-reduced harmonics and subharmonics 1–17 are indicated in '''bold'''.  


<div><div style="display: inline-grid; margin-right: 25px;">
<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Dubhe
|-
|-
! rowspan="2" | &#35; !! rowspan="2" | Cents&#42; !! colspan="1" | Approximate Ratios
! # !! Cents* !! Approximate Ratios
|-
|-
! 3.5.7.17 subgroup
| 0 || 0.0 || '''1/1'''
|-
|-
| -12 || 1160.5 || 49/25, 243/125
| 1 || 220.3 || 17/15, 135/119, 175/153
|-
|-
| -11 || 1380.8 || 189/85
| 2 || 440.6 || '''9/7''', 35/27
|-
|-
| -10 || 1601.1 || 63/25
| 3 || 660.9 || 25/17, 51/35, 119/81
|-
| 4 || 881.1 || '''5/3''', 81/49
|-
| 5 || 1101.4 || '''17/9''', 225/119, 459/245
|-
| 6 || 1321.7 || 15/7, 175/81
|-
| 7 || 1542.0 || 17/7, 125/51
|-
| 8 || 1762.2 || 25/9, 135/49
|-
| 9 || 80.5 || 51/49, 85/81, 125/119
|-
| 10 || 300.8 || 25/21
|-
| 11 || 521.1 || 85/63
|-
|-
| -9 || 1821.4 || 49/17, 243/85, 357/125
| 12 || 741.4 || 75/49, 125/81
|-
|-
| -8 || 139.7 || 27/25, 49/45
| 13 || 961.7 || 85/49
|-
|-
| -7 || 360.0 || 21/17, 153/125
| 14 || 1181.9 || 125/63
|-
|-
| -6 || 580.3 || 7/5, 243/175
| 15 || 1402.2 || 425/189
|-
|-
| -5 || 800.6 || 27/17, 119/75, 245/153
| 16 || 1622.5 || 125/49
|-
|-
| -4 || 1020.8 || 9/5, '''49/27'''
| 17 || 1842.8 || 425/147
|}
* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
</div>
 
<div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
|-
|-
| -3 || 1241.1 || 35/17, 51/25
! # !! Cents* !! Approximate Ratios
|-
|-
| -2 || 1461.4 || '''7/3''', 81/35
| 0 || 0.0 || '''1/1'''
|-
|-
| -1 || 1681.7 || 45/17, 119/45, 459/175
| -1 || 1681.7 || 45/17, 119/45, 459/175
|-
|-
| 0 || 0.0 || '''1/1''', 245/243
| -2 || 1461.4 || '''7/3''', 81/35
|-
| -3 || 1241.1 || 35/17, 51/25, 243/119
|-
|-
| 1 || 220.3 || 17/15, 135/119, 175/153
| -4 || 1020.8 || '''9/5''', 49/27
|-
|-
| 2 || 440.6 || 9/7, '''35/27'''
| -5 || 800.6 || '''27/17''', 119/75, 245/153
|-
|-
| 3 || 660.9 || 25/17, 51/35
| -6 || 580.3 || 7/5, 243/175
|-
|-
| 4 || 881.1 || '''5/3''', 81/49
| -7 || 360.0 || 21/17, 153/125
|-
|-
| 5 || 1101.4 || '''17/9''', 225/119, 459/245
| -8 || 139.7 || 27/25, 49/45
|-
|-
| 6 || 1321.7 || 15/7, '''175/81'''
| -9 || 1821.4 || 49/17, 243/85, 357/125
|-
|-
| 7 || 1542.0 || 17/7, 125/51
| -10 || 1601.1 || 63/25
|-
|-
| 8 || 1762.2 || '''25/9''', 135/49
| -11 || 1380.9 || 189/85
|-
|-
| 9 || 80.5 || 51/49, '''85/81''', 125/119
| -12 || 1160.6 || 49/25, 243/125
|-
|-
| 10 || 300.8 || 25/21
| -13 || 940.3 || 147/85
|-
|-
| 11 || 521.1 || 85/63
| -14 || 720.0 || 189/125
|-
|-
| 12 || 741.4 || 75/49, '''125/81'''
| -15 || 499.8 || 567/425
|-
|-
| 13 || 961.7 || 85/49
| -16 || 279.5 || 147/125
|-
|-
| 14 || 1181.9 || 125/63
| -17 || 59.2 || 441/425
{{table notes|cols=4
| In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
}}
|}
|}
* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
</div>
</div>


{{stub}}
== Tuning spectrum ==
{| class="wikitable center-all left-4"
! Edt<br>Generator
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator<br>(¢)
! Comments
|-
|
| [[17/15]]
| 216.687
| Untempered tuning
|-
| [[35edt|4\35]]
|
| 217.366
|
|-
|
| [[7/3]]
| 217.542
| 1/2-fidesma
|-
| [[61edt|7\61]]
|
| 218.257
|
|-
| [[87edt|10\87]]
|
| 218.616
|
|-
|
| [[49/45]]
| 219.316
|
|-
|
| [[17/7]]
| 219.447
|
|-
| [[26edt|3\26]]
|
| 219.456
| Double [[Bohlen-Pierce]]
|-
|
| [[7/5]]
| 219.907
|
|-
|
|
| 220.170
| [[Delta-rational chord|DR]] 3:5:7
|-
|
| [[17/9]]
| 220.209
|
|-
| [[95edt|11\95]]
|
| 220.226
|
|-
|
| [[25/21]]
| 220.380
|
|-
| [[69edt|8\69]]
|
| 220.518
|
|-
|
| [[5/3]]
| 221.090
|
|-
| [[43edt|5\43]]
|
| 221.158
|
|-
| [[60edt|7\60]]
|
| 221.895
|
|-
|
| 175/81
| 222.272
|
|-
| [[77edt|9\77]]
|
| 222.306
|
|-
|
| [[25/17]]
| 222.557
|
|-
| [[17edt|2\17]]
|
| 223.759
|
|-
|
| [[35/27]]
| 224.637
|
|}
 
[[Category:Dubhe| ]] <!-- main article -->
[[Category:BPS]]
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]
[[Category:Sensamagic clan]]
Retrieved from "https://en.xen.wiki/w/Dubhe"