Dubhe: Difference between revisions

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== Interval chain ==
== 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 1–17 are indicated in '''bold'''. All ratios are considered to be in the 287-[[odd limit#Nonoctave equaves|throdd limit]].


<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
|-
| -11 || 1380.8 || 189/85
|-
| -10 || 1601.1 || 63/25
|-
| -9 || 1821.4 || 49/17, 243/85, 357/125
|-
| -8 || 139.7 || 27/25, 49/45
|-
| -7 || 360.0 || 21/17, 153/125
|-
| -6 || 580.3 || 7/5, 243/175
|-
| -5 || 800.6 || 27/17, 119/75, 245/153
|-
| -4 || 1020.8 || 9/5, '''49/27'''
|-
| -3 || 1241.1 || 35/17, 51/25, 243/119
|-
| -2 || 1461.4 || '''7/3''', 81/35
|-
| -1 || 1681.7 || 45/17, 119/45, 459/175
|- style="background-color: #C7BFDF;"
| 0 || 0.0 || '''1/1''', 245/243
|-
|-
| 1 || 220.3 || 17/15, 135/119, 175/153
| 1 || 220.3 || 17/15, 135/119, 175/153
|-
|-
| 2 || 440.6 || 9/7, '''35/27'''
| 2 || 440.6 || '''9/7''', 35/27
|-
|-
| 3 || 660.9 || 25/17, 51/35, '''119/81'''
| 3 || 660.9 || 25/17, 51/35, 119/81
|-
|-
| 4 || 881.1 || '''5/3''', 81/49
| 4 || 881.1 || '''5/3''', 81/49
Line 50: Line 22:
| 5 || 1101.4 || '''17/9''', 225/119, 459/245
| 5 || 1101.4 || '''17/9''', 225/119, 459/245
|-
|-
| 6 || 1321.7 || 15/7, '''175/81'''
| 6 || 1321.7 || 15/7, 175/81
|-
|-
| 7 || 1542.0 || 17/7, 125/51
| 7 || 1542.0 || 17/7, 125/51
|-
|-
| 8 || 1762.2 || '''25/9''', 135/49
| 8 || 1762.2 || 25/9, 135/49
|-
|-
| 9 || 80.5 || 51/49, '''85/81''', 125/119
| 9 || 80.5 || 51/49, 85/81, 125/119
|-
|-
| 10 || 300.8 || 25/21
| 10 || 300.8 || 25/21
Line 62: Line 34:
| 11 || 521.1 || 85/63
| 11 || 521.1 || 85/63
|-
|-
| 12 || 741.4 || 75/49, '''125/81'''
| 12 || 741.4 || 75/49, 125/81
|-
|-
| 13 || 961.7 || 85/49
| 13 || 961.7 || 85/49
|-
|-
| 14 || 1181.9 || 125/63
| 14 || 1181.9 || 125/63
{{table notes|cols=4
| In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
}}
|}
|}
</div>
* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]


== Tuning spectrum ==
== Tuning spectrum ==

Revision as of 16:40, 3 October 2024

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 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 scales of 8L 1s and 9L 8s against the tritave. The "canonical" tuning for the generator is 3\26edt, representing the double Bohlen-Pierce scale, but a range of other tunings are valid, including 2\17edt and 5\43edt.

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

Interval chain

In the below, tritave-reduced harmonics 1–17 are indicated in bold. All ratios are considered to be in the 287-throdd limit.

# Cents* Approximate Ratios
0 0.0 1/1
1 220.3 17/15, 135/119, 175/153
2 440.6 9/7, 35/27
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
12 741.4 75/49, 125/81
13 961.7 85/49
14 1181.9 125/63
  • In 3.5.7-targeted DKW tuning for BPS

Tuning spectrum

Edt
Generator 		Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
17/15 216.687 Untempered tuning
4\35 217.366
7/3 217.542 1/2-fidesma
7\61 218.257
10\87 218.616
49/45 219.316
17/7 219.447
3\26 219.456 Double Bohlen-Pierce
7/5 219.907
220.170 DR 3:5:7
17/9 220.209
11\95 220.226
25/21 220.380
8\69 220.518
5/3 221.090
5\43 221.158
7\60 221.895
175/81 222.272
9\77 222.306
25/17 222.557
2\17 223.759
35/27 224.637
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