Dubhe: Difference between revisions
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→Interval chains: I don't see why negative steps are included nor why they're not shown up to the same number as the positive steps. -table note template in favor of plain notes. There's also way too many bolded stuff |
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For technical data, see ''[[No-twos subgroup temperaments#Dubhe]]''. | For technical data, see ''[[No-twos subgroup temperaments#Dubhe]]''. | ||
== Interval | == Interval chains == | ||
In the below, tritave-reduced harmonics | 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]]. | ||
<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 | |||
|- | |- | ||
! # !! Cents | ! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="1" | Approximate Ratios | ||
|- | |- | ||
| 0 || 0.0 || '''1/1''' | ! 3.5.7.17 subgroup | ||
|- | |||
| -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 | |||
|- | |||
| 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 || | | 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 22: | Line 50: | ||
| 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 34: | Line 62: | ||
| 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> | |||
== Tuning spectrum == | == Tuning spectrum == | ||
Revision as of 14:55, 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 chains
In the below, tritave-reduced harmonics below 243 are indicated in bold. All ratios are considered to be in the 287-throdd limit.
| # | Cents* | Approximate Ratios |
|---|---|---|
| 3.5.7.17 subgroup | ||
| -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 |
| 0 | 0.0 | 1/1, 245/243 |
| 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 |
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 |