Shallowtone: Difference between revisions
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'''Shallowtone''' is a [[ | {{Infobox regtemp | ||
| Title = Shallowtone | |||
| Subgroups = 2.3.5, 2.3.5.7 | |||
| Comma basis = [[295245/262144]] (5-limit)<br>[[36/35]], [[295245/262144]] (7-limit) | |||
| Mapping = 1; 1 -10 12 | |||
| Edo join 1 = 7 | Edo join 2 = 30b | |||
| Generators = 3/2 | |||
| Generators tuning = 681.2 | |||
| Optimization method = CWE | |||
| MOS scales = [[2L 3s]], [[2L 5s]], [[7L 2s]] | |||
| Odd limit 1 = 5 | Mistuning 1 = 18.7 | Complexity 1 = 16 | |||
| Odd limit 2 = 9 | Mistuning 2 = 45.3 | Complexity 2 = 23 | |||
}} | |||
'''Shallowtone''' is a [[regular temperament|temperament]] where the [[shallowtone comma]] is [[tempering out|tempered out]]. It is generated by a flattened [[3/2|perfect fifth]], typically sharper than in [[mavila]] but flatter than in [[7edo]], and [[5/4]] is reached by minus ten fifths [[octave reduction|octave-reduced]], which is an augmented third (C–E𝄪) in melodic antidiatonic notation and a diminished third (C–E𝄫) in harmonic antidiatonic notation. This gives it a high [[error]], although not as much as mavila. | |||
The only reasonable [[extension]] to the [[7-limit]] tempers out [[36/35]]. Additionally, there is a weak extension [[Greenwoodmic temperaments #Semishallowtone|semishallowtone]] with a half-octave period, tempering out [[405/392]]. | |||
The name was coined by [[User:CompactStar|CompactStar]] in 2024. It refers to how it is an opposite to [[deeptone]] (which is on the other side of 7edo), as well as because it is just below the diatonic "surface". | The name was coined by [[User:CompactStar|CompactStar]] in 2024. It refers to how it is an opposite to [[deeptone]] (which is on the other side of 7edo), as well as because it is just below the diatonic "surface". | ||
See [[ | See [[Mint temperaments #Shallowtone]] for technical data. | ||
== Interval chain == | == Interval chain == | ||
| Line 10: | Line 25: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! rowspan="2" | | ! rowspan="2" | # | ||
! rowspan="2" | Cents* | ! rowspan="2" | Cents* | ||
! colspan="2" | Approximate ratios | ! colspan="2" | Approximate ratios | ||
| Line 97: | Line 112: | ||
| [[117/80]], [[189/128]] | | [[117/80]], [[189/128]] | ||
|} | |} | ||
<nowiki />* In 5-limit CTE tuning | <nowiki/>* In 5-limit CTE tuning, octave reduced | ||
[[Category:Shallowtone| ]] <!-- main article --> | [[Category:Shallowtone| ]] <!-- main article --> | ||
[[Category:Rank-2 temperaments]] | [[Category:Rank-2 temperaments]] | ||
[[Category:Mint temperaments]] | [[Category:Mint temperaments]] | ||
[[Category:Exotemperaments]] | |||
Latest revision as of 02:37, 6 July 2026
| Shallowtone |
36/35, 295245/262144 (7-limit)
9-odd-limit: 45.3 ¢
9-odd-limit: 23 notes
Shallowtone is a temperament where the shallowtone comma is tempered out. It is generated by a flattened perfect fifth, typically sharper than in mavila but flatter than in 7edo, and 5/4 is reached by minus ten fifths octave-reduced, which is an augmented third (C–E𝄪) in melodic antidiatonic notation and a diminished third (C–E𝄫) in harmonic antidiatonic notation. This gives it a high error, although not as much as mavila.
The only reasonable extension to the 7-limit tempers out 36/35. Additionally, there is a weak extension semishallowtone with a half-octave period, tempering out 405/392.
The name was coined by CompactStar in 2024. It refers to how it is an opposite to deeptone (which is on the other side of 7edo), as well as because it is just below the diatonic "surface".
See Mint temperaments #Shallowtone for technical data.
Interval chain
In the following table, prime harmonics are labeled in bold.
| # | Cents* | Approximate ratios | |
|---|---|---|---|
| 5-limit | 13-limit extension | ||
| 0 | 0.000 | 1/1 | |
| 1 | 681.801 | 3/2 | 13/9 |
| 2 | 163.602 | 9/8 | 11/10, 13/12, 35/32, 81/70, 8192/8019 |
| 3 | 845.403 | 27/16 | 13/8, 117/70, 4096/2673 |
| 4 | 327.204 | 81/64 | 39/32, 99/80, 1024/891 |
| 5 | 1009.0511 | 243/128 | 117/64, 512/297 |
| 6 | 490.806 | 512/405, 729/512 | 128/99 |
| 7 | 1172.607 | 256/135 | 64/33 |
| 8 | 654.408 | 64/45 | 16/11, 112/81 |
| 9 | 136.209 | 16/15 | 12/11, 28/27 |
| 10 | 818.010 | 8/5 | 14/9, 18/11, 64/39 |
| 11 | 299.811 | 6/5 | 7/6, 27/22, 63/52 |
| 12 | 981.612 | 9/5 | 7/4, 26/15, 81/44 |
| 13 | 463.413 | 27/20 | 13/10, 21/16, 243/176 |
| 14 | 1145.214 | 81/40 | 39/20, 63/32, 729/352 |
| 15 | 627.015 | 243/160 | 117/80, 189/128 |
* In 5-limit CTE tuning, octave reduced