26edo: Difference between revisions
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{{EDO intro|26}} | |||
== Theory == | == Theory == | ||
26edo tempers out [[81/80]] in the [[5-limit]], making it a [[meantone]] tuning with a very flat fifth. | 26edo tempers out [[81/80]] in the [[5-limit]], making it a [[meantone]] tuning with a very flat fifth. | ||
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|- | |- | ||
! Degrees | ! Degrees | ||
! [[ | ! [[Cent]]s | ||
! Approximate Ratios* | ! Approximate Ratios* | ||
! Interval<br>Name | ! Interval<br>Name | ||
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For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]]. | For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]]. | ||
== | == Approximation to JI == | ||
=== 15-odd-limit interval mappings === | === 15-odd-limit interval mappings === | ||
The following table shows how [[15-odd-limit intervals]] are represented in 26edo. Prime harmonics are in '''bold'''; intervals with a non-[[consistent]] mapping are in ''italic''. | The following table shows how [[15-odd-limit intervals]] are represented in 26edo. Prime harmonics are in '''bold'''; intervals with a non-[[consistent]] mapping are in ''italic''. | ||
{| class="wikitable center-all" | {| class="wikitable mw-collapsible mw-collapsed center-all" | ||
|+ | |+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent) | ||
|- | |- | ||
! Interval, complement | ! Interval, complement | ||
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| 21.823 | | 21.823 | ||
|} | |} | ||
{{15-odd-limit|26}} | |||
== Approximation to irrational intervals == | |||
After [[13edo#Phi vibes|13edo]], the weird coïncidences continue: [[11/7#Proximity with π/2|acoustic π/2]] (17\26) is just in between [[13edo#Phi vibes|the ϕ intervals provided by 13edo]] (16\26 for [[Logarithmic phi|logarithmic ϕ]]/2, and 18\26 for [[Acoustic phi|acoustic ϕ]]). | After [[13edo#Phi vibes|13edo]], the weird coïncidences continue: [[11/7#Proximity with π/2|acoustic π/2]] (17\26) is just in between [[13edo#Phi vibes|the ϕ intervals provided by 13edo]] (16\26 for [[Logarithmic phi|logarithmic ϕ]]/2, and 18\26 for [[Acoustic phi|acoustic ϕ]]). | ||
Not until 1076edo do we find a better | Not until 1076edo do we find a better edo in terms of relative error on these intervals (which is not a very relevant edo for logarithmic ϕ, since 1076 does not belong to the Fibonacci sequence). | ||
However, it should be noted that [[Logarithmic constants VS acoustic constants|from an acoustic perspective]], acoustic π and acoustic ϕ are both better represented on [[23edo]]. | However, it should be noted that [[Logarithmic constants VS acoustic constants|from an acoustic perspective]], acoustic π and acoustic ϕ are both better represented on [[23edo]]. | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable" | {| class="wikitable" | ||
! rowspan="2" |[[Just intonation subgroup|Subgroup]] | ! rowspan="2" | [[Just intonation subgroup|Subgroup]] | ||
! rowspan="2" |[[Comma basis|Comma List]] | ! rowspan="2" | [[Comma basis|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
8ve Stretch (¢) | ! colspan="2" | Tuning Error | ||
! colspan="2" |Tuning Error | |||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|[-41 26⟩ | | [-41 26⟩ | ||
|[⟨26 41]] | | [⟨26 41]] | ||
| +3.043 | | +3.043 | ||
|3.05 | | 3.05 | ||
|6.61 | | 6.61 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|81/80, 78125/73728 | | 81/80, 78125/73728 | ||
|[⟨26 41 60]] | | [⟨26 41 60]] | ||
| +4.489 | | +4.489 | ||
|3.22 | | 3.22 | ||
|6.98 | | 6.98 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|50/49, 81/80, 405/392 | | 50/49, 81/80, 405/392 | ||
|[⟨26 41 60 73]] | | [⟨26 41 60 73]] | ||
| +3.324 | | +3.324 | ||
|3.44 | | 3.44 | ||
|7.45 | | 7.45 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|45/44, 50/49, 81/80, 99/98 | | 45/44, 50/49, 81/80, 99/98 | ||
|[⟨26 41 60 73 90]] | | [⟨26 41 60 73 90]] | ||
| +2.509 | | +2.509 | ||
|3.48 | | 3.48 | ||
|7.53 | | 7.53 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|45/44, 50/49, 65/64, 78/77, 81/80 | | 45/44, 50/49, 65/64, 78/77, 81/80 | ||
|[⟨26 41 60 73 90 96]] | | [⟨26 41 60 73 90 96]] | ||
| +2.531 | | +2.531 | ||
|3.17 | | 3.17 | ||
|6.87 | | 6.87 | ||
|- | |- | ||
|2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
|45/44, 50/49, 65/64 78/77, 81/80, 85/84 | | 45/44, 50/49, 65/64 78/77, 81/80, 85/84 | ||
|[⟨26 41 60 73 90 96 106]] | | [⟨26 41 60 73 90 96 106]] | ||
| +2.613 | | +2.613 | ||
|2.94 | | 2.94 | ||
|6.38 | | 6.38 | ||
|- | |- | ||
|2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
|45/44, 50/49, 57/56, 65/64, 78/77, 81/80, 85/84 | | 45/44, 50/49, 57/56, 65/64, 78/77, 81/80, 85/84 | ||
|[⟨26 41 60 73 90 96 106 110]] | | [⟨26 41 60 73 90 96 106 110]] | ||
| +2.894 | | +2.894 | ||
|2.85 | | 2.85 | ||
|6.18 | | 6.18 | ||
|} | |} | ||
26et is lower in relative error than any previous equal temperaments in the [[17-limit|17-]], [[19-limit|19-]], [[23-limit|23-]], and [[29-limit]] (using the 26i val for the 23- and 29-limit). The next equal temperaments performing better in those subgroups are [[27edo|27eg]], 27eg, [[29edo|29g]], and [[46edo|46]], respectively. | 26et is lower in relative error than any previous equal temperaments in the [[17-limit|17-]], [[19-limit|19-]], [[23-limit|23-]], and [[29-limit]] (using the 26i val for the 23- and 29-limit). The next equal temperaments performing better in those subgroups are [[27edo|27eg]], 27eg, [[29edo|29g]], and [[46edo|46]], respectively. | ||
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[[Gamelismic_clan#Unidec-Hendec|Hendec]], the 13-limit 26&46 temperament with generator ~10/9, concentrates the intervals of greatest accuracy in 26et into the lower ranges of complexity. It has a period of half an octave, with 13/12 reachable by four generators, 8/7 by two, 14/11 by one, 10/9 by one, and 11/8 by three. All of these are tuned to within 2.5 cents of accuracy. | [[Gamelismic_clan#Unidec-Hendec|Hendec]], the 13-limit 26&46 temperament with generator ~10/9, concentrates the intervals of greatest accuracy in 26et into the lower ranges of complexity. It has a period of half an octave, with 13/12 reachable by four generators, 8/7 by two, 14/11 by one, 10/9 by one, and 11/8 by three. All of these are tuned to within 2.5 cents of accuracy. | ||
== Commas == | === Commas === | ||
26et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 26 41 60 73 90 96 }}.) | 26et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 26 41 60 73 90 96 }}.) | ||
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<references/> | <references/> | ||
== Orgone | == Scales == | ||
[[ | === Orgone temperament === | ||
[[Andrew Heathwaite]] first proposed [[Orgonia|orgone]] temperament to take advantage of 26edo's excellent 11 and 7 approximations. 7 degrees of 26edo is a wide minor third of approximately 323.077 cents, and that interval taken as a generator produces 7-tone and 11-tone MOS scales: | |||
The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. [[MOSScales|MOS]] of type [[4L_3s|4L 3s (mish)]]. | The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. [[MOSScales|MOS]] of type [[4L_3s|4L 3s (mish)]]. | ||
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The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates [[16/11|16:11]] and 3g approximates [[7/4|7:4]] (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents. | The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates [[16/11|16:11]] and 3g approximates [[7/4|7:4]] (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents. | ||
[[File:orgone_heptatonic.jpg|alt=orgone_heptatonic.jpg|orgone_heptatonic.jpg]] | [[File:orgone_heptatonic.jpg|alt=orgone_heptatonic.jpg|orgone_heptatonic.jpg]] | ||
== Additional | === Additional scalar bases available === | ||
Since the perfect 5th in | Since the perfect 5th in 26edo spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12). The former approach produces MOS at 1L+4s, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (though further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval). | ||
-Igs | -Igs | ||
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* [[Lumatone mapping for 26edo]] | * [[Lumatone mapping for 26edo]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||
[[Category:Twentuning]] | [[Category:Twentuning]] | ||