24edo: Difference between revisions
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The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). The 24-tone interval of 550 cents is 1.3 cents flatter than 11/8 and is almost indistinguishable from it. In addition, the interval approximating 11/9 is 7 steps which is exactly half the perfect fifth. Additionally, like [[22edo]], 24edo tempers out the [[quartisma]], linking the otherwise sub-par [[7-limit]] harmonies with those of the [[11-limit]]. | The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). The 24-tone interval of 550 cents is 1.3 cents flatter than 11/8 and is almost indistinguishable from it. In addition, the interval approximating 11/9 is 7 steps which is exactly half the perfect fifth. Additionally, like [[22edo]], 24edo tempers out the [[quartisma]], linking the otherwise sub-par [[7-limit]] harmonies with those of the [[11-limit]]. | ||
The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11, 17, and 19. Expanding this, one will find that 24edo is consistent in the no-7s 19-odd-limit, though the 2.3.11.17.19 [[subgroup]] is where it is the most accurate. | The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11, 17, and 19. Expanding this, one will find that 24edo is consistent in the no-7s 19-odd-limit, though the 2.3.11.17.19 [[subgroup]] is where it is the most accurate. | ||
Its step, at 50 cents, is notable for having some of the highest [[harmonic entropy]] possible, making it, in theory, one of the most dissonant intervals possible (using the relatively common values of ''a'' = 2 and ''s'' = 1%, the peak occurs at around 46.4 cents). Intervals less than 40 cents tend to be perceived as being closer to a unison, and thus, more consonant as a result, while intervals larger than approximately 60 cents are often perceived as having less "tension", and thus are also considered to be more consonant. | Its step, at 50 cents, is notable for having some of the highest [[harmonic entropy]] possible, making it, in theory, one of the most dissonant intervals possible (using the relatively common values of {{nowrap|''a'' {{=}} 2 and {{nowrap|''s'' {{=}} 1%}}, the peak occurs at around 46.4 cents). Intervals less than 40 cents tend to be perceived as being closer to a unison, and thus, more consonant as a result, while intervals larger than approximately 60 cents are often perceived as having less "tension", and thus are also considered to be more consonant. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.11.17.19 [[subgroup]]}}</ref> | ! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.11.17.19 [[subgroup]]}}</ref> | ||
! colspan="3" | [[Ups and downs notation]] | ! colspan="3" | [[Ups and downs notation]] | ||
! colspan="3" | [[SKULO interval names|SKULO notation]] (U or S = 1) | ! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U or S {{=}} 1)}} | ||
! [[24edo solfege|Solfege]] | ! [[24edo solfege|Solfege]] | ||
|- | |- | ||
| Line 456: | Line 456: | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
| | | style="width: 40px;" | [[File:HeQu1.svg|21px|center]] | ||
| A "semisharp" accidental comprising one half of a regular musical sharp symbol. | | A "semisharp" or "half-sharp" accidental comprising one half of a regular musical sharp symbol. | ||
|} | |} | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
| | | style="width: 40px;" | [[File:HeQu3.svg|32px|center]] | ||
| A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp. | | A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp. | ||
|} | |} | ||
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{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
| | | style="width: 40px;" | [[File:HeQd1.svg|22px|center]] | ||
| A "semiflat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left. | | A "semiflat" or "half-flat" accidental, comprising a flat symbol mirrored horizontally so that the lobe is facing left. | ||
|} | |} | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
| | | style="width: 40px;" | [[File:HeQd3.svg|36px|center]] | ||
| A "flat and a half" or "sesquiflat" accidental, comprising | | A "flat and a half" or "sesquiflat" accidental, comprising a half-flat symbol and a regular flat symbol placed back to back. | ||
|} | |} | ||
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! rowspan="2" | [[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br />8ve Stretch (¢) | ||
! colspan="2" | Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
| Line 575: | Line 575: | ||
{| class="commatable wikitable center-1 center-2 right-4 center-5" | {| class="commatable wikitable center-1 center-2 right-4 center-5" | ||
|- | |- | ||
! [[Harmonic limit| Prime<br>Limit]] | ! [[Harmonic limit| Prime<br />Limit]] | ||
! [[Ratio]]<ref group="note">{{rd}}</ref> | ! [[Ratio]]<ref group="note">{{rd}}</ref> | ||
! [[Monzo]] | ! [[Monzo]] | ||
| Line 933: | Line 933: | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|- | |- | ||
! Periods<br>per 8ve | ! Periods<br />per 8ve | ||
! Generator | ! Generator | ||
! Name | ! Name | ||
| Line 1,008: | Line 1,008: | ||
| {{dash|0, 5, 14|s=hair|d=med}} | | {{dash|0, 5, 14|s=hair|d=med}} | ||
| {{dash|C, E{{sesquiflat2}}, G|s=hair|d=med}} | | {{dash|C, E{{sesquiflat2}}, G|s=hair|d=med}} | ||
| Cvm<br>Cm({{demiflat2}}3), Cmin({{demiflat2}}3) | | Cvm<br />Cm({{demiflat2}}3), Cmin({{demiflat2}}3) | ||
| C subminor<br>C minor semiflat-three | | C subminor<br />C minor semiflat-three | ||
|- | |- | ||
| 10:12:15 | | 10:12:15 | ||
| Line 1,032: | Line 1,032: | ||
| {{dash|0, 9, 14|s=hair|d=med}} | | {{dash|0, 9, 14|s=hair|d=med}} | ||
| {{dash|C, E{{demisharp2}}, G|s=hair|d=med}} | | {{dash|C, E{{demisharp2}}, G|s=hair|d=med}} | ||
| C^<br>C({{demisharp2}}3), Cmaj({{demisharp2}}3) | | C^<br />C({{demisharp2}}3), Cmaj({{demisharp2}}3) | ||
| C supermajor<br>C major semisharp-three | | C supermajor<br />C major semisharp-three | ||
|} | |} | ||
| Line 1,062: | Line 1,062: | ||
| {{dash|1, vb3, 5, vb7|s=hair|d=med}} | | {{dash|1, vb3, 5, vb7|s=hair|d=med}} | ||
| {{dash|C, E{{sesquiflat2}}, G, B{{sesquiflat2}}|s=hair|d=med}} | | {{dash|C, E{{sesquiflat2}}, G, B{{sesquiflat2}}|s=hair|d=med}} | ||
| smin7<br>min7({{demiflat2}}3, {{demiflat2}}7) | | smin7<br />min7({{demiflat2}}3, {{demiflat2}}7) | ||
| Subminor seven<br>Minor seven semiflat-three semiflat-seven | | Subminor seven<br />Minor seven semiflat-three semiflat-seven | ||
| | | | ||
|- | |- | ||
| Line 1,090: | Line 1,090: | ||
| {{dash|1, b3, 5, b7|s=hair|d=med}} | | {{dash|1, b3, 5, b7|s=hair|d=med}} | ||
| {{dash|C, E{{demisharp2}}, G, B{{demisharp2}}|s=hair|d=med}} | | {{dash|C, E{{demisharp2}}, G, B{{demisharp2}}|s=hair|d=med}} | ||
| smaj7<br>maj7({{demisharp2}}3, {{demisharp2}}7) | | smaj7<br />maj7({{demisharp2}}3, {{demisharp2}}7) | ||
| Supermajor seven<br>Major seven semisharp-three semisharp-seven | | Supermajor seven<br />Major seven semisharp-three semisharp-seven | ||
| | | | ||
|- | |- | ||
| Line 1,104: | Line 1,104: | ||
| {{dash|1, 3, 5, vb7|s=hair|d=med}} | | {{dash|1, 3, 5, vb7|s=hair|d=med}} | ||
| {{dash|C, E, G, B{{sesquiflat2}}|s=hair|d=med}} | | {{dash|C, E, G, B{{sesquiflat2}}|s=hair|d=med}} | ||
| h7<br>7({{demiflat2}}7) | | h7<br />7({{demiflat2}}7) | ||
| Harmonic seven<br>Dominant 7 semiflat-seven | | Harmonic seven<br />Dominant 7 semiflat-seven | ||
| | | | ||
|- | |- | ||
| Line 1,112: | Line 1,112: | ||
| {{dash|C, E{{sesquiflat2}}, G, B♭|s=hair|d=med}} | | {{dash|C, E{{sesquiflat2}}, G, B♭|s=hair|d=med}} | ||
| min7({{demiflat2}}3) | | min7({{demiflat2}}3) | ||
| Arto<br>Minor seven semiflat-three | | Arto<br />Minor seven semiflat-three | ||
| [[File:arto tetrad on C.mp3]] | | [[File:arto tetrad on C.mp3]] | ||
|- | |- | ||
| Line 1,118: | Line 1,118: | ||
| {{dash|1, ^3, 5, vb7|s=hair|d=med}} | | {{dash|1, ^3, 5, vb7|s=hair|d=med}} | ||
| {{dash|C, E{{demisharp2}}, G, B{{sesquiflat2}}|s=hair|d=med}} | | {{dash|C, E{{demisharp2}}, G, B{{sesquiflat2}}|s=hair|d=med}} | ||
| h7({{demisharp2}}3)<br>7({{demisharp2}}3, {{demiflat2}}7) | | h7({{demisharp2}}3)<br />7({{demisharp2}}3, {{demiflat2}}7) | ||
| Tendo<br>Harmonic seven semisharp-three<br>Dominant seven semisharp-three semiflat-seven | | Tendo<br />Harmonic seven semisharp-three<br />Dominant seven semisharp-three semiflat-seven | ||
| | | | ||
|} | |} | ||