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| <span style="display: block; text-align: right;">[[23平均律|日本語]]</span>
| | {{interwiki |
| __FORCETOC__
| | | de = 23edo |
| -----
| | | en = 23edo |
| | | es = 23 EDO |
| | | ja = 23平均律 |
| | }} |
| | {{Infobox ET}} |
| | {{Wikipedia|23 equal temperament}} |
| | {{ED intro}} |
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| =<span style="background-color: #ffffff; color: #00b7e3; font-family: 'Times New Roman',Times,serif; font-size: 113%;">23 tone equal temperament</span>= | | == Theory == |
| | 23edo is significant in that it is the last edo that has no [[5L 2s|diatonic]] perfect fifths and not even [[5edo]] or [[7edo]] fifths. It is also the last edo that fails to approximate the [[3/1|3rd]], [[5/1|5th]], [[7/1|7th]], and [[11/1|11th]] [[harmonic]]s within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them ([[5/3]], [[7/3]], [[11/3]], [[7/5]], [[11/5]], [[11/7]]) and combinations of them ([[15/8]], [[21/16]], [[33/32]], [[35/32]], [[55/32]], [[77/64]]) very well. The lowest harmonics well-approximated by 23edo are [[9/1|9]], [[13/1|13]], [[15/1|15]], [[17/1|17]], [[21/1|21]], [[23/1|23]], [[31/1|31]], [[33/1|33]] and [[35/1|35]]. |
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| '''''23-tET''''', or '''''23-EDO''''', is a tempered musical system which divides the [[Octave|octave]] into 23 equal parts of approximately 52.173913 cents, which is also called with the neologism Icositriphony ''(Icositrifonía)''. It has good approximations for [[5/3|5/3]], [[11/7|11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just_intonation_subgroup|just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit|17-limit]] [[46edo|46edo]], the larger 17-limit [[k*N_subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th [[prime_numbers|prime]] edo, following [[19edo|19edo]] and coming before [[29edo|29edo]].
| | === Mapping === |
| | As with [[9edo]], [[16edo]], and [[25edo]], one way to treat 23edo is as a tuning of the [[mavila]] temperament, tempering out the "comma" of [[135/128]] and equating three acute [[4/3]]'s with 5/1 (related to the Armodue system). This means mapping "[[3/2]]" to 13 degrees of 23, and results in a 7-note [[2L 5s|antidiatonic]] scale of 3–3–4–3–3–3–4 (in steps of 23edo), which extends to a 9-note [[7L 2s|superdiatonic]] scale (3–3–3–1–3–3–3–3–1). One can notate 23edo using the [[Armodue]] system, but just like notating 17edo with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23edo, the "Armodue 6th" is sharper than it is in 16edo, just like the diatonic 5th in 17edo is sharper than in 12edo. In other words, 2b is lower in pitch than 1#, just like how in 17edo Eb is lower than D#. |
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| ==<span style="font-size: 1.4em;">Intervals</span>==
| | However, one can also map 3/2 to 14 degrees of 23edo without significantly increasing the error, taking us to a [[7-limit]] temperament where two broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo]] and [[18edo]] and produces [[mos scale]]s of 5 and 8 notes: 5–5–4–5–4 ([[3L 2s|antipentic]]) and 4–1–4–1–4–4–1–4 (the "quartertone" version of the [[Easley Blackwood Jr.|Blackwood]]/[[Paul Rapoport|Rapoport]]/[[Erv Wilson|Wilson]] 13edo "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23edo a sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0–4–9 degrees of 23edo as an approximation to 16:18:21, and 0–5–9 as 1/(16:18:21); both of these triads are abundant in the 8-note mos scale. |
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| [[File:Ciclo_Icositrifonía.png|alt=Ciclo Icositrifonía.png|491x490px|link=Harmony_of_23edo]] | | 23edo has good approximations for [[5/3]], [[11/7]], 13 and 17, among many others, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of no-19's [[23-limit]] [[46edo]], the larger no-19's 23-limit [[k*N subgroups|2*23 subgroup]] 2.9.15.21.33.13.17.23 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does no-19's 23-limit 46edo, and may be regarded as a basis for analyzing the harmony of 23edo so far, as approximations to just intervals goes. If one dares to take advantage of this harmony by using 23edo as a period, you get [[icositritonic]], a [[23rd-octave temperaments|23rd-octave temperament]], so that the harmony of 23edo is adequately explained by what harmonies you can achieve using only periods and zero generators. |
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| | See ''[[Harmony of 23edo]]'' for more details. |
| | |
| | === Odd harmonics === |
| | {{Harmonics in equal|23}} |
| | |
| | === Octave stretch === |
| | Some approximations can be improved by octave stretching. See ''[[23edo and octave stretching]]'' for more details. |
| | |
| | === Subsets and supersets === |
| | 23edo is the 9th [[prime edo]], following [[19edo]] and coming before [[29edo]], so it does not contain any nontrivial subset edos, though it contains [[23ed4]]. 46edo, which doubles it, considerably improves most of its approximations of lower harmonics. |
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| | === Miscellany === |
| | 23edo was proposed by ethnomusicologist {{w|Erich von Hornbostel}} as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe. |
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| 23edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.
| | == Selected just intervals == |
| | {{Q-odd-limit intervals|23}} |
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| The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 23edo "on the fly".
| | == Notation == |
| | ===Conventional notation === |
| | {{Mavila}} |
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| Armodue notation is a nonatonic notation that uses the numbers 1-9 as note names.
| | ===Sagittal notation=== |
| | ====Best fifth notation==== |
| | This notation uses the same sagittal sequence as EDOs [[28edo#Sagittal notation|28]] and [[33edo#Sagittal notation|33]]. |
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| {| class="wikitable"
| | <imagemap> |
| |-
| | File:23-EDO_Sagittal.svg |
| | style="text-align:center;" | <span style="color: #660000;">[[Degree|Degree]]*</span>
| | desc none |
| | style="text-align:center;" | [[cent|Cent]]s
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | style="text-align:center;" | Approximate
| | rect 367 0 527 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 367 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]] |
| | default [[File:23-EDO_Sagittal.svg]] |
| | </imagemap> |
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| Ratios **
| | ====Second-best fifth notation==== |
| | colspan="2" style="text-align:center;" | Major wider
| | This notation uses the same sagittal sequence as EDOs [[30edo#Sagittal notation|30]], [[37edo#Sagittal notation|37]], and [[44edo#Sagittal notation|44]]. |
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| than minor
| | <imagemap> |
| | colspan="2" style="text-align:center;" | Major narrower | | File:23b_Sagittal.svg |
| | desc none |
| | rect 80 0 300 50 [[Sagittal_notation]] |
| | rect 375 0 535 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] |
| | rect 20 80 375 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]] |
| | default [[File:23b_Sagittal.svg]] |
| | </imagemap> |
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| than minor
| | === Armodue notation === |
| | style="text-align:center;" | Armodue
| | Armodue notation is a nonatonic notation that uses the numbers 1-9 as note names. |
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| Notation
| | {| class="wikitable center-all right-1 right-3 left-10" |
| | | Notes | |
| |- | | |- |
| | style="text-align:center;" | 0
| | ! [[Degree]] |
| | style="text-align:center;" | 0
| | ! [[Cent]]s |
| | style="text-align:center;" | 1/1
| | ! Approximate <br> Ratios <ref>Based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible.</ref> |
| | style="text-align:center;" | P1
| | ! colspan="2" | Major wider <br> than minor |
| | style="text-align:center;" | D
| | ! colspan="2" | Major narrower <br> than minor |
| | style="text-align:center;" | P1
| | ! Armodue <br> Notation |
| | style="text-align:center;" | D
| | ! Notes |
| | style="text-align:center;" | 1
| |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 1 | | | 0 |
| | style="text-align:center;" | 52.174 | | | 0.000 |
| | style="text-align:center;" | 33/32, 34/33 | | | 1/1 |
| | style="text-align:center;" | A1 | | | P1 || D |
| | style="text-align:center;" | D#
| | | P1 || D |
| | style="text-align:center;" | d1 | | | 1 |
| | style="text-align:center;" | Db | | | |
| | style="text-align:center;" | 2b | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 2 | | | 1 |
| | style="text-align:center;" | 104.348 | | | 52.174 |
| | style="text-align:center;" | 17/16, 16/15, 18/17 | | | 33/32, 34/33 |
| | style="text-align:center;" | d2 | | | A1 || D# |
| | style="text-align:center;" | Eb | | | d1 || Db |
| | style="text-align:center;" | A2 | | | 2b |
| | style="text-align:center;" | E# | | | |
| | style="text-align:center;" | 1# | |
| | | Less than 1 cent off 17/16 | |
| |- | | |- |
| | style="text-align:center;" | 3 | | | 2 |
| | style="text-align:center;" | 156.522 | | | 104.348 |
| | style="text-align:center;" | 11/10, 12/11, 35/32 | | | 17/16, 16/15, 18/17 |
| | style="text-align:center;" | m2 | | | d2 || Eb |
| | style="text-align:center;" | E | | | A2 || E# |
| | style="text-align:center;" | M2 | | | 1# |
| | style="text-align:center;" | E
| | | Less than 1 cent off [[17/16]] |
| | style="text-align:center;" | 2 | |
| | | | |
| |- | | |- |
| | style="text-align:center;" | 4· | | | 3 |
| | style="text-align:center;" | 208.696 | | | 156.522 |
| | style="text-align:center;" | 9/8, 44/39 | | | 11/10, 12/11, 35/32 |
| | style="text-align:center;" | M2 | | | m2 || E |
| | style="text-align:center;" | E# | | | M2 || E |
| | style="text-align:center;" | m2 | | | 2 |
| | style="text-align:center;" | Eb | | | |
| | style="text-align:center;" | 3b | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 5 | | | 4 |
| | style="text-align:center;" | 260.870 | | | 208.696 |
| | style="text-align:center;" | 7/6, 15/13, 29/25 | | | 9/8, 44/39 |
| | style="text-align:center;" | A2, d3 | | | M2 || E# |
| | style="text-align:center;" | Ex, Fbb | | | m2 || Eb |
| | style="text-align:center;" | d2, A3 | | | 3b |
| | style="text-align:center;" | Ebb, Fx | | | |
| | style="text-align:center;" | 2# | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 6 | | | 5 |
| | style="text-align:center;" | 313.043 | | | 260.870 |
| | style="text-align:center;" | 6/5 | | | 7/6, 15/13, 29/25 |
| | style="text-align:center;" | m3 | | | A2, d3 || Ex, Fbb |
| | style="text-align:center;" | Fb | | | d2, A3 || Ebb, Fx |
| | style="text-align:center;" | M3 | | | 2# |
| | style="text-align:center;" | F# | | | |
| | style="text-align:center;" | 3 | |
| | | Much better 6/5 than 12-edo
| |
| |- | | |- |
| | style="text-align:center;" | 7· | | | 6 |
| | style="text-align:center;" | 365.217 | | | 313.043 |
| | style="text-align:center;" | 16/13, 21/17, 26/21 | | | 6/5 |
| | style="text-align:center;" | M3 | | | m3 || Fb |
| | style="text-align:center;" | F | | | M3 || F# |
| | style="text-align:center;" | m3 | | | 3 |
| | style="text-align:center;" | F
| | | Much better [[6/5]] than 12-edo |
| | style="text-align:center;" | 4b | |
| | | | |
| |- | | |- |
| | style="text-align:center;" | 8 | | | 7 |
| | style="text-align:center;" | 417.391 | | | 365.217 |
| | style="text-align:center;" | 14/11, 33/26 | | | 16/13, 21/17, 26/21 |
| | style="text-align:center;" | A3 | | | M3 || F |
| | style="text-align:center;" | F#
| | | m3 || F |
| | style="text-align:center;" | d3 | | | 4b |
| | style="text-align:center;" | Fb | | | |
| | style="text-align:center;" | 3# | |
| | | Practically just 14/11 | |
| |- | | |- |
| | style="text-align:center;" | 9 | | | 8 |
| | style="text-align:center;" | 469.565 | | | 417.391 |
| | style="text-align:center;" | 21/16, 17/13 | | | 14/11, 33/26 |
| | style="text-align:center;" | d4 | | | A3 || F# |
| | style="text-align:center;" | Gb | | | d3 || Fb |
| | style="text-align:center;" | A4 | | | 3# |
| | style="text-align:center;" | G# | | | Practically just [[14/11]] |
| | style="text-align:center;" | 4 | |
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| |
| |- | | |- |
| | style="text-align:center;" | 10· | | | 9 |
| | style="text-align:center;" | 521.739 | | | 469.565 |
| | style="text-align:center;" | 23/17, 88/65, 256/189 | | | 21/16, 17/13 |
| | style="text-align:center;" | P4 | | | d4 || Gb |
| | style="text-align:center;" | G | | | A4 || G# |
| | style="text-align:center;" | P4 | | | 4 |
| | style="text-align:center;" | G
| | | |
| | style="text-align:center;" | 5 | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 11 | | | 10 |
| | style="text-align:center;" | 573.913 | | | 521.739 |
| | style="text-align:center;" | 7/5, 32/23, 46/33 | | | 23/17, 88/65, 256/189 |
| | style="text-align:center;" | A4 | | | P4 || G |
| | style="text-align:center;" | G# | | | P4 || G |
| | style="text-align:center;" | d4 | | | 5 |
| | style="text-align:center;" | Gb | | | |
| | style="text-align:center;" | 6b | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 12 | | | 11 |
| | style="text-align:center;" | 626.087 | | | 573.913 |
| | style="text-align:center;" | 10/7, 23/16, 33/23 | | | 7/5, 32/23, 46/33 |
| | style="text-align:center;" | d5 | | | A4 || G# |
| | style="text-align:center;" | Ab | | | d4 || Gb |
| | style="text-align:center;" | A5 | | | 6b |
| | style="text-align:center;" | A# | | | |
| | style="text-align:center;" | 5# | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 13· | | | 12 |
| | style="text-align:center;" | 678.261 | | | 626.087 |
| | style="text-align:center;" | 34/23, 65/44, 189/128 | | | 10/7, 23/16, 33/23 |
| | style="text-align:center;" | P5 | | | d5 || Ab |
| | style="text-align:center;" | A | | | A5 || A# |
| | style="text-align:center;" | P5 | | | 5# |
| | style="text-align:center;" | A | | | |
| | style="text-align:center;" | 6 | |
| | | Great Hornbostel generator | |
| |- | | |- |
| | style="text-align:center;" | 14 | | | 13 |
| | style="text-align:center;" | 730.435 | | | 678.261 |
| | style="text-align:center;" | 32/21, 26/17 | | | 34/23, 65/44, 189/128 |
| | style="text-align:center;" | A5 | | | P5 || A |
| | style="text-align:center;" | A# | | | P5 || A |
| | style="text-align:center;" | d5 | | | 6 |
| | style="text-align:center;" | Ab | | | Great Hornbostel generator |
| | style="text-align:center;" | 7b | |
| | | | |
| |- | | |- |
| | style="text-align:center;" | 15 | | | 14 |
| | style="text-align:center;" | 782.609 | | | 730.435 |
| | style="text-align:center;" | 11/7, 52/33 | | | 32/21, 26/17 |
| | style="text-align:center;" | d6 | | | A5 || A# |
| | style="text-align:center;" | Bb | | | d5 || Ab |
| | style="text-align:center;" | A6 | | | 7b |
| | style="text-align:center;" | B# | | | |
| | style="text-align:center;" | 6# | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 16· | | | 15 |
| | style="text-align:center;" | 834.783 | | | 782.609 |
| | style="text-align:center;" | 13/8, 34/21, 21/13 | | | 11/7, 52/33 |
| | style="text-align:center;" | m6 | | | d6 || Bb |
| | style="text-align:center;" | B | | | A6 || B# |
| | style="text-align:center;" | M6 | | | 6# |
| | style="text-align:center;" | B | | | Practically just [[11/7]] |
| | style="text-align:center;" | 7 | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 17 | | | 16 |
| | style="text-align:center;" | 886.957 | | | 834.783 |
| | style="text-align:center;" | 5/3 | | | 13/8, 34/21, 21/13 |
| | style="text-align:center;" | M6 | | | m6 || B |
| | style="text-align:center;" | B# | | | M6 || B |
| | style="text-align:center;" | m6 | | | 7 |
| | style="text-align:center;" | Bb | | | |
| | style="text-align:center;" | 8b | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 18 | | | 17 |
| | style="text-align:center;" | 939.130 | | | 886.957 |
| | style="text-align:center;" | 12/7, 26/15, 50/29 | | | 5/3 |
| | style="text-align:center;" | A6, d7 | | | M6 || B# |
| | style="text-align:center;" | Bx, Cbb | | | m6 || Bb |
| | style="text-align:center;" | d6, A7
| | | 8b |
| | style="text-align:center;" | Bbb, Cx | | | Much better [[5/3]] than 12-edo |
| | style="text-align:center;" | 7# | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 19· | | | 18 |
| | style="text-align:center;" | 991.304 | | | 939.130 |
| | style="text-align:center;" | 16/9, 39/22 | | | 12/7, 26/15, 50/29 |
| | style="text-align:center;" | m7 | | | A6, d7 || Bx, Cbb |
| | style="text-align:center;" | Cb | | | d6, A7 || Bbb, Cx |
| | style="text-align:center;" | M7 | | | 7# |
| | style="text-align:center;" | C# | | | |
| | style="text-align:center;" | 8
| |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 20 | | | 19 |
| | style="text-align:center;" | 1043.478 | | | 991.304 |
| | style="text-align:center;" | 11/6, 20/11, 64/35 | | | 16/9, 39/22 |
| | style="text-align:center;" | M7 | | | m7 || Cb |
| | style="text-align:center;" | C | | | M7 || C# |
| | style="text-align:center;" | m7 | | | 8 |
| | style="text-align:center;" | C | | | |
| | style="text-align:center;" | 9b | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 21 | | | 20 |
| | style="text-align:center;" | 1095.652 | | | 1043.478 |
| | style="text-align:center;" | 15/8, 17/9, 32/17 | | | 11/6, 20/11, 64/35 |
| | style="text-align:center;" | A7 | | | M7 || C |
| | style="text-align:center;" | C# | | | m7 || C |
| | style="text-align:center;" | d7 | | | 9b |
| | style="text-align:center;" | Cb | | | |
| | style="text-align:center;" | 8# | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 22 | | | 21 |
| | style="text-align:center;" | 1147.826 | | | 1095.652 |
| | style="text-align:center;" | 33/17, 64/33 | | | 15/8, 17/9, 32/17 |
| | style="text-align:center;" | d8 | | | A7 || C# |
| | style="text-align:center;" | Db | | | d7 || Cb |
| | style="text-align:center;" | A8 | | | 8# |
| | style="text-align:center;" | D# | | | Less than 1 cent off [[32/17]] |
| | style="text-align:center;" | 9 | |
| | |
| |
| |- | | |- |
| | style="text-align:center;" | 23·· | | | 22 |
| | style="text-align:center;" | 1200 | | | 1147.826 |
| | style="text-align:center;" | 2/1
| | | 33/17, 64/33 |
| | style="text-align:center;" | P8
| | | d8 || Db |
| | style="text-align:center;" | D | | | A8 || D# |
| | style="text-align:center;" | P8
| | | 9 |
| | style="text-align:center;" | D | | | |
| | style="text-align:center;" | 1
| | |- |
| | |
| | | 23 |
| | | 1200.000 |
| | | 2/1 |
| | | P8 || D |
| | | P8 || D |
| | | 1 |
| | | |
| |} | | |} |
| <ul><li>the dots indicate which frets on a 23-edo guitar would have dots.<ul><li>based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible.</li></ul></li></ul>
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|
| The chart below shows some of the [[MOSScales|Moment of Symmetry (MOS)]] modes of [[Mavila|Mavila]] available in 23edo, mainly Pentatonic(5-note), anti-diatonic(7-note), 9- and 16-note MOSs. Here the outer ring represents individual step of 23edo itself, while the rings moving inward represent 16, 9, 7 and 5 note MOSs:
| | <references/> |
|
| |
|
| [[File:23edoMavilaMOS.jpg|alt=23edoMavilaMOS.jpg|23edoMavilaMOS.jpg]]
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|
| 23-EDO was proposed by ethnomusicologist [http://en.wikipedia.org/wiki/Erich_von_Hornbostel Erich von Hornbostel] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.
| | [[File:Ciclo_Icositrifonía.png|alt=Ciclo Icositrifonía.png|491x490px|link=Harmony_of_23edo]] |
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|
| 23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, 11/3, 7/5, 11/7, and 11/5) very well. The lowest harmonics well-approximated by 23-EDO are 13, 17, 21, and 23. See [[Harmony_of_23edo|here]] for more details. Also note that some approximations can be improved by [[23edo_and_octave_stretching|octave stretching]].
| | == Approximation to irrational intervals == |
| | 23edo has good approximations of [[acoustic phi]] on 16\23, and [[pi]] on 38\23. Not until [[72edo|72]] do we find a better edo in terms of absolute error, and not until [[749edo|749]] do we find one in terms of relative error. |
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| As with[[9edo| 9-EDO]], [[16edo|16-EDO]], and [[25edo|25-EDO]], one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute [[4/3|4/3]]'s with 5/1 (related to the Armodue system). This means mapping '[[3/2|3/2]]' to 13 degrees of 23, and results in a 7 notes [[2L_5s|Anti-diatonic scale]] of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes [[7L_2s|Superdiatonic scale]] (3 3 3 1 3 3 3 3 1). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.
| | {| class="wikitable center-all" |
| | | |+Direct approximation |
| However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a [[7-limit|7-limit]] temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo|13-EDO]] and [[18edo|18-EDO]] and produces [[MOSScales|MOS scales]] of 5 and 8 notes: 5 5 4 5 4 (the [[3L_2s|"anti-pentatonic"]]) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29 Rapoport]/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale.
| | |- |
| | | ! Interval |
| ==Kosmorsky's Sephiroth modes==
| | ! Error (abs, [[Cent|¢]]) |
| I would argue that the most significant modes of 23 edo are those of the 2 2 2 3 2 2 3 2 2 3 scale ([[3L_7s|3L 7s fair mosh]]); This is derived from extending the ~1/3 comma tempered 13th Harmonic, two of which add up to the 21st harmonic and three add up to the 17th harmonic almost perfectly. Interestingly, the chord 8:13:21:34 is a fragment of the fibonacci sequence.
| | |- |
| | | | π |
| Notated in ascending (standard) form. I have named these 10 modes according to the Sephiroth as follows:
| | | 0.813 |
| | | |- |
| 2 2 2 3 2 2 3 2 2 3 - Mode Keter
| | | π/ϕ |
| | | | 0.879 |
| 2 2 3 2 2 3 2 2 3 2 - Chesed
| | |- |
| | | | ϕ |
| 2 3 2 2 3 2 2 3 2 2 - Netzach
| | | 1.692 |
| | | |} |
| 3 2 2 3 2 2 3 2 2 2 - Malkuth
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| | |
| 2 2 3 2 2 3 2 2 2 3 - Binah
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| 2 3 2 2 3 2 2 2 3 2 - Tiferet
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| 3 2 2 3 2 2 2 3 2 2 - Yesod
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| 2 2 3 2 2 2 3 2 2 3 - Chokmah
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| 2 3 2 2 2 3 2 2 3 2 - Gevurah
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| | |
| 3 2 2 2 3 2 2 3 2 2 - Hod
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| | |
| =Music=
| |
| [http://soonlabel.com/xenharmonic/archives/2460 Chromatic canon, by Claudi Meneghin]
| |
| | |
| <span style=""><span style="">''[http://home.vicnet.net.au/%7Eepoetry/family.mp3 The Family Supper]''</span></span> by [[Warren_Burt|Warren Burt]]
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| <span style=""><span style="">''[http://www.youtube.com/watch?v=Hqst8MaRiYM Icositriphonic Heptatonic MOS]''</span></span> by [[Igliashon_Jones|Igliashon Jones]]
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| <span style=""><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20His%20Wandering%20Kinship%20with%20Ashes.mp3 His Wandering Kinship with Ashes]''</span></span> by Igliashon Jones
| | == Regular temperament properties == |
| | === Uniform maps === |
| | {{Uniform map|edo=23}} |
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| <span style=""><span style="">''[http://www.nonoctave.com/tunes/CosmicChamber.mp3 Cosmic Chamber]''</span></span> by [[X._J._Scott|X. J. Scott]]
| | === Commas === |
| | 23et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 23 36 53 65 80 85 }}. Also note the discussion above, where there are some commas mentioned that are not in the standard comma list (e.g., 28/27). |
|
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| <span style=""><span style="">''[http://www.nonoctave.com/tunes/Daisies.mp3 Daisies on the Beach]''</span></span> by X. J. Scott
| | {| class="commatable wikitable center-all left-3 right-4 left-6" |
| | |
| <span style="background-position: 100% 50%; cursor: pointer; padding-right: 10px;"><span style=""><span style="">''[http://www.akjmusic.com/audio/boogie_pie.mp3 Boogie Pie]''</span></span></span>by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]
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| <span style=""><span style="">''[http://clones.soonlabel.com/public/micro/23edo/daily20110619_23edo_23_chilled.mp3 23 Chilled]''</span></span> by [[Chris_Vaisvil|Chris Vaisvil]]
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| <span style=""><span style="">''[http://www.seraph.it/dep/det/DesertWinds.mp3 Desert Winds]''</span></span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/926007c7483e4abc5a48d582c0667947-105.html blog entry])
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| <span style=""><span style="">''[http://www.seraph.it/dep/det/23Laments.mp3 23 Laments]''</span></span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/b2bf6f252efd467ee36ecc332a4872ac-106.html blog entry])
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| <span style="">''[http://www.seraph.it/dep/det/Doomsday23.mp3 Doomsday 23]''</span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/add481fdf4ae8c3afe56a0d2cb6dd672-164.html blog entry])
| |
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| <span style="">''[http://www.seraph.it/dep/int/Adagio23ForStrings.mp3 Barber’s Adagio For Strings in 23ED2]''</span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/9e630d3f8ba93ab8264a3862dac950ce-192.html blog entry])
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| <span style="">''[http://www.seraph.it/dep/det/Nubian%20Dance.mp3 Nubian Dance]''</span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/694f0a26d29cd2a215f37754dd8428c3-237.html blog entry])
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| ''Allegro Moderato'' by Easley Blackwood
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| [http://andrewheathwaite.bandcamp.com/track/pentaswing Pentaswing] [[Technical_Notes_for_Newbeams#Track notes:-Pentaswing|Notes]] by [[Andrew_Heathwaite|Andrew Heathwaite]]
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| <span style="">''[http://micro.soonlabel.com/MOS/20120418-9mos-mindaugas.mp3 Mindaugas Rex Lithuaniae]''</span> by [http://chrisvaisvil.com/?p=2267 Chris Vaisvil] (in Superpelog-9 tuning)
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| ''[http://micro.soonlabel.com/23edo/Tutim_Dennsuul/T%fatim%20Dennsuul%20-%20Indigorange.mp3 Indigorange]'' by Tutim Dennsuul
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| ''[http://micro.soonlabel.com/23edo/Tutim_Dennsuul/T%fatim%20Dennsuul%20-%20Wignud.mp3 Wignud]'' by Tutim Dennsuul
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| ''[http://micro.soonlabel.com/23edo/Tutim_Dennsuul/Tutim%20Dennsuul%20-%20%20Harid.mp3 Harid]'' by Tutim Dennsuul
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| ''[https://soundcloud.com/nanovibrationalrelations/a-rest-in-the-desert-23edo A Rest In The Desert]'' [http://micro.soonlabel.com/gene_ward_smith/Others/Mcandrew/A%20Rest%20In%20The%20Desert%20(23edo).mp3 play] by Gary Mcandrew
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| ''[http://spectropolrecords.bandcamp.com/track/jacky-ligon-numenoctagon Numenoctagon]'' by Jacky Ligon (on spectropolrecords)
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| [http://www.dubbhism.com/2014/02/out-now-ligon-sevish-dubshot-23.html#more 23 (album)] by Jacky Ligon, Sevish & Tony Dubshot ([https://soundcloud.com/ism-studio/sets/ligon-sevish-dubshot-23 soundcloud])
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| [https://soundcloud.com/shunya-kiyokawa/23edo-klavier8 23EDO Klavier8] by [[Shunya_Kiyokawa|Shunya Kiyokawa]]
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| =Commas=
| |
| 23 EDO tempers out the following commas. (Note: This assumes the val < 23 36 53 65 80 85 |.) Also note the discussion above, where there are some commas mentioned that are not in the standard comma list (e.g., 28/27).
| |
| | |
| {| class="wikitable"
| |
| |- | | |- |
| ! | Comma | | ! [[Harmonic limit|Prime<br>limit]] |
| ! | Monzo | | ! [[Ratio]] |
| ! | Value (Cents) | | ! [[Monzo]] |
| ! | Name 1 | | ! [[Cents]] |
| ! | Name 2 | | ! [[Color name]] |
| ! | Name 3
| | ! Name(s) |
| |- | | |- |
| | style="text-align:center;" | 135/128 | | | 5 |
| | | | -7 3 1 > | | | [[135/128]] |
| | style="text-align:right;" | 92.18
| | | {{monzo| -7 3 1 }} |
| | style="text-align:center;" | Major Chroma | | | 92.18 |
| | style="text-align:center;" | Major Limma | | | Layobi |
| | style="text-align:center;" | Pelogic Comma
| | | Mavila comma, major chroma |
| |- | | |- |
| | style="text-align:center;" | 15625/15552 | | | 5 |
| | | | -6 -5 6 > | | | [[15625/15552]] |
| | style="text-align:right;" | 8.11
| | | {{monzo| -6 -5 6 }} |
| | style="text-align:center;" | Kleisma | | | 8.11 |
| | style="text-align:center;" | Semicomma Majeur
| | | Tribiyo |
| | style="text-align:center;" |
| | | Kleisma, semicomma majeur |
| |- | | |- |
| | style="text-align:center;" | 36/35 | | | 7 |
| | | | 2 2 -1 -1 > | | | [[36/35]] |
| | style="text-align:right;" | 48.77
| | | {{monzo| 2 2 -1 -1 }} |
| | style="text-align:center;" | Septimal Quarter Tone | | | 48.77 |
| | style="text-align:center;" | | | | Rugu |
| | style="text-align:center;" |
| | | Mint comma, septimal quartertone |
| |- | | |- |
| | style="text-align:center;" | 525/512 | | | 7 |
| | | | -9 1 2 1 > | | | [[525/512]] |
| | style="text-align:right;" | 43.41
| | | {{monzo| -9 1 2 1 }} |
| | style="text-align:center;" | Avicennma | | | 43.41 |
| | style="text-align:center;" | Avicenna's Enharmonic Diesis
| | | Lazoyoyo |
| | style="text-align:center;" |
| | | Avicennma, Avicenna's enharmonic diesis |
| |- | | |- |
| | style="text-align:center;" | 4000/3969 | | | 7 |
| | | | 5 -4 3 -2 > | | | [[4000/3969]] |
| | style="text-align:right;" | 13.47
| | | {{monzo| 5 -4 3 -2 }} |
| | style="text-align:center;" | Octagar | | | 13.47 |
| | style="text-align:center;" |
| | | Rurutriyo |
| | style="text-align:center;" |
| | | Octagar comma |
| |- | | |- |
| | style="text-align:center;" | 6144/6125 | | | 7 |
| | | | 11 1 -3 -2 > | | | [[6144/6125]] |
| | style="text-align:right;" | 5.36
| | | {{monzo| 11 1 -3 -2 }} |
| | style="text-align:center;" | Porwell | | | 5.36 |
| | style="text-align:center;" |
| | | Sarurutrigu |
| | style="text-align:center;" |
| | | Porwell comma |
| |- | | |- |
| | style="text-align:center;" | 100/99 | | | 11 |
| | | | 2 -2 2 0 -1 > | | | [[100/99]] |
| | style="text-align:right;" | 17.40
| | | {{monzo| 2 -2 2 0 -1 }} |
| | style="text-align:center;" | Ptolemisma | | | 17.40 |
| | style="text-align:center;" |
| | | Luyoyo |
| | style="text-align:center;" |
| | | Ptolemisma |
| |- | | |- |
| | style="text-align:center;" | 441/440 | | | 11 |
| | | | -3 2 -1 2 -1 > | | | [[441/440]] |
| | style="text-align:right;" | 3.93
| | | {{monzo| -3 2 -1 2 -1 }} |
| | style="text-align:center;" | Werckisma | | | 3.93 |
| | style="text-align:center;" |
| | | Luzozogu |
| | style="text-align:center;" |
| | | Werckisma |
| |} | | |} |
|
| |
|
| ='''23 tone [[Equal_Modes|Equal Modes]]:'''= | | == Scales == |
| | |
| | Important [[mos]]ses include: |
| | |
| | * Mavila 2L5s 4334333 (13\23, 1\1) |
| | * Mavila 7L2s 133313333 (13\23, 1\1) |
| | * Sephiroth 3L4s 2525252 (7\23, 1\1) |
| | * [[Semiquartal]] 5L4s 332323232 (5\23, 1\1) |
| | |
| | The chart below shows some of the mos modes of [[mavila]] available in 23edo, mainly Pentatonic (5-note), antidiatonic (7-note), 9- and 16-note mosses. Here the outer ring represents individual step of 23edo itself, while the rings moving inward represent 16, 9, 7 and 5 note mosses: |
| | |
| | [[File:23edoMavilaMOS.jpg|alt=23edoMavilaMOS.jpg|23edoMavilaMOS.jpg]] |
| | |
| | === 23-tone mos scales === |
|
| |
|
| {| class="wikitable" | | {| class="wikitable" |
| | ! [[MOS scale]] |
| | ! Name |
| |- | | |- |
| | | 10 10 3
| | | 10 10 3 |
| | |
| | | |
| |-
| |
| | | 9 9 5
| |
| | |
| |
| |-
| |
| | | 8 8 7
| |
| | |
| |
| |-
| |
| | | 7 7 7 2
| |
| | |
| |
| |-
| |
| | | 7 2 7 7
| |
| | |
| |
| |-
| |
| | | 6 6 6 5
| |
| | |
| |
| |-
| |
| | | 6 5 6 6
| |
| | |
| |
| |-
| |
| | | 5 4 5 5 4
| |
| | | [[3L_2s|3L 2s (father)]]
| |
| |-
| |
| | | 5 4 5 4 5
| |
| | |
| |
| |-
| |
| | | 7 1 7 7 1
| |
| | |
| |
| |- | | |- |
| | | 7 1 7 1 7 | | | 9 9 5 |
| | |
| | | |
| |- | | |- |
| | | 5 5 5 5 3 | | | 8 8 7 |
| | | [[4L_1s|4L 1s (bug)]] | | | |
| |- | | |- |
| | | 5 3 5 5 5 | | | 7 7 7 2 |
| | |
| | | |
| |- | | |- |
| | | 4 4 4 4 4 3 | | | 6 6 6 5 |
| | | [[5L_1s|5L 1s (Grumpy hexatonic)]] | | | |
| |- | | |- |
| | | 4 3 4 4 4 4 | | | 5 4 5 5 4 |
| | | | | | [[3L 2s|3L 2s (oneiro-pentatonic)]] |
| |- | | |- |
| | | 5 1 5 1 5 1 5 | | | 5 4 5 4 5 |
| | | [[4L_3s|4L 3s (mish)]] | | | |
| |- | | |- |
| | | 3 3 3 5 3 3 3 | | | 7 1 7 7 1 |
| | | [[1L_6s|1L 6s (Happy heptatonic)]] | | | |
| |- | | |- |
| | | 4 3 3 3 3 3 4 | | | 7 1 7 1 7 |
| | | [[2L_5s|2L 5s (mavila, anti-diatonic)]] | | | |
| |- | | |- |
| | | 3 4 3 3 4 3 3 | | | 5 5 5 5 3 |
| | | | | | [[4L 1s|4L 1s (bug pentatonic)]] |
| |- | | |- |
| | | 3 3 4 3 3 3 4 | | | 4 4 4 4 4 3 |
| | | | | | [[5L 1s|5L 1s (machinoid)]] |
| |- | | |- |
| | | 3 3 3 4 3 3 4 | | | 5 1 5 1 5 1 5 |
| | | | | | [[4L 3s|4L 3s (smitonic)]] |
| |- | | |- |
| | | 3 3 3 4 3 4 3 | | | 3 3 3 5 3 3 3 |
| | | | | | [[1L 6s|1L 6s (antiarcheotonic)]] |
| |- | | |- |
| | | 2 5 2 5 2 5 2 | | | 4 3 3 3 3 3 4 |
| | | [[3L_4s|3L 4s (mosh)]] | | | |
| |- | | |- |
| | | 4 1 4 4 1 4 4 1 | | | 3 3 4 3 3 3 4 |
| | | [[5L_3s|5L 3s (unfair father)]]
| | | [[2L 5s|2L 5s (mavila, anti-diatonic)]] |
| |- | | |- |
| | | 3 3 3 3 3 3 3 2 | | | 4 3 3 3 3 4 3 |
| | | [[7L_1s|7L 1s (Grumpy octatonic)]] | | | |
| |- | | |- |
| | | 3 2 3 3 3 3 3 3 | | | 2 5 2 5 2 5 2 |
| | | | | | [[3L 4s|3L 4s (mosh)]] |
| |- | | |- |
| | | '''3 3 3 1 3 3 3 3 1''' | | | 4 1 4 4 1 4 4 1 |
| | | [[7L_2s|7L 2s (mavila superdiatonic)]]
| | | [[5L 3s|5L 3s (oneirotonic)]] |
| |- | | |- |
| | | 3 3 1 3 3 3 1 3 3
| | | 3 3 3 3 3 3 3 2 |
| | | | | | [[7L 1s|7L 1s (porcupoid)]] |
| |- | | |- |
| | | 3 2 3 2 3 2 3 2 3 | | | 3 3 3 1 3 3 3 3 1 |
| | | [[5L_4s|5L 4s (unfair bug)]]
| | |[[7L 2s|7L 2s (mavila superdiatonic)]] |
| |- | | |- |
| | | 2 2 2 3 2 2 3 2 2 3 | | | 3 2 3 2 3 2 3 2 3 |
| | | Mode Keter | | | [[5L 4s|5L 4s (bug semiquartal)]] |
| |- | | |- |
| | | 2 2 3 2 2 3 2 2 3 2 | | | 3 2 2 3 2 2 3 2 2 2 |
| | | Chesed | | | [[3L 7s|3L 7s (sephiroid)]] |
| |- | | |- |
| | | 2 3 2 2 3 2 2 3 2 2 | | | 4 1 1 4 1 1 4 1 1 4 1 |
| | | Netzach | | | [[4L 7s|4L 7s (kleistonic)]] |
| |- | | |- |
| | | 3 2 2 3 2 2 3 2 2 2 | | | 3 1 3 1 3 1 3 1 3 1 3 |
| | | Malkuth | | | Palestine 11 |
| |- | | |- |
| | | 2 2 3 2 2 3 2 2 2 3 | | | 3 1 1 3 1 3 1 1 3 1 3 1 1 |
| | | Binah | | | [[5L 8s|5L 8s (ateamtonic)]] |
| |- | | |- |
| | | 2 3 2 2 3 2 2 2 3 2 | | | 2 2 2 2 1 2 2 2 1 2 2 2 1 |
| | | Tiferet | | | [[10L 3s|10L 3s (luachoid)]] |
| |- | | |- |
| | | 3 2 2 3 2 2 2 3 2 2 | | | 2 2 1 2 2 1 2 2 1 2 2 1 2 1 |
| | | Yesod | | | [[9L 5s]] (Brittle [[Titanium]]) |
| |- | | |- |
| | | 2 2 3 2 2 2 3 2 2 3 | | | 2 1 2 2 1 2 2 1 2 2 1 2 2 1 |
| | | Chokmah | | | Palestine 14 |
| |- | | |- |
| | | 2 3 2 2 2 3 2 2 3 2 | | | 1 1 1 4 1 1 1 1 4 1 1 1 1 4 |
| | | Gevurah | | | [[3L 11s]] |
| |- | | |- |
| | | 3 2 2 2 3 2 2 3 2 2 | | | 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 |
| | | Hod | | | [[4L 11s|4L 11s (mynoid)]] |
| |- | | |- |
| | | '''3 1 3 1 3 1 3 1 3 1 3''' | | | 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 |
| | | Palestine 11 | | | [[8L 7s]] |
| |- | | |- |
| | | 2 <span style="font-size: 12.8000001907349px;">2 2 2 1 2 2 2 1 2 2 2 1</span>
| | | 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 1 |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Mode Tishrei</span> | | | [[7L 9s|7L 9s (mavila chromatic)]] |
| |- | | |- |
| | | 2 <span style="font-size: 12.8000001907349px;">2 2 1 2 2 2 1 2 2 2 1 2</span>
| | | 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Cheshvan</span> | | | Palestine 17 |
| |- | | |- |
| | | 2 <span style="font-size: 12.8000001907349px;">2 1 2 2 2 1 2 2 2 1 2 2</span>
| | | 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Kislev</span> | | | [[5L 13s]] |
| |- | | |- |
| | | 2 <span style="font-size: 12.8000001907349px;">1 2 2 2 1 2 2 2 1 2 2 2</span>
| | | 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Tevet</span>
| | | [[4L 15s]] |
| |-
| |
| | | 1 <span style="font-size: 12.8000001907349px;">2 2 2 1 2 2 2 1 2 2 2 2</span>
| |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Shvat</span>
| |
| |-
| |
| | | 2 2 2 1 2 2 2 1 2 2 2 2 1
| |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Adar minor</span>
| |
| |-
| |
| | | <span style="font-size: 12.8000001907349px;">2 2 1 2 2 2 1 2 2 2 2 1 2</span>
| |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Adar major</span>
| |
| |-
| |
| | | <span style="font-size: 12.8000001907349px;">2 1 2 2 2 1 2 2 2 2 1 2 2</span>
| |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Nisan</span>
| |
| |-
| |
| | | <span style="font-size: 12.8000001907349px;">1 2 2 2 1 2 2 2 2 1 2 2 2</span>
| |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Iyar</span>
| |
| |-
| |
| | | <span style="font-size: 12.8000001907349px;">2 2 2 1 2 2 2 2 1 2 2 2 1</span>
| |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Sivan</span>
| |
| |-
| |
| | | <span style="font-size: 12.8000001907349px;">2 2 1 2 2 2 2 1 2 2 2 1 2</span>
| |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Tammuz</span>
| |
| |-
| |
| | | <span style="font-size: 12.8000001907349px;">2 1 2 2 2 2 1 2 2 2 1 2 2</span> | |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Av</span>
| |
| |-
| |
| | | <span style="font-size: 12.8000001907349px;">1 2 2 2 2 1 2 2 2 1 2 2 1</span>
| |
| | | <span style="background-color: #ffffff; font-size: 12.8000001907349px;">Elul</span>
| |
| |-
| |
| | | 2 2 1 2 2 1 2 2 1 2 2 1 2 1
| |
| | |
| |
| |-
| |
| | | '''2 1 2 2 1 2 2 1 2 2 1 2 2 1'''
| |
| | | Palestine 14
| |
| |-
| |
| | | 1 1 1 4 1 1 1 1 4 1 1 1 1 4
| |
| | |
| |
| |-
| |
| | | 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
| |
| | |
| |
| |-
| |
| | | '''2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1'''
| |
| | | Palestine 17
| |
| |-
| |
| | | 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1
| |
| | |
| |
| |} | | |} |
|
| |
|
| =Books= | | While [[35edo]] is the largest edo without a nondegenerate [[5L 2s]] scale, it has both degenerate cases (the equalised 7edo and the collapsed 5edo). |
| [[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG]] | | |
| | 23edo is the largest edo without any form of 5L 2s, including the degenerate cases. |
| | |
| | === Kosmorsky's Sephiroth modes === |
| | |
| | Kosmorsky has argued that the most significant modes of 23 edo are those of the 2 2 2 3 2 2 3 2 2 3 scale ([[3L 7s|3L 7s fair mosh]]); This is derived from extending the ~1/3 comma tempered 13th Harmonic, two of which add up to the 21st harmonic and three add up to the 17th harmonic almost perfectly. Interestingly, the chord 8:13:21:34 is a fragment of the fibonacci sequence. |
| | |
| | Notated in ascending (standard) form. I have named these 10 modes according to the Sephiroth as follows: |
| | |
| | 2 2 2 3 2 2 3 2 2 3 - Mode Keter |
| | |
| | 2 2 3 2 2 3 2 2 3 2 - Chesed |
| | |
| | 2 3 2 2 3 2 2 3 2 2 - Netzach |
| | |
| | 3 2 2 3 2 2 3 2 2 2 - Malkuth |
| | |
| | 2 2 3 2 2 3 2 2 2 3 - Binah |
| | |
| | 2 3 2 2 3 2 2 2 3 2 - Tiferet |
| | |
| | 3 2 2 3 2 2 2 3 2 2 - Yesod |
| | |
| | 2 2 3 2 2 2 3 2 2 3 - Chokmah |
|
| |
|
| =Instruments= | | 2 3 2 2 2 3 2 2 3 2 - Gevurah |
| | |
| | 3 2 2 2 3 2 2 3 2 2 - Hod |
| | |
| | === Miscellaneous === |
| | 5 5 1 2 5 5 - [[Antipental blues]] (approximated from [[Dwarf17marv]]) |
| | |
| | 7 2 4 6 4 - Arcade (approximated from [[32afdo]]) |
| | |
| | 6 4 1 2 2 6 2 - [[Blackened skies]] (approximated from [[Compton]] in [[72edo]]) |
| | |
| | 5 5 3 7 3 - Geode (approximated from [[6afdo]]) |
| | |
| | 5 4 2 2 4 2 4 - Lost phantom (approximated from [[Mavila]] in [[30edo]]) |
| | |
| | 6 4 2 1 5 1 4 - [[Lost spirit]] (approximated from [[Meantone]] in [[31edo]]) |
| | |
| | 5 2 6 6 4 - Mechanical (approximated from [[31afdo]]) |
| | |
| | 5 4 4 2 8 - Mushroom (approximated from [[30afdo]]) |
| | |
| | 6 4 3 7 3 - Nightdrive (approximated from [[Mavila]] in [[30edo]]) |
| | |
| | 6 4 1 2 6 4 - Pelagic (approximated from [[Mavila]] in [[30edo]]) |
| | |
| | 2 3 8 2 8 - Approximation of [[Pelog]] lima |
| | |
| | 4 3 6 6 4 - Springwater (approximated from [[8afdo]]) |
| | |
| | 2 5 2 4 6 4 - Starship (approximated from [[68ifdo]]) |
| | |
| | 2 4 6 1 10 - Tightrope (this is the original/default tuning) |
| | |
| | 6 7 4 2 4 - Underpass (approximated from [[10afdo]]) |
| | |
| | 2 5 6 6 4 - Volcanic (approximated from [[16afdo]]) |
| | |
| | == Instruments == |
|
| |
|
| [[File:Icositriphonic_Guitar.PNG|alt=Icositriphonic_Guitar.PNG|601x305px|Icositriphonic_Guitar.PNG]] | | [[File:Icositriphonic_Guitar.PNG|alt=Icositriphonic_Guitar.PNG|601x305px|Icositriphonic_Guitar.PNG]] |
| Line 647: |
Line 588: |
| This movie is a series of still shots Chris took during the process of making a 23 edo guitar in a stick like form. At the end the guitar is played without effects etc. and the open string tuning is sounded - which starts with a normal E and then adjusted to the 9th / 7th fret unison, like a typical 12edo guitar fashion. | | This movie is a series of still shots Chris took during the process of making a 23 edo guitar in a stick like form. At the end the guitar is played without effects etc. and the open string tuning is sounded - which starts with a normal E and then adjusted to the 9th / 7th fret unison, like a typical 12edo guitar fashion. |
|
| |
|
| <youtube>K4iO7k152og</youtube> [[Category:11/7]] | | <youtube>K4iO7k152og</youtube> |
| [[Category:23-tone]]
| | |
| [[Category:23edo]] | | === Lumatone === |
| [[Category:5/3]]
| | See: [[Lumatone mapping for 23edo]] |
| [[Category:edo]]
| | |
| [[Category:guitar]]
| | == Music == |
| [[Category:intervals]] | | {{Main|23edo/Music}} |
| [[Category:keyboard]] | | {{Catrel|23edo tracks}} |
| [[Category:listen]]
| | |
| [[Category:mavila]]
| | == Further reading == |
| [[Category:modes]]
| | [[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG|thumb|''Icosikaitriphonic Scales for Guitar'' cover art.]] |
| [[Category:prime_edo]] | | * [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosikaitriphonic Scales for Guitar: A Repository of Theory, Reference Materials, and Scale Charts for Xentonal Families]''. 2010. |
| [[Category:subgroup]] | | |
| [[Category:theory]] | | [[Category:23-tone scales]] |
| [[Category:todo:unify_precision]] | | [[Category:Guitar]] |
| [[Category:twentuning]] | | [[Category:Mavila]] |
| | [[Category:Modes]] |
| | [[Category:Twentuning]] |