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The ''5-limit'' consists of all [[Just_intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called [http://en.wikipedia.org/wiki/Regular_number regular numbers]. Some examples of 5-limit intervals are [[5/4|5/4]], [[6/5|6/5]], [[10/9|10/9]] and [[81/80|81/80]]. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, [[4/3|4/3]], [[3/2|3/2]], [[8/5|8/5]], [[5/3|5/3]], 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.
The '''5-limit''' consists of all [[just intonation]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called [http://en.wikipedia.org/wiki/Regular_number regular numbers]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]]. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, [[4/3]], [[3/2]], [[8/5]], [[5/3]], [[2/1]]. Approximating these ratios has been basic to Western common-practice music since the Renaissance.


The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a [http://en.wikipedia.org/wiki/Hexagonal_lattice hexagonal lattice] or as a [http://en.wikipedia.org/wiki/Square_lattice square lattice]; this can be done automatically by [http://www.huygens-fokker.org/scala/ Scala]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [http://en.wikipedia.org/wiki/Hexagonal_tiling hexagonal tiling].
The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a [http://en.wikipedia.org/wiki/Hexagonal_lattice hexagonal lattice] or as a [http://en.wikipedia.org/wiki/Square_lattice square lattice]; this can be done automatically by [http://www.huygens-fokker.org/scala/ Scala]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [http://en.wikipedia.org/wiki/Hexagonal_tiling hexagonal tiling].


[[EDO|EDO]]s which do relatively well in approximating the 5-limit are [[2edo|2edo]], [[3edo|3edo]], [[7edo|7edo]], [[9edo|9edo]], [[10edo|10edo]], [[12edo|12edo]], [[19edo|19edo]], [[22edo|22edo]], [[31edo|31edo]], [[34edo|34edo]], [[53edo|53edo]], [[118edo|118edo]] and [[289edo|289edo]].
[[EDO]]s which do relatively well in approximating the 5-limit are [[2edo]], [[3edo]], [[7edo]], [[9edo]], [[10edo]], [[12edo]], [[19edo]], [[22edo]], [[31edo]], [[34edo]], [[53edo]], [[118edo]] and [[289edo]].


==Syntonic Comma Pairs==
== Syntonic Comma Pairs ==


A significant interval in 5-limit JI is [[81/80|81/80]], the syntonic comma or Didymus' comma, which measures about 21.5¢. Although it rarely appears as an interval in a scale, it represents the difference between many 5-limit intervals and a nearby [[3-limit|3-limit]] (Pythagorean) interval. 81/80 is tempered out in [[12edo|12edo]], [[Meantone|meantone]], and many other related systems, meaning that those 5- and 3-limit distinctions are obliterated and one interval stands in for each. Living in a largely [[12edo|12edo]] musical culture from birth, we are not accustomed to distinguishing two different major thirds, two different minor seconds, etc. Below is a list of some common intervals involving 3 and 5 which are distinguished by 81/80. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). '''Bold''' fractions are simplest for this interval category.
A significant interval in 5-limit JI is [[81/80|81/80]], the syntonic comma or Didymus' comma, which measures about 21.5¢. Although it rarely appears as an interval in a scale, it represents the difference between many 5-limit intervals and a nearby [[3-limit|3-limit]] (Pythagorean) interval. 81/80 is tempered out in [[12edo|12edo]], [[Meantone|meantone]], and many other related systems, meaning that those 5- and 3-limit distinctions are obliterated and one interval stands in for each. Living in a largely [[12edo|12edo]] musical culture from birth, we are not accustomed to distinguishing two different major thirds, two different minor seconds, etc. Below is a list of some common intervals involving 3 and 5 which are distinguished by 81/80. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). '''Bold''' fractions are simplest for this interval category.
Line 16: Line 16:
! colspan="3" |yoyo or gugu interval (6561/6400)
! colspan="3" |yoyo or gugu interval (6561/6400)
|-
|-
! colspan="2" |[[Kite's color notation|interval name]]
! colspan="2" |[[Kite's color notation|Color name]]
! | ratio
! ratio
! | cents
! cents
! |
!  
! |
!  
! colspan="2" |[[Kite's color notation|interval name]]
! colspan="2" |[[Kite's color notation|Color name]]
! | ratio
! ratio
! | cents
! cents
![[Kite's color notation|interval]]
! [[Kite's color notation|Color]]
! | ratio
! ratio
! | cents
! cents
|-
|-
|w1
| w1
|wa unison
| wa unison
| | '''[[1/1|1/1]]'''
| '''[[1/1]]'''
| | '''0.000'''
| '''0.000'''
| | unison
| unison
| | C
| C
|g1
| g1
|gu comma
| gu comma
| | [[81/80|81/80]]
| [[81/80]]
| | 21.506
| 21.506
|Lgg1
| Lgg1
| | [[6561/6400|6561/6400]]
| [[6561/6400]]
| | 43.013
| 43.013
|-
|-
|Lw1
| Lw1
|large wa 1sn
| large wa 1sn
| | [[2187/2048|2187/2048]]
| [[2187/2048]]
| | 113.685
| 113.685
| | aug. unison
| aug. unison
| | C#
| C#
|Ly1
| Ly1
|large yo 1sn
| large yo 1sn
| | [[135/128|135/128]]
| [[135/128]]
| | 92.179
| 92.179
|yy1
| yy1
| | '''[[25/24|25/24]]'''
| '''[[25/24]]'''
| | '''70.672'''
| '''70.672'''
|-
|-
|sw2
| sw2
|small wa 2nd
| small wa 2nd
| | [[256/243|256/243]]
| [[256/243]]
| | 90.225
| 90.225
| | minor 2nd
| minor 2nd
| | Db
| Db
|g2
| g2
|gu 2nd
| gu 2nd
| | '''[[16/15|16/15]]'''
| '''[[16/15]]'''
| | '''111.731'''
| '''111.731'''
|gg2
| gg2
| | [[27/25|27/25]]
| [[27/25]]
| | 133.238
| 133.238
|-
|-
|w2
| w2
|wa 2nd
| wa 2nd
| | '''[[9/8|9/8]]'''
| '''[[9/8]]'''
| | '''203.910'''
| '''203.910'''
| | major 2nd
| major 2nd
| | D
| D
|y2
| y2
|yo 2nd
| yo 2nd
| | [[10/9|10/9]]
| [[10/9]]
| | 182.404
| 182.404
|syy2
| syy2
| | [[800/729|800/729]]
| [[800/729]]
| | 160.897
| 160.897
|-
|-
|Lw2
| Lw2
|large wa 2nd
| large wa 2nd
| | [[19683/16384|19683/16384]]
| [[19683/16384]]
| | 317.595
| 317.595
| | aug. 2nd
| aug. 2nd
| | D#
| D#
|Ly2
| Ly2
|large yo 2nd
| large yo 2nd
| | [[1215/1024|1215/1024]]
| [[1215/1024]]
| | 296.089
| 296.089
|yy2
| yy2
| | '''[[75/64|75/64]]'''
| '''[[75/64]]'''
| | '''274.582'''
| '''274.582'''
|-
|-
|w3
| w3
|wa 3rd
| wa 3rd
| | [[32/27|32/27]]
| [[32/27]]
| | 294.135
| 294.135
| | minor 3rd
| minor 3rd
| | Eb
| Eb
|g3
| g3
|gu 3rd
| gu 3rd
| | '''[[6/5|6/5]]'''
| '''[[6/5]]'''
| | '''315.641'''
| '''315.641'''
|gg3
| gg3
| | [[243/200|243/200]]
| [[243/200]]
| | 337.148
| 337.148
|-
|-
|Lw3
| Lw3
|large wa 3rd
| large wa 3rd
| | [[81/64|81/64]]
| [[81/64]]
| | 407.820
| 407.820
| | major 3rd
| major 3rd
| | E
| E
|y3
| y3
|yo 3rd
| yo 3rd
| | '''[[5/4|5/4]]'''
| '''[[5/4]]'''
| | '''386.314'''
| '''386.314'''
|yy3
| yy3
| | [[100/81|100/81]]
| [[100/81]]
| | 364.807
| 364.807
|-
|-
|sw4
| sw4
|small wa 4th
| small wa 4th
| | [[8192/6561|8192/6561]]
| [[8192/6561]]
| | 384.360
| 384.360
| | dim. fourth
| dim. fourth
| | Fb
| Fb
|sg4
| sg4
|small gu 4th
| small gu 4th
| | [[512/405|512/405]]
| [[512/405]]
| | 405.866
| 405.866
|gg4
| gg4
| | '''[[32/25|32/25]]'''
| '''[[32/25]]'''
| | '''427.373'''
| '''427.373'''
|-
|-
|w4
| w4
|wa 4th
| wa 4th
| | '''[[4/3|4/3]]'''
| '''[[4/3]]'''
| | '''498.045'''
| '''498.045'''
| | fourth
| fourth
| | F
| F
|g4
| g4
|gu 4th
| gu 4th
| | [[27/20|27/20]]
| [[27/20]]
| | 519.551
| 519.551
|Lgg4
| Lgg4
| | [[2187/1600|2187/1600]]
| [[2187/1600]]
| | 541.058
| 541.058
|-
|-
|Lw4
| Lw4
|large wa 4th
| large wa 4th
| | [[729/512|729/512]]
| [[729/512]]
| | 611.730
| 611.730
| | aug. fourth
| aug. fourth
| | F#
| F#
|y4
| y4
|yo 4th
| yo 4th
| | [[45/32|45/32]]
| [[45/32]]
| | 590.224
| 590.224
|yy4
| yy4
| | '''[[25/18|25/18]]'''
| '''[[25/18]]'''
| | '''568.717'''
| '''568.717'''
|-
|-
|sw5
| sw5
|small wa 5th
| small wa 5th
| | [[1024/729|1024/729]]
| [[1024/729]]
| | 588.270
| 588.270
| | dim. fifth
| dim. fifth
| | Gb
| Gb
|g5
| g5
|gu 5th
| gu 5th
| | [[64/45|64/45]]
| [[64/45]]
| | 609.776
| 609.776
|gg5
| gg5
| | '''[[36/25|36/25]]'''
| '''[[36/25]]'''
| | '''631.283'''
| '''631.283'''
|-
|-
|w5
| w5
|wa 5th
| wa 5th
| | '''[[3/2|3/2]]'''
| '''[[3/2]]'''
| | '''701.955'''
| '''701.955'''
| | fifth
| fifth
| | G
| G
|y5
| y5
|yo 5th
| yo 5th
| | [[40/27|40/27]]
| [[40/27]]
| | 680.449
| 680.449
|syy5
| syy5
| | [[3200/2187|3200/2187]]
| [[3200/2187]]
| | 658.942
| 658.942
|-
|-
|Lw5
| Lw5
|large wa 5th
| large wa 5th
| | [[6561/4096|6561/4096]]
| [[6561/4096]]
| | 815.640
| 815.640
| | aug. fifth
| aug. fifth
| | G#
| G#
|Ly5
| Ly5
|large yo 5th
| large yo 5th
| | [[405/256|405/256]]
| [[405/256]]
| | 794.134
| 794.134
|yy5
| yy5
| | '''[[25/16|25/16]]'''
| '''[[25/16]]'''
| | '''772.627'''
| '''772.627'''
|-
|-
|sw6
| sw6
|small wa 6th
| small wa 6th
| | [[128/81|128/81]]
| [[128/81]]
| | 792.180
| 792.180
| | minor 6th
| minor 6th
| | Ab
| Ab
|g6
| g6
|gu 6th
| gu 6th
| | '''[[8/5|8/5]]'''
| '''[[8/5]]'''
| | '''813.686'''
| '''813.686'''
|gg6
| gg6
| | [[81/50|81/50]]
| [[81/50]]
| | 835.193
| 835.193
|-
|-
|w6
| w6
|wa 6th
| wa 6th
| | [[27/16|27/16]]
| [[27/16]]
| | 905.865
| 905.865
| | major 6th
| major 6th
| | A
| A
|y6
| y6
|yo 6th
| yo 6th
| | '''[[5/3|5/3]]'''
| '''[[5/3]]'''
| | '''884.359'''
| '''884.359'''
|yy6
| yy6
| | [[400/243|400/243]]
| [[400/243]]
| | 862.852
| 862.852
|-
|-
|sw7
| sw7
|small wa 7th
| small wa 7th
| | [[32768/19683|32768/19683]]
| [[32768/19683]]
| | 882.405
| 882.405
| | dim. 7th
| dim. 7th
| | Bbb
| Bbb
|sg7
| sg7
|small gu 7th
| small gu 7th
| | [[2048/1215|2048/1215]]
| [[2048/1215]]
| | 903.911
| 903.911
|gg7
| gg7
| | '''[[128/75|128/75]]'''
| '''[[128/75]]'''
| | '''925.418'''
| '''925.418'''
|-
|-
|w7
| w7
|wa 7th
| wa 7th
| | [[16/9|16/9]]
| [[16/9]]
| | 996.090
| 996.090
| | minor 7th
| minor 7th
| | Bb
| Bb
|g7
| g7
|gu 7th
| gu 7th
| | '''[[9/5|9/5]]'''
| '''[[9/5]]'''
| | '''1017.596'''
| '''1017.596'''
|Lgg7
| Lgg7
| | [[729/400|729/400]]
| [[729/400]]
| | 1039.103
| 1039.103
|-
|-
|Lw7
| Lw7
|large wa 7th
| large wa 7th
| | [[243/128|243/128]]
| [[243/128]]
| | 1109.775
| 1109.775
| | major 7th
| major 7th
| | B
| B
|y7
| y7
|yo 7th
| yo 7th
| | '''[[15/8|15/8]]'''
| '''[[15/8]]'''
| | '''1088.269'''
| '''1088.269'''
|yy7
| yy7
| | [[50/27|50/27]]
| [[50/27]]
| | 1066.762
| 1066.762
|-
|-
|sw8
| sw8
|small wa 8ve
| small wa 8ve
| | [[4096/2187|4096/2187]]
| [[4096/2187]]
| | 1086.315
| 1086.315
| | dim. octave
| dim. octave
| | Cb
| Cb
|sg8
| sg8
|small gu 8ve
| small gu 8ve
| | [[256/135|256/135]]
| [[256/135]]
| | 1107.821
| 1107.821
|gg8
| gg8
| | '''[[48/25|48/25]]'''
| '''[[48/25]]'''
| | '''1129.328'''
| '''1129.328'''
|-
|-
|w8
| w8
|wa 8ve
| wa 8ve
| | '''[[2/1|2/1]]'''
| '''[[2/1]]'''
| | '''1200.000'''
| '''1200.000'''
| | octave
| octave
| | C
| C
|y8
| y8
|yo 8ve
| yo 8ve
| | [[160/81|160/81]]
| [[160/81]]
| | 1178.494
| 1178.494
|syy8
| syy8
| | [[12800/6561|12800/6561]]
| [[12800/6561]]
| | 1156.987
| 1156.987
|}
|}


Line 313: Line 313:
See [[Harmonic_Limit|Harmonic Limit]]
See [[Harmonic_Limit|Harmonic Limit]]


=Music=
== Music ==
[http://clones.soonlabel.com/public/micro/just/Duodene/duodene2.mp3 Duodene2] by [[Chris_Vaisvil|Chris Vaisvil]]
* [http://clones.soonlabel.com/public/micro/just/Duodene/duodene2.mp3 Duodene2] by [[Chris Vaisvil]]
 
* [http://micro.soonlabel.com/just/Ariels-JI/ariels-12-tone-ji.mp3 Ariel's 12-tone JI] by [[Chris Vaisvil]]
[http://micro.soonlabel.com/just/Ariels-JI/ariels-12-tone-ji.mp3 Ariel's 12-tone JI] by Chris Vaisvil
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/The%20Ballad%20of%20Jed%20Clampett.mp3 The Ballad of Jed Clampett] by [http://en.wikipedia.org/wiki/Paul_Henning Paul Henning]
 
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/Do%20Wah%20Diddy.mp3 Do Wah Diddy Diddy] by [http://en.wikipedia.org/wiki/Jeff_Barry Jeff Barry] and [http://en.wikipedia.org/wiki/Ellie_Greenwich Ellie Greenwich]
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/The%20Ballad%20of%20Jed%20Clampett.mp3 The Ballad of Jed Clampett] by [http://en.wikipedia.org/wiki/Paul_Henning Paul Henning]
* [https://soundcloud.com/williamcopper/0511_1 Symphony 4, first movement] by [http://www.williamcopper.com William Copper]
 
* [http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Magnificat0465.mp3 Magnificat] by [http://www.williamcopper.com William Copper]
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/Do%20Wah%20Diddy.mp3 Do Wah Diddy Diddy] by [http://en.wikipedia.org/wiki/Jeff_Barry Barry] and [http://en.wikipedia.org/wiki/Ellie_Greenwich Greenwich]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Catch%20for%20Woodwind%20Quintet-0570.mp3 Catch for Woodwin Quintet] by [http://www.hartenshield.com/william_copper.html William Copper]
 
[https://soundcloud.com/williamcopper/0511_1 Symphony 4, first movement] by [http://www.williamcopper.com William Copper]
 
[http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Magnificat0465.mp3 Magnificat] by [http://www.williamcopper.com William Copper]
 
[http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Catch%20for%20Woodwind%20Quintet-0570.mp3 Catch for Woodwin Quintet] by [http://www.hartenshield.com/william_copper.html William Copper]


[[Category:5-limit| ]] <!-- main article -->
[[Category:5-limit| ]] <!-- main article -->
[[Category:example]]
[[Category:Interval]]
[[Category:interval]]
[[Category:Lattice]]
[[Category:lattice]]
[[Category:Listen]]
[[Category:limit]]
[[Category:Sound example]]
[[Category:listen]]
[[Category:Prime limit]]
[[Category:prime_limit]]
[[Category:Rank 3]]
[[Category:rank_3]]

Revision as of 22:02, 31 October 2018

The 5-limit consists of all just intonation intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called regular numbers. Some examples of 5-limit intervals are 5/4, 6/5, 10/9 and 81/80. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.

The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a hexagonal lattice or as a square lattice; this can be done automatically by Scala. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a hexagonal tiling.

EDOs which do relatively well in approximating the 5-limit are 2edo, 3edo, 7edo, 9edo, 10edo, 12edo, 19edo, 22edo, 31edo, 34edo, 53edo, 118edo and 289edo.

Syntonic Comma Pairs

A significant interval in 5-limit JI is 81/80, the syntonic comma or Didymus' comma, which measures about 21.5¢. Although it rarely appears as an interval in a scale, it represents the difference between many 5-limit intervals and a nearby 3-limit (Pythagorean) interval. 81/80 is tempered out in 12edo, meantone, and many other related systems, meaning that those 5- and 3-limit distinctions are obliterated and one interval stands in for each. Living in a largely 12edo musical culture from birth, we are not accustomed to distinguishing two different major thirds, two different minor seconds, etc. Below is a list of some common intervals involving 3 and 5 which are distinguished by 81/80. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). Bold fractions are simplest for this interval category.

wa (3-limit) interval interval category yo or gu (5-limit) interval (81/80) yoyo or gugu interval (6561/6400)
Color name ratio cents Color name ratio cents Color ratio cents
w1 wa unison 1/1 0.000 unison C g1 gu comma 81/80 21.506 Lgg1 6561/6400 43.013
Lw1 large wa 1sn 2187/2048 113.685 aug. unison C# Ly1 large yo 1sn 135/128 92.179 yy1 25/24 70.672
sw2 small wa 2nd 256/243 90.225 minor 2nd Db g2 gu 2nd 16/15 111.731 gg2 27/25 133.238
w2 wa 2nd 9/8 203.910 major 2nd D y2 yo 2nd 10/9 182.404 syy2 800/729 160.897
Lw2 large wa 2nd 19683/16384 317.595 aug. 2nd D# Ly2 large yo 2nd 1215/1024 296.089 yy2 75/64 274.582
w3 wa 3rd 32/27 294.135 minor 3rd Eb g3 gu 3rd 6/5 315.641 gg3 243/200 337.148
Lw3 large wa 3rd 81/64 407.820 major 3rd E y3 yo 3rd 5/4 386.314 yy3 100/81 364.807
sw4 small wa 4th 8192/6561 384.360 dim. fourth Fb sg4 small gu 4th 512/405 405.866 gg4 32/25 427.373
w4 wa 4th 4/3 498.045 fourth F g4 gu 4th 27/20 519.551 Lgg4 2187/1600 541.058
Lw4 large wa 4th 729/512 611.730 aug. fourth F# y4 yo 4th 45/32 590.224 yy4 25/18 568.717
sw5 small wa 5th 1024/729 588.270 dim. fifth Gb g5 gu 5th 64/45 609.776 gg5 36/25 631.283
w5 wa 5th 3/2 701.955 fifth G y5 yo 5th 40/27 680.449 syy5 3200/2187 658.942
Lw5 large wa 5th 6561/4096 815.640 aug. fifth G# Ly5 large yo 5th 405/256 794.134 yy5 25/16 772.627
sw6 small wa 6th 128/81 792.180 minor 6th Ab g6 gu 6th 8/5 813.686 gg6 81/50 835.193
w6 wa 6th 27/16 905.865 major 6th A y6 yo 6th 5/3 884.359 yy6 400/243 862.852
sw7 small wa 7th 32768/19683 882.405 dim. 7th Bbb sg7 small gu 7th 2048/1215 903.911 gg7 128/75 925.418
w7 wa 7th 16/9 996.090 minor 7th Bb g7 gu 7th 9/5 1017.596 Lgg7 729/400 1039.103
Lw7 large wa 7th 243/128 1109.775 major 7th B y7 yo 7th 15/8 1088.269 yy7 50/27 1066.762
sw8 small wa 8ve 4096/2187 1086.315 dim. octave Cb sg8 small gu 8ve 256/135 1107.821 gg8 48/25 1129.328
w8 wa 8ve 2/1 1200.000 octave C y8 yo 8ve 160/81 1178.494 syy8 12800/6561 1156.987

It is important to note that 5-limit music does not mean favoring intervals of 5 over intervals of 3. It means allowing for both 3's and 5's in generating harmonic material, and so it is an interplay between both. The 5-limit includes the 3-limit -- a work in 5-limit JI will utilize intervals from both sides of the chart above.

See Harmonic Limit

Music

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