81/80: Difference between revisions

(One intermediate revision by one other user not shown)

Line 5:

{{Infobox Interval

| Name = syntonic comma, Didymus' comma, meantone comma, Ptolemaic comma

| Color name = g1, ~~Gu comma~~, <br/> ~~gu unison~~

| Color name = g1, gu unison,<br/>gM, guma

| Comma = yes

| Sound = audacity pluck 81 80.wav

Line 11:

The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the [[64:81:96]] ~~chord~~ is quite dissonant, but flattening the third by 81/80 ~~gives~~ the much more consonant [[4:5:6]]. However, adding/subtracting this comma to/from the [[4/3|perfect fourth]], [[3/2|fifth]], or [[2/1|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to ~~5-limit just intonation~~ will create ~~such wolves~~; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones~~, and~~ also equates [[9/8]] ~~and~~ [[10/9]], giving a tuning for the [[tone|whole tone]] which is intermediate between ~~10/9 and 9/8~~; hence the name "meantone".

The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the pythagorean major triad, [[64:81:96]], is quite dissonant, but flattening the 81/64 major third by 81/80 leads to the much more consonant [[4:5:6]] chord, with a 5/4 major third in place of 81/64. However, adding/subtracting this comma to/from the [[4/3|perfect fourth]], [[3/2|fifth]], or [[2/1|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to make intervals and chords pure in one key will create wolf intervals in others; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones. This also equates [[10/9]] with [[9/8]], giving a tuning for the [[tone|whole tone]] which is intermediate between them; hence the name "meantone".

81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].

@@ Line 5: / Line 5: @@
 {{Infobox Interval
 | Name = syntonic comma, Didymus' comma, meantone comma, Ptolemaic comma
-| Color name = g1, Gu comma, <br/> gu unison
+| Color name = g1, gu unison,<br/>gM, guma
 | Comma = yes
 | Sound = audacity pluck 81 80.wav
@@ Line 11: / Line 11: @@
 {{Wikipedia|Syntonic comma}}
-The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the [[64:81:96]] chord is quite dissonant, but flattening the third by 81/80 gives the much more consonant [[4:5:6]]. However, adding/subtracting this comma to/from the [[4/3|perfect fourth]], [[3/2|fifth]], or [[2/1|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones, and also equates [[9/8]] and [[10/9]], giving a tuning for the [[tone|whole tone]] which is intermediate between 10/9 and 9/8; hence the name "meantone".
+The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the pythagorean major triad, [[64:81:96]], is quite dissonant, but flattening the 81/64 major third by 81/80 leads to the much more consonant [[4:5:6]] chord, with a 5/4 major third in place of 81/64. However, adding/subtracting this comma to/from the [[4/3|perfect fourth]], [[3/2|fifth]], or [[2/1|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to make intervals and chords pure in one key will create wolf intervals in others; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones. This also equates [[10/9]] with [[9/8]], giving a tuning for the [[tone|whole tone]] which is intermediate between them; hence the name "meantone".
 /80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].