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Trouble reading roman numeral notation with flats


What does the roman numeral notation mean for this borrowed chord?Harmony and roman numeral analysis: how to deal with chromaticism?Roman numeral notation for a suspended chord?Putting Roman numeral chord notation into practiceStrikethrough and Roman numeral analysis in SchoenbergRoman numeral chord notation in minor scale?Roman Numeral Chords with SlashOrigin of Roman Numeral AnalysisRoman numeral anaysis helpUsing Roman Numeral Notation with Notes in the Bass (not figured bass)













1















I am working my way through an article on transformations in rock harmony, but I am getting stuck trying to read the author's roman numeral analysis. I would really appreciate some help understanding how to read flattened roman numerals. Here is an example of what I mean:



enter image description here



Thank you!










share|improve this question


























    1















    I am working my way through an article on transformations in rock harmony, but I am getting stuck trying to read the author's roman numeral analysis. I would really appreciate some help understanding how to read flattened roman numerals. Here is an example of what I mean:



    enter image description here



    Thank you!










    share|improve this question
























      1












      1








      1








      I am working my way through an article on transformations in rock harmony, but I am getting stuck trying to read the author's roman numeral analysis. I would really appreciate some help understanding how to read flattened roman numerals. Here is an example of what I mean:



      enter image description here



      Thank you!










      share|improve this question














      I am working my way through an article on transformations in rock harmony, but I am getting stuck trying to read the author's roman numeral analysis. I would really appreciate some help understanding how to read flattened roman numerals. Here is an example of what I mean:



      enter image description here



      Thank you!







      roman-numerals functional-harmony






      share|improve this question













      share|improve this question











      share|improve this question




      share|improve this question










      asked 3 hours ago









      286642286642

      1338




      1338




















          5 Answers
          5






          active

          oldest

          votes


















          1














          TLDR;



          a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling




          In the usual Roman numeral analysis we have these conventions:



          First we have to give a label for a key like E: for E major or Em: for E minor. Then we have these points...



          • capital numeral means major triad

          • lower case numeral means minor triad

          • 'o' after numeral means diminished triad

          • '7' means a diatonic seventh above root

          • a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling

          That last rule is the crux of your question.



          In E major the diatonic chord of the sixth degree is C# minor and gets labelled vi.



          In E minor the diatonic chord of the sixth degree is C major and gets labelled VI.



          If the VI from E minor is used in E major the chord root is altered from C# to C natural. (This is called chord borrowing.) When in E major that change in the chord root is made explicit in the Roman numeral figure by using the flat to show the change of chord root: bVI.



          The reason your book's example is confusing is because the key signature on the staff is E minor, but the analysis key shows E which is read as E major (it should be Em: to mean minor) and then the Roman numeral figures are labeled with flats which makes things seem like the assumed key is E major. Basically the labeling is a mess and it makes confusing the use of sharps and flats on Roman numeral figures.



          As others point out, the book doesn't even label the chords as 7th chords!



          The book seems sloppy.



          @Tim makes a good point that Roman numeral analysis can be tricky with rock music. The real conundrum to me is the final chord. It's an E dominant 7th with a G natural on top. That is a very bluesy sound. It's a very familiar sound, but Roman numerals are a poor way to label it. There is no way to indicate the simultaneous G# and G natural... and the minor 7th. About the only thing the I label gets right is a root on E and a perfect fifth of B. Tread carefully with harmonic analysis of rock music. You might find a lot of inappropriate labels.






          share|improve this answer























          • Thank you for the great response! Your answer was very clear--I appreciate it!

            – 286642
            39 mins ago


















          2














          Are you confused because the song's in E minor, but the Roman numeral chord names are relative to E MAJOR? Yes, that's how it works. The numbers only care that E is the tonic. E minor triad is i, E major triad is I. The uppercase Roman number doesn't mean 'diatonic', 'in key' or 'correct', it just means 'major'.



          Similarly for modified numerals. G is the 'correct' third note of E minor, but G B D is called ♭III. That's ♭ because G is the flattened 3rd of E major, III because it;s a major triad.



          G B♭ D would be ♭iii. That's ♭ because it's built on the flat 3rd of E major, iii because it's a minor triad.



          I repeat, these names would be the same whether we were in E major, E minor or E anything-else.






          share|improve this answer






























            1














            RNA isn't designed for analysing 'rock' music. It sort of works when non-diatonic chords need naming, as we use 'b' in certain circumstances. As in key E. There is no D and no C, only D# and C#. So, we have to call the D chord bVII and C chord bVI. It gets as close as we can.



            The bIII represents G, again because there's no G in key E: G# is iii (III) but we need the flattened G#, making G. Thus bIII.






            share|improve this answer






























              1














              The trouble is:



              The key sign is G-major/e-minor:
              (in e-minor the flat should be a natural sign to explain the VI and VII degree are not of the harmonic scale!)



              Indeed this progression is notated in respect of E-major!



              D - D - D - C - G - C - D - E



              If you transpose this figure to C you will understand that the flats are concerning



              Bb - Bb - Bb - Ab - Eb - Ab - Bb - G - C (with the blue note b10)






              share|improve this answer






























                1














                The notation seems a bit confusing. The first chord is a D7 in the key of either E-Major or e-minor; the author isn't quite so clear. The key signature indicates e-minor in which case the chord is VII (no flat needed.) If the author shows keys in ALL CAPS (which has been outdated since 1834), the chord VII7 but that disagrees with the single sharp signature. The author also leaves off the seventh indicator.



                Perhaps we can assume the author uses E irrespective of Major or minor mode and that he leaves off the seventh. This still leaved the bVII vs VII so it's still still. The (V) must mean the seventh of G.



                The author shows a descending scale (the caratted numerals above some of the chords) so perhaps that may help. I think he's using the Roman numerals associated with E-Major but the score is written as e-minor; he's assuming that all chords have sevenths (OK, but limiting). Then we have (if the score were in e-minor): VII7, VII7, VII7, VI7, III, VII7, VI7, V7, IM7 (at least as I would notate it.)






                share|improve this answer






















                  Your Answer








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                  5 Answers
                  5






                  active

                  oldest

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                  5 Answers
                  5






                  active

                  oldest

                  votes









                  active

                  oldest

                  votes






                  active

                  oldest

                  votes









                  1














                  TLDR;



                  a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling




                  In the usual Roman numeral analysis we have these conventions:



                  First we have to give a label for a key like E: for E major or Em: for E minor. Then we have these points...



                  • capital numeral means major triad

                  • lower case numeral means minor triad

                  • 'o' after numeral means diminished triad

                  • '7' means a diatonic seventh above root

                  • a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling

                  That last rule is the crux of your question.



                  In E major the diatonic chord of the sixth degree is C# minor and gets labelled vi.



                  In E minor the diatonic chord of the sixth degree is C major and gets labelled VI.



                  If the VI from E minor is used in E major the chord root is altered from C# to C natural. (This is called chord borrowing.) When in E major that change in the chord root is made explicit in the Roman numeral figure by using the flat to show the change of chord root: bVI.



                  The reason your book's example is confusing is because the key signature on the staff is E minor, but the analysis key shows E which is read as E major (it should be Em: to mean minor) and then the Roman numeral figures are labeled with flats which makes things seem like the assumed key is E major. Basically the labeling is a mess and it makes confusing the use of sharps and flats on Roman numeral figures.



                  As others point out, the book doesn't even label the chords as 7th chords!



                  The book seems sloppy.



                  @Tim makes a good point that Roman numeral analysis can be tricky with rock music. The real conundrum to me is the final chord. It's an E dominant 7th with a G natural on top. That is a very bluesy sound. It's a very familiar sound, but Roman numerals are a poor way to label it. There is no way to indicate the simultaneous G# and G natural... and the minor 7th. About the only thing the I label gets right is a root on E and a perfect fifth of B. Tread carefully with harmonic analysis of rock music. You might find a lot of inappropriate labels.






                  share|improve this answer























                  • Thank you for the great response! Your answer was very clear--I appreciate it!

                    – 286642
                    39 mins ago















                  1














                  TLDR;



                  a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling




                  In the usual Roman numeral analysis we have these conventions:



                  First we have to give a label for a key like E: for E major or Em: for E minor. Then we have these points...



                  • capital numeral means major triad

                  • lower case numeral means minor triad

                  • 'o' after numeral means diminished triad

                  • '7' means a diatonic seventh above root

                  • a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling

                  That last rule is the crux of your question.



                  In E major the diatonic chord of the sixth degree is C# minor and gets labelled vi.



                  In E minor the diatonic chord of the sixth degree is C major and gets labelled VI.



                  If the VI from E minor is used in E major the chord root is altered from C# to C natural. (This is called chord borrowing.) When in E major that change in the chord root is made explicit in the Roman numeral figure by using the flat to show the change of chord root: bVI.



                  The reason your book's example is confusing is because the key signature on the staff is E minor, but the analysis key shows E which is read as E major (it should be Em: to mean minor) and then the Roman numeral figures are labeled with flats which makes things seem like the assumed key is E major. Basically the labeling is a mess and it makes confusing the use of sharps and flats on Roman numeral figures.



                  As others point out, the book doesn't even label the chords as 7th chords!



                  The book seems sloppy.



                  @Tim makes a good point that Roman numeral analysis can be tricky with rock music. The real conundrum to me is the final chord. It's an E dominant 7th with a G natural on top. That is a very bluesy sound. It's a very familiar sound, but Roman numerals are a poor way to label it. There is no way to indicate the simultaneous G# and G natural... and the minor 7th. About the only thing the I label gets right is a root on E and a perfect fifth of B. Tread carefully with harmonic analysis of rock music. You might find a lot of inappropriate labels.






                  share|improve this answer























                  • Thank you for the great response! Your answer was very clear--I appreciate it!

                    – 286642
                    39 mins ago













                  1












                  1








                  1







                  TLDR;



                  a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling




                  In the usual Roman numeral analysis we have these conventions:



                  First we have to give a label for a key like E: for E major or Em: for E minor. Then we have these points...



                  • capital numeral means major triad

                  • lower case numeral means minor triad

                  • 'o' after numeral means diminished triad

                  • '7' means a diatonic seventh above root

                  • a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling

                  That last rule is the crux of your question.



                  In E major the diatonic chord of the sixth degree is C# minor and gets labelled vi.



                  In E minor the diatonic chord of the sixth degree is C major and gets labelled VI.



                  If the VI from E minor is used in E major the chord root is altered from C# to C natural. (This is called chord borrowing.) When in E major that change in the chord root is made explicit in the Roman numeral figure by using the flat to show the change of chord root: bVI.



                  The reason your book's example is confusing is because the key signature on the staff is E minor, but the analysis key shows E which is read as E major (it should be Em: to mean minor) and then the Roman numeral figures are labeled with flats which makes things seem like the assumed key is E major. Basically the labeling is a mess and it makes confusing the use of sharps and flats on Roman numeral figures.



                  As others point out, the book doesn't even label the chords as 7th chords!



                  The book seems sloppy.



                  @Tim makes a good point that Roman numeral analysis can be tricky with rock music. The real conundrum to me is the final chord. It's an E dominant 7th with a G natural on top. That is a very bluesy sound. It's a very familiar sound, but Roman numerals are a poor way to label it. There is no way to indicate the simultaneous G# and G natural... and the minor 7th. About the only thing the I label gets right is a root on E and a perfect fifth of B. Tread carefully with harmonic analysis of rock music. You might find a lot of inappropriate labels.






                  share|improve this answer













                  TLDR;



                  a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling




                  In the usual Roman numeral analysis we have these conventions:



                  First we have to give a label for a key like E: for E major or Em: for E minor. Then we have these points...



                  • capital numeral means major triad

                  • lower case numeral means minor triad

                  • 'o' after numeral means diminished triad

                  • '7' means a diatonic seventh above root

                  • a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling

                  That last rule is the crux of your question.



                  In E major the diatonic chord of the sixth degree is C# minor and gets labelled vi.



                  In E minor the diatonic chord of the sixth degree is C major and gets labelled VI.



                  If the VI from E minor is used in E major the chord root is altered from C# to C natural. (This is called chord borrowing.) When in E major that change in the chord root is made explicit in the Roman numeral figure by using the flat to show the change of chord root: bVI.



                  The reason your book's example is confusing is because the key signature on the staff is E minor, but the analysis key shows E which is read as E major (it should be Em: to mean minor) and then the Roman numeral figures are labeled with flats which makes things seem like the assumed key is E major. Basically the labeling is a mess and it makes confusing the use of sharps and flats on Roman numeral figures.



                  As others point out, the book doesn't even label the chords as 7th chords!



                  The book seems sloppy.



                  @Tim makes a good point that Roman numeral analysis can be tricky with rock music. The real conundrum to me is the final chord. It's an E dominant 7th with a G natural on top. That is a very bluesy sound. It's a very familiar sound, but Roman numerals are a poor way to label it. There is no way to indicate the simultaneous G# and G natural... and the minor 7th. About the only thing the I label gets right is a root on E and a perfect fifth of B. Tread carefully with harmonic analysis of rock music. You might find a lot of inappropriate labels.







                  share|improve this answer












                  share|improve this answer



                  share|improve this answer










                  answered 51 mins ago









                  Michael CurtisMichael Curtis

                  9,574534




                  9,574534












                  • Thank you for the great response! Your answer was very clear--I appreciate it!

                    – 286642
                    39 mins ago

















                  • Thank you for the great response! Your answer was very clear--I appreciate it!

                    – 286642
                    39 mins ago
















                  Thank you for the great response! Your answer was very clear--I appreciate it!

                  – 286642
                  39 mins ago





                  Thank you for the great response! Your answer was very clear--I appreciate it!

                  – 286642
                  39 mins ago











                  2














                  Are you confused because the song's in E minor, but the Roman numeral chord names are relative to E MAJOR? Yes, that's how it works. The numbers only care that E is the tonic. E minor triad is i, E major triad is I. The uppercase Roman number doesn't mean 'diatonic', 'in key' or 'correct', it just means 'major'.



                  Similarly for modified numerals. G is the 'correct' third note of E minor, but G B D is called ♭III. That's ♭ because G is the flattened 3rd of E major, III because it;s a major triad.



                  G B♭ D would be ♭iii. That's ♭ because it's built on the flat 3rd of E major, iii because it's a minor triad.



                  I repeat, these names would be the same whether we were in E major, E minor or E anything-else.






                  share|improve this answer



























                    2














                    Are you confused because the song's in E minor, but the Roman numeral chord names are relative to E MAJOR? Yes, that's how it works. The numbers only care that E is the tonic. E minor triad is i, E major triad is I. The uppercase Roman number doesn't mean 'diatonic', 'in key' or 'correct', it just means 'major'.



                    Similarly for modified numerals. G is the 'correct' third note of E minor, but G B D is called ♭III. That's ♭ because G is the flattened 3rd of E major, III because it;s a major triad.



                    G B♭ D would be ♭iii. That's ♭ because it's built on the flat 3rd of E major, iii because it's a minor triad.



                    I repeat, these names would be the same whether we were in E major, E minor or E anything-else.






                    share|improve this answer

























                      2












                      2








                      2







                      Are you confused because the song's in E minor, but the Roman numeral chord names are relative to E MAJOR? Yes, that's how it works. The numbers only care that E is the tonic. E minor triad is i, E major triad is I. The uppercase Roman number doesn't mean 'diatonic', 'in key' or 'correct', it just means 'major'.



                      Similarly for modified numerals. G is the 'correct' third note of E minor, but G B D is called ♭III. That's ♭ because G is the flattened 3rd of E major, III because it;s a major triad.



                      G B♭ D would be ♭iii. That's ♭ because it's built on the flat 3rd of E major, iii because it's a minor triad.



                      I repeat, these names would be the same whether we were in E major, E minor or E anything-else.






                      share|improve this answer













                      Are you confused because the song's in E minor, but the Roman numeral chord names are relative to E MAJOR? Yes, that's how it works. The numbers only care that E is the tonic. E minor triad is i, E major triad is I. The uppercase Roman number doesn't mean 'diatonic', 'in key' or 'correct', it just means 'major'.



                      Similarly for modified numerals. G is the 'correct' third note of E minor, but G B D is called ♭III. That's ♭ because G is the flattened 3rd of E major, III because it;s a major triad.



                      G B♭ D would be ♭iii. That's ♭ because it's built on the flat 3rd of E major, iii because it's a minor triad.



                      I repeat, these names would be the same whether we were in E major, E minor or E anything-else.







                      share|improve this answer












                      share|improve this answer



                      share|improve this answer










                      answered 2 hours ago









                      Laurence PayneLaurence Payne

                      36.5k1670




                      36.5k1670





















                          1














                          RNA isn't designed for analysing 'rock' music. It sort of works when non-diatonic chords need naming, as we use 'b' in certain circumstances. As in key E. There is no D and no C, only D# and C#. So, we have to call the D chord bVII and C chord bVI. It gets as close as we can.



                          The bIII represents G, again because there's no G in key E: G# is iii (III) but we need the flattened G#, making G. Thus bIII.






                          share|improve this answer



























                            1














                            RNA isn't designed for analysing 'rock' music. It sort of works when non-diatonic chords need naming, as we use 'b' in certain circumstances. As in key E. There is no D and no C, only D# and C#. So, we have to call the D chord bVII and C chord bVI. It gets as close as we can.



                            The bIII represents G, again because there's no G in key E: G# is iii (III) but we need the flattened G#, making G. Thus bIII.






                            share|improve this answer

























                              1












                              1








                              1







                              RNA isn't designed for analysing 'rock' music. It sort of works when non-diatonic chords need naming, as we use 'b' in certain circumstances. As in key E. There is no D and no C, only D# and C#. So, we have to call the D chord bVII and C chord bVI. It gets as close as we can.



                              The bIII represents G, again because there's no G in key E: G# is iii (III) but we need the flattened G#, making G. Thus bIII.






                              share|improve this answer













                              RNA isn't designed for analysing 'rock' music. It sort of works when non-diatonic chords need naming, as we use 'b' in certain circumstances. As in key E. There is no D and no C, only D# and C#. So, we have to call the D chord bVII and C chord bVI. It gets as close as we can.



                              The bIII represents G, again because there's no G in key E: G# is iii (III) but we need the flattened G#, making G. Thus bIII.







                              share|improve this answer












                              share|improve this answer



                              share|improve this answer










                              answered 2 hours ago









                              TimTim

                              103k10107259




                              103k10107259





















                                  1














                                  The trouble is:



                                  The key sign is G-major/e-minor:
                                  (in e-minor the flat should be a natural sign to explain the VI and VII degree are not of the harmonic scale!)



                                  Indeed this progression is notated in respect of E-major!



                                  D - D - D - C - G - C - D - E



                                  If you transpose this figure to C you will understand that the flats are concerning



                                  Bb - Bb - Bb - Ab - Eb - Ab - Bb - G - C (with the blue note b10)






                                  share|improve this answer



























                                    1














                                    The trouble is:



                                    The key sign is G-major/e-minor:
                                    (in e-minor the flat should be a natural sign to explain the VI and VII degree are not of the harmonic scale!)



                                    Indeed this progression is notated in respect of E-major!



                                    D - D - D - C - G - C - D - E



                                    If you transpose this figure to C you will understand that the flats are concerning



                                    Bb - Bb - Bb - Ab - Eb - Ab - Bb - G - C (with the blue note b10)






                                    share|improve this answer

























                                      1












                                      1








                                      1







                                      The trouble is:



                                      The key sign is G-major/e-minor:
                                      (in e-minor the flat should be a natural sign to explain the VI and VII degree are not of the harmonic scale!)



                                      Indeed this progression is notated in respect of E-major!



                                      D - D - D - C - G - C - D - E



                                      If you transpose this figure to C you will understand that the flats are concerning



                                      Bb - Bb - Bb - Ab - Eb - Ab - Bb - G - C (with the blue note b10)






                                      share|improve this answer













                                      The trouble is:



                                      The key sign is G-major/e-minor:
                                      (in e-minor the flat should be a natural sign to explain the VI and VII degree are not of the harmonic scale!)



                                      Indeed this progression is notated in respect of E-major!



                                      D - D - D - C - G - C - D - E



                                      If you transpose this figure to C you will understand that the flats are concerning



                                      Bb - Bb - Bb - Ab - Eb - Ab - Bb - G - C (with the blue note b10)







                                      share|improve this answer












                                      share|improve this answer



                                      share|improve this answer










                                      answered 2 hours ago









                                      Albrecht HügliAlbrecht Hügli

                                      3,392220




                                      3,392220





















                                          1














                                          The notation seems a bit confusing. The first chord is a D7 in the key of either E-Major or e-minor; the author isn't quite so clear. The key signature indicates e-minor in which case the chord is VII (no flat needed.) If the author shows keys in ALL CAPS (which has been outdated since 1834), the chord VII7 but that disagrees with the single sharp signature. The author also leaves off the seventh indicator.



                                          Perhaps we can assume the author uses E irrespective of Major or minor mode and that he leaves off the seventh. This still leaved the bVII vs VII so it's still still. The (V) must mean the seventh of G.



                                          The author shows a descending scale (the caratted numerals above some of the chords) so perhaps that may help. I think he's using the Roman numerals associated with E-Major but the score is written as e-minor; he's assuming that all chords have sevenths (OK, but limiting). Then we have (if the score were in e-minor): VII7, VII7, VII7, VI7, III, VII7, VI7, V7, IM7 (at least as I would notate it.)






                                          share|improve this answer



























                                            1














                                            The notation seems a bit confusing. The first chord is a D7 in the key of either E-Major or e-minor; the author isn't quite so clear. The key signature indicates e-minor in which case the chord is VII (no flat needed.) If the author shows keys in ALL CAPS (which has been outdated since 1834), the chord VII7 but that disagrees with the single sharp signature. The author also leaves off the seventh indicator.



                                            Perhaps we can assume the author uses E irrespective of Major or minor mode and that he leaves off the seventh. This still leaved the bVII vs VII so it's still still. The (V) must mean the seventh of G.



                                            The author shows a descending scale (the caratted numerals above some of the chords) so perhaps that may help. I think he's using the Roman numerals associated with E-Major but the score is written as e-minor; he's assuming that all chords have sevenths (OK, but limiting). Then we have (if the score were in e-minor): VII7, VII7, VII7, VI7, III, VII7, VI7, V7, IM7 (at least as I would notate it.)






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                                              The notation seems a bit confusing. The first chord is a D7 in the key of either E-Major or e-minor; the author isn't quite so clear. The key signature indicates e-minor in which case the chord is VII (no flat needed.) If the author shows keys in ALL CAPS (which has been outdated since 1834), the chord VII7 but that disagrees with the single sharp signature. The author also leaves off the seventh indicator.



                                              Perhaps we can assume the author uses E irrespective of Major or minor mode and that he leaves off the seventh. This still leaved the bVII vs VII so it's still still. The (V) must mean the seventh of G.



                                              The author shows a descending scale (the caratted numerals above some of the chords) so perhaps that may help. I think he's using the Roman numerals associated with E-Major but the score is written as e-minor; he's assuming that all chords have sevenths (OK, but limiting). Then we have (if the score were in e-minor): VII7, VII7, VII7, VI7, III, VII7, VI7, V7, IM7 (at least as I would notate it.)






                                              share|improve this answer













                                              The notation seems a bit confusing. The first chord is a D7 in the key of either E-Major or e-minor; the author isn't quite so clear. The key signature indicates e-minor in which case the chord is VII (no flat needed.) If the author shows keys in ALL CAPS (which has been outdated since 1834), the chord VII7 but that disagrees with the single sharp signature. The author also leaves off the seventh indicator.



                                              Perhaps we can assume the author uses E irrespective of Major or minor mode and that he leaves off the seventh. This still leaved the bVII vs VII so it's still still. The (V) must mean the seventh of G.



                                              The author shows a descending scale (the caratted numerals above some of the chords) so perhaps that may help. I think he's using the Roman numerals associated with E-Major but the score is written as e-minor; he's assuming that all chords have sevenths (OK, but limiting). Then we have (if the score were in e-minor): VII7, VII7, VII7, VI7, III, VII7, VI7, V7, IM7 (at least as I would notate it.)







                                              share|improve this answer












                                              share|improve this answer



                                              share|improve this answer










                                              answered 2 hours ago









                                              ttwttw

                                              8,908932




                                              8,908932



























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