Please, more Schumann harmony! There are so many harmonic techniques that deserved to be discussed in his works which are under-discussed, and that I would love to learn about!
I think you're correct! [As a 'side benefit' consideration too, the Pianist won't be flustered by the Bb to F# Bass-line Diminished Fourth in Bar 8 whereas Vocalists are more readily 'flubbed up' by oddly (singly or doubly) augmented or diminished melodic intervals because they have to recognize and hear the interval in their head first in order to generate the next pitch correctly. (And our sight-singing courses generally don't drill music students in singing intervals that are 'weirder' than Augmented Sixths either.) So if any very weirdly-notated intervals are going to be used in a Piano-accompanied song it's better to avoid giving them to the Vocalist.]
There are work-arounds in vocal music where an enharmonic shift is aided by a parenthetical respelling of a held note. I think I first saw that in some early Vaughan Williams choral pieces, but I'm sure you can find earlier examples.
As a singer, the flat notation is definitely easier to sight read, and there are cases in other pieces where the composer writes the wrong note spellings for the voice for the sake of easiness in reading (a practice which makes me very happy lol) As for the piano, I'd guess reading a B dominant b9 also causes less headaches than a Cb dominant b9, as you said. I think everyone comes out happier this way!
Love Schumann and love your videos, I think you nailed it here; the visual satisfaction of contrary motion in the voice leading makes perfect sense...please keep it up with the gory details that I crave!
I don't know if this is the perfect video to introduce someone to the wonders of Music Theory - or the perfect video to scare them off of it forever! Maybe both - depending on the person! 😂
Mostly agree... but I think preserving the flats in the singing line is just for the ease of singers... I often have to make that kind of a choice (correct theory vs visual practicity) and the reallity is that scale is more important than harmony for the singer intonation. We pianists can deal with bizzare chords always, singers go out of tune. I also agree with the idea of contrary motion, similar to an omnibus progression in miniature. There's a lot of tritonal relationship in this progression. Flat sixth degree becomes II dominant of E minir that goes to the Vth that goes to the flat sixth of E minor and becomes the II dominant that goes to the Vth.
Context is everything. As a composer I like to fool the ear. Bitonality and higher order tonalities are interesting for this purpose, where a resolution can be made to occur on just about anything. Anyway, nice work - you are the expert.
Hello, Mister Kent, I read that you are a composer. I am one myself. I don't take any composing lessons, I'd really appreciate If you had the time to listen to the Symphony I'm writing. The first and second movement are already on my channel. It would be really great to get some feedback from someone, who has more experience than myself
Love it! please make more! Schumann is so fascinating. Super nit-picky, I know, but it was personally hard to think about the harmony when the recording was in a different key from the score displayed. My sole criticism
Excellent discussion. I wonder, however, if Schumann wrote the vocal part as a thing by itself, notating it to make it flow as a stand-alone single line. Your own examples on how the piano part would change if Schumann had tried to use the same notation style in both piano and voice show that if it's grateful for one, it's not for the other. If you hide the piano part and look at the vocal part alone, it makes perfect sense as notated. Ditto if you hide the vocal part and look at the piano part alone. So by notating the two instruments in distinct ways, I think Schumann is trying to make it easier for the singer and the pianist to read their own parts independently. At least, that's my own theory.
This simpler explanation was my immediate thought, as well. Side note: I think Elaine Gould recommends making this consideration take precedence over the overall consistency (if necessary). Interesting stuff, nonetheless.
Do you have an explanation for some of Bartók's weird choice of enharmonic spellings? For example in his first string quartet, second movement the first violin plays melodic fragments D-Eb, D-D#, Cx-D# all within four seconds. Such examples can be found all over his music ranging from his other string quartets to orchestral pieces such as his Concerto for orchestra.
Three possibilities come to mind, although I'm sure there are more: 1) Bartók may have wanted to preserve an aural distinction between sharps and flats, which is easily (and even intuitively) accomplished in an ensemble of fretless string instruments. C sharp would be higher than a D flat, as one is on the way up to D and the other on the way down to C. Feldman would do something like this, intentionally throwing in bizarre spellings to induce microtonal variation without notating specifics. 2) Bartók may be interested in showing which notes are part of whatever collection of pitches he's using at that moment, so weird spellings emerge as he tries to show us which notes are neighbor tones. This is especially true for his later music. 3) Strange spellings in one instrument may be a result of how a larger harmony is being spelled.
This reminded me of some of my less stellar moments in college music theory. For mere music lovers interested in how harmony can create affect, passages such as the chordal breakdown starting at 2:10 should have been played out loud to aurally illustrate the possible directions music could have taken. Leonard Bernstein would have done so here and at other similar instances in the video. Making everything audible is essential to making such discussions even partially accessible to those who have no command of tonal-analysis terminology. Otherwise the whole can be just as incomprehensible as a lecture on the equations of General Relativity.
I have read several books that argue equal temperament was not as universally accepted in this era as is taught today, and in fact there is proof that a lot of instruments were still being tuned in sixth and fifth comma mean-tone temp in this era. If that was the case an a sharp would actually be lower than a b flat (counter intuitive I know) so...I dunno. It Would not effect enharmonic moments like a c flat and b natural but, just something I wanted to toss in there as a half formed thought. There is a surprising amount of proof pianos were not being universally tuned in strict equal temperament until into the 20th. Regardless, this explanation makes sense to me and is certainly at least part of what he was thinking when he made these decisions.
The trouble with that is that pianos had to be tuned in one system or another, and a singer of any era would have a tough time tuning something that's so close to sounding like a perfect 2:1 even if the piano's not in equal temperament. If voice leading like this were in a string quartet, I'd be totally on board with looking at contemporary tuning theories to make an educated guess at some interval ratios!
@@ClassicalNerd Oh yeah, of course I agree with that, I was more meaning where Schumann's head may have been at while notating the music. I mean, we do have things like "expressive intonation" but as you say, that is more a string player thing. I am certainly not suggesting the vocalist was expected to fine tune to that degree.
Hello, Could you please make a video about the composer Gerald Finzi ? His cello concerto is truly a masterpiece. I really like what you are doing. Thank you very much !
I wanted to see what you guys in the comments would think about something, I recently wrote a piece using a tonal melody as a joke at the beginning, Sarcastically I mean. Would People see this as obvious or just assume it is boring or pretty?
An a-tonal melody you mean? Or A tonal melody that’s soothing and not harsh to the ear? I mean it depends upon the listener as always if they like it or not. Doesn’t necessarily matter if they know how you compose first or not but it would help
@@AtomizedSoundI mean a tonal melody, an a VERY tonal melody equipped with EXTREMELY john Williams over the top beautiful harmony. It’s on my Chanel if you wanna check it out. It’s the piano trio movement 4. I was thinking about more of a well educated listener. I’ve showed it to some of my non-classical friends amd they just said it was very pretty, to the more dissonant parts they just said it sounded like a horror movie(your everyday person hearing Schoenberg for the first time kinfdof reaction. I’m sure you’ve seen that.)
My explanation would be that the F# 7(Gb 7) is the augmented sixth of B-flat major, which also serves as secondary dominant in e minor. The C7 at bar 10 is the augmented sixth of e minor and secondary dominant of B-flat major at the same time.
Brilliant interpretation. As a composer (and semi-theorist), I guess Schumann focused on voice leading more than the traditional functional harmonic progression (as you said in the video?). Thank you for the great content!
Yeah…. I’m still not convinced about the logic of Schumann’s d9s. I’m the vocal line, I suppose the strange accidentals work for the sake of proper voice-leading (as was mentioned), but the enharmonics in the piano part really do not make sense to me. Perhaps there is something I’m missing. I mean, it is Schumann after all. 😂
Speaking of weird enharmonic spellings, how did Ferruccio Busoni actually use the synthetic scales he wrote about in his "Sketch for a New Esthetic of Music"? Some of the scales he listed had weird enharmonic spellings like C Db Eb Fb G A B C C Db Eb F Gb A B C C D Eb Fb Gb A B C C D Eb Fb G A# B C C D Eb Fb G# A B C C Db Eb F# G# A Bb C I listened to Busoni's late works like Fantasia Contrappuntistica, Sonatina no. 2 and Toccata, for example, and I couldn't find any analysis of how Busoni's synthetic scales are used in context.
I don't see much weird about these scales in terms of enharmonics. If you're asking why he wrote, say, Fb instead of E, consider that all of these scales have some version of notes A-G-standard diatonic practice. It's the same reason that F# Major has an E#, for instance. As to how he _used_ them, I must admit a blind spot here, as I've never done a video about Busoni and he's 33rd on the request list as of this writing. From what I _do_ know, I think it's safe to put him in a moderately "futurist" category in terms of tonal theory, as he was also interested in microtonality (although, as I understand it, on a more theoretical than practical level, similar to Schoenberg).
@@ClassicalNerdI think I figured it out. He generated synthetic scales and all his other musical ideas from a three note cell much like Bartók did in his music. For instance, in his Sonatina No. 2, there's the main melodic cell (C F# B), a seven note chord (C F F# Bb B E G#) and a synthetic scale that overshoots the octave (Bb C D E F# G A B) which is derived from the main chord.
@@skern49 -- both were developed for a time and context very different from the present. And as practitioners have adapted each communication system across the years and centuries to satisfy changing requirements (e.g., pronunciation, temperament, harmonic preferences, publishing technologies, audience), irregularities and complexities have accumulated. Both systems are still functional, yes, obviously, so my rhetorical use of the word "desperately" isn't true. However, each is harder to use and to learn than it has to be. Alternative notational systems, were new writing systems designed from scratch today, would very likely be better suited to current needs and conventions.
"Dissonance" is a nebulous concept that can refer to a number of different things-in this case, it's a matter of spelling an aural consonance (a pure 2:1 ratio in 12-EDO) as an interval that is _always_ dissonant.
@@ClassicalNerd Dissonance is always about sound: sonare = sounding. You could also notate a diminished second, the inversion of augmented octave, but it will still sound like a prime. A notated interval can not be in itself dissonant, the resulting sound can, but a 2;1 ratio sounds like a perfect inteval no matter how it's written.
I'm afraid all I can do is refer you to the opening lines of this video. Functional harmony treats some intervals as though they are dissonances regardless of their interval ratios in 12-EDO space, hence my first example here.
@@ClassicalNerd "Dissonances that sound like consonances"..... That makes no sense. The fact that you say so doesn't make it automatically true. Just like there is no diminished 4th nor a diminished sixth a diminished 9th is just enharmonic switching for the sake of it. By the way, I studied classical harmony and counterpoint at conservatory, graduated in 1987 working as a teacher since then. Can you please link to any proof? Some music theory site? I checked some stuff on Wikipedia. It seems to be a mathematical thing in different tunings but in 12 tone equal tuning the ratio is, as you said, 2:1, which we HEAR as perfect consonance. Schumann wrote it like he did for reading purposes, I'm sure. Please convince me I am wrong if you can but not by just saying so ;-)
“Only the following are designated as dissonances: the major and minor seconds, the major and minor sevenths, the ninths, etc.; in addition, all diminished and augmented intervals, thus, diminished and augmented octaves, fourths, fifths, etc.” _Theory of Harmony [Harmonielehre],_ 3rd edition, by Arnold Schoenberg, trans. Roy Carter (1978, University of California Press), p. 22 “Dissonances in tonal music are: M2, m2, M7, m7, P4, and all diminished and augmented intervals.” _Music in Theory and Practice, Vol. 1,_ 8th edition, by Bruce Benward and Marilyn Saker (2003, McGraw Hill), p. 390
Please, more Schumann harmony! There are so many harmonic techniques that deserved to be discussed in his works which are under-discussed, and that I would love to learn about!
6:38 i would think another good reason not to have B to A# in the voice is the second chord clearly functions as a dominant and not an augmented 6th.
I think you're correct! [As a 'side benefit' consideration too, the Pianist won't be flustered by the Bb to F# Bass-line Diminished Fourth in Bar 8 whereas Vocalists are more readily 'flubbed up' by oddly (singly or doubly) augmented or diminished melodic intervals because they have to recognize and hear the interval in their head first in order to generate the next pitch correctly. (And our sight-singing courses generally don't drill music students in singing intervals that are 'weirder' than Augmented Sixths either.) So if any very weirdly-notated intervals are going to be used in a Piano-accompanied song it's better to avoid giving them to the Vocalist.]
There are work-arounds in vocal music where an enharmonic shift is aided by a parenthetical respelling of a held note. I think I first saw that in some early Vaughan Williams choral pieces, but I'm sure you can find earlier examples.
As a singer, the flat notation is definitely easier to sight read, and there are cases in other pieces where the composer writes the wrong note spellings for the voice for the sake of easiness in reading (a practice which makes me very happy lol)
As for the piano, I'd guess reading a B dominant b9 also causes less headaches than a Cb dominant b9, as you said. I think everyone comes out happier this way!
This makes sense...
Love Schumann and love your videos, I think you nailed it here; the visual satisfaction of contrary motion in the voice leading makes perfect sense...please keep it up with the gory details that I crave!
Excellent job discussing this highly unusual passage! Can't wait for more similar
This was brilliant. Thank you!
Thank you,Thomas, for a new video. Alles Gute.🌹🌹🌹🌹
I LOVE am leuchtenden sommermorgen, might even be my favourite Schumann lied but I hadn't noticed this about it and wow that's incredibly cool
I want to quit my job and spend the rest of my life listening to you talk about music
Very well spotted and explained. Thank you!
I don't know if this is the perfect video to introduce someone to the wonders of Music Theory - or the perfect video to scare them off of it forever! Maybe both - depending on the person! 😂
I'm having a thought atm. And my theory stems from a bunch of notes put together, not diminished and augmented notes. Just me.
I won't let anyone at all scare me off. Come on, it's Robert Schumann. His pieces are awesome.
This is awesome!!! Great content
this schumann guy seems like an eccentric fellow
Mostly agree... but I think preserving the flats in the singing line is just for the ease of singers... I often have to make that kind of a choice (correct theory vs visual practicity) and the reallity is that scale is more important than harmony for the singer intonation. We pianists can deal with bizzare chords always, singers go out of tune. I also agree with the idea of contrary motion, similar to an omnibus progression in miniature. There's a lot of tritonal relationship in this progression. Flat sixth degree becomes II dominant of E minir that goes to the Vth that goes to the flat sixth of E minor and becomes the II dominant that goes to the Vth.
Loved this!
Context is everything. As a composer I like to fool the ear. Bitonality and higher order tonalities are interesting for this purpose, where a resolution can be made to occur on just about anything. Anyway, nice work - you are the expert.
Hello, Mister Kent,
I read that you are a composer. I am one myself. I don't take any composing lessons, I'd really appreciate If you had the time to listen to the Symphony I'm writing. The first and second movement are already on my channel. It would be really great to get some feedback from someone, who has more experience than myself
Gracias maestro por una excelente explicación, claramente Schumann exploraba con la tonalidad.
Thank you 🍻🍻
Love it! please make more! Schumann is so fascinating. Super nit-picky, I know, but it was personally hard to think about the harmony when the recording was in a different key from the score displayed. My sole criticism
I'm just thankful that there is _any_ recording in the public domain.
Yeah
The wonders of relative pitch 🎉
Excellent discussion. I wonder, however, if Schumann wrote the vocal part as a thing by itself, notating it to make it flow as a stand-alone single line. Your own examples on how the piano part would change if Schumann had tried to use the same notation style in both piano and voice show that if it's grateful for one, it's not for the other. If you hide the piano part and look at the vocal part alone, it makes perfect sense as notated. Ditto if you hide the vocal part and look at the piano part alone.
So by notating the two instruments in distinct ways, I think Schumann is trying to make it easier for the singer and the pianist to read their own parts independently.
At least, that's my own theory.
This simpler explanation was my immediate thought, as well. Side note: I think Elaine Gould recommends making this consideration take precedence over the overall consistency (if necessary).
Interesting stuff, nonetheless.
Do you have an explanation for some of Bartók's weird choice of enharmonic spellings? For example in his first string quartet, second movement the first violin plays melodic fragments D-Eb, D-D#, Cx-D# all within four seconds.
Such examples can be found all over his music ranging from his other string quartets to orchestral pieces such as his Concerto for orchestra.
Three possibilities come to mind, although I'm sure there are more:
1) Bartók may have wanted to preserve an aural distinction between sharps and flats, which is easily (and even intuitively) accomplished in an ensemble of fretless string instruments. C sharp would be higher than a D flat, as one is on the way up to D and the other on the way down to C. Feldman would do something like this, intentionally throwing in bizarre spellings to induce microtonal variation without notating specifics.
2) Bartók may be interested in showing which notes are part of whatever collection of pitches he's using at that moment, so weird spellings emerge as he tries to show us which notes are neighbor tones. This is especially true for his later music.
3) Strange spellings in one instrument may be a result of how a larger harmony is being spelled.
@@ClassicalNerd The last word here is "spelled" in a wrong way. It should be spelt correctly. Any larger harmony here?
This reminded me of some of my less stellar moments in college music theory. For mere music lovers interested in how harmony can create affect, passages such as the chordal breakdown starting at 2:10 should have been played out loud to aurally illustrate the possible directions music could have taken. Leonard Bernstein would have done so here and at other similar instances in the video. Making everything audible is essential to making such discussions even partially accessible to those who have no command of tonal-analysis terminology. Otherwise the whole can be just as incomprehensible as a lecture on the equations of General Relativity.
I have read several books that argue equal temperament was not as universally accepted in this era as is taught today, and in fact there is proof that a lot of instruments were still being tuned in sixth and fifth comma mean-tone temp in this era. If that was the case an a sharp would actually be lower than a b flat (counter intuitive I know) so...I dunno. It Would not effect enharmonic moments like a c flat and b natural but, just something I wanted to toss in there as a half formed thought. There is a surprising amount of proof pianos were not being universally tuned in strict equal temperament until into the 20th. Regardless, this explanation makes sense to me and is certainly at least part of what he was thinking when he made these decisions.
The trouble with that is that pianos had to be tuned in one system or another, and a singer of any era would have a tough time tuning something that's so close to sounding like a perfect 2:1 even if the piano's not in equal temperament. If voice leading like this were in a string quartet, I'd be totally on board with looking at contemporary tuning theories to make an educated guess at some interval ratios!
@@ClassicalNerd Oh yeah, of course I agree with that, I was more meaning where Schumann's head may have been at while notating the music. I mean, we do have things like "expressive intonation" but as you say, that is more a string player thing. I am certainly not suggesting the vocalist was expected to fine tune to that degree.
Hello,
Could you please make a video about the composer Gerald Finzi ? His cello concerto is truly a masterpiece.
I really like what you are doing. Thank you very much !
Duly noted: www.lentovivace.com/classicalnerd.html
I wanted to see what you guys in the comments would think about something, I recently wrote a piece using a tonal melody as a joke at the beginning, Sarcastically I mean. Would People see this as obvious or just assume it is boring or pretty?
People who know you as an established artist will see the joke, everyone else will assume it's just boring.
@@lollertoaster very good point thanks. Now I just have to become known as an established artist. Lol
An a-tonal melody you mean? Or A tonal melody that’s soothing and not harsh to the ear? I mean it depends upon the listener as always if they like it or not. Doesn’t necessarily matter if they know how you compose first or not but it would help
@@AtomizedSoundI mean a tonal melody, an a VERY tonal melody equipped with EXTREMELY john Williams over the top beautiful harmony. It’s on my Chanel if you wanna check it out. It’s the piano trio movement 4. I was thinking about more of a well educated listener. I’ve showed it to some of my non-classical friends amd they just said it was very pretty, to the more dissonant parts they just said it sounded like a horror movie(your everyday person hearing Schoenberg for the first time kinfdof reaction. I’m sure you’ve seen that.)
My explanation would be that the F# 7(Gb 7) is the augmented sixth of B-flat major, which also serves as secondary dominant in e minor.
The C7 at bar 10 is the augmented sixth of e minor and secondary dominant of B-flat major at the same time.
Brilliant interpretation. As a composer (and semi-theorist), I guess Schumann focused on voice leading more than the traditional functional harmonic progression (as you said in the video?). Thank you for the great content!
Yeah…. I’m still not convinced about the logic of Schumann’s d9s. I’m the vocal line, I suppose the strange accidentals work for the sake of proper voice-leading (as was mentioned), but the enharmonics in the piano part really do not make sense to me. Perhaps there is something I’m missing. I mean, it is Schumann after all. 😂
The piece is notated in Bb major.....then why do I hear Ab major????
perfect pitch strikes again
Yeah, a tenor recording would have been less irritating. Just take it as notated in Bb…
@@mariusfelixlange6709 I wasn't irritated per se......it was just a curious observation.....
Could you make a Video about alexander glazunov?
Duly noted: www.lentovivace.com/classicalnerd.html
Speaking of weird enharmonic spellings, how did Ferruccio Busoni actually use the synthetic scales he wrote about in his "Sketch for a New Esthetic of Music"? Some of the scales he listed had weird enharmonic spellings like
C Db Eb Fb G A B C
C Db Eb F Gb A B C
C D Eb Fb Gb A B C
C D Eb Fb G A# B C
C D Eb Fb G# A B C
C Db Eb F# G# A Bb C
I listened to Busoni's late works like Fantasia Contrappuntistica, Sonatina no. 2 and Toccata, for example, and I couldn't find any analysis of how Busoni's synthetic scales are used in context.
I don't see much weird about these scales in terms of enharmonics. If you're asking why he wrote, say, Fb instead of E, consider that all of these scales have some version of notes A-G-standard diatonic practice. It's the same reason that F# Major has an E#, for instance.
As to how he _used_ them, I must admit a blind spot here, as I've never done a video about Busoni and he's 33rd on the request list as of this writing. From what I _do_ know, I think it's safe to put him in a moderately "futurist" category in terms of tonal theory, as he was also interested in microtonality (although, as I understand it, on a more theoretical than practical level, similar to Schoenberg).
@@ClassicalNerdI think I figured it out. He generated synthetic scales and all his other musical ideas from a three note cell much like Bartók did in his music. For instance, in his Sonatina No. 2, there's the main melodic cell (C F# B), a seven note chord (C F F# Bb B E G#) and a synthetic scale that overshoots the octave (Bb C D E F# G A B) which is derived from the main chord.
Can you do Bernd Alois Zimmermann?
Duly noted: www.lentovivace.com/classicalnerd.html
How did you first discover Robert Schumann
Can't say I remember.
I love how discussing his spelling choices "he didnt care" or "he didnt have time for thinking" or "he made a mistake" werent even considered
The sketches confirm that it was an intentional choice.
@@ClassicalNerd with a 100% certainty, surely
Two writing systems desperately require reform: English and Western musical notation.
why?
@@skern49 -- both were developed for a time and context very different from the present. And as practitioners have adapted each communication system across the years and centuries to satisfy changing requirements (e.g., pronunciation, temperament, harmonic preferences, publishing technologies, audience), irregularities and complexities have accumulated. Both systems are still functional, yes, obviously, so my rhetorical use of the word "desperately" isn't true. However, each is harder to use and to learn than it has to be. Alternative notational systems, were new writing systems designed from scratch today, would very likely be better suited to current needs and conventions.
So it's only a reading thing, it has nothing to do with dissonance in sound.
"Dissonance" is a nebulous concept that can refer to a number of different things-in this case, it's a matter of spelling an aural consonance (a pure 2:1 ratio in 12-EDO) as an interval that is _always_ dissonant.
@@ClassicalNerd Dissonance is always about sound: sonare = sounding. You could also notate a diminished second, the inversion of augmented octave, but it will still sound like a prime. A notated interval can not be in itself dissonant, the resulting sound can, but a 2;1 ratio sounds like a perfect inteval no matter how it's written.
I'm afraid all I can do is refer you to the opening lines of this video. Functional harmony treats some intervals as though they are dissonances regardless of their interval ratios in 12-EDO space, hence my first example here.
@@ClassicalNerd "Dissonances that sound like consonances"..... That makes no sense. The fact that you say so doesn't make it automatically true. Just like there is no diminished 4th nor a diminished sixth a diminished 9th is just enharmonic switching for the sake of it. By the way, I studied classical harmony and counterpoint at conservatory, graduated in 1987 working as a teacher since then. Can you please link to any proof? Some music theory site? I checked some stuff on Wikipedia. It seems to be a mathematical thing in different tunings but in 12 tone equal tuning the ratio is, as you said, 2:1, which we HEAR as perfect consonance. Schumann wrote it like he did for reading purposes, I'm sure. Please convince me I am wrong if you can but not by just saying so ;-)
“Only the following are designated as dissonances: the major and minor seconds, the major and minor sevenths, the ninths, etc.; in addition, all diminished and augmented intervals, thus, diminished and augmented octaves, fourths, fifths, etc.” _Theory of Harmony [Harmonielehre],_ 3rd edition, by Arnold Schoenberg, trans. Roy Carter (1978, University of California Press), p. 22
“Dissonances in tonal music are: M2, m2, M7, m7, P4, and all diminished and augmented intervals.” _Music in Theory and Practice, Vol. 1,_ 8th edition, by Bruce Benward and Marilyn Saker (2003, McGraw Hill), p. 390