Thank you, I think it was a good choice here because the video covered a some especially complicated material. But in an ideal world, RU-vid video essays would end with something a little less tedious!
Your timing is perfect. We are currently working through fugal theory in my counterpoint class. Thanks as always for sharing with us. I know how much work goes into these videos, and it is greatly appreciated.
What a treasure - piece of information... And the reason is clear. This is clearly an explanation of how things started in chronological order, that leave no space for misunderstood or leap of logic. So persuasive. It can almost be translated into algorithm that could solve the following problems: 1) in which hexachord a given subject belongs to and , 2)as a result, which (real/tonal) answers correspond to it! Things clearly explained... Thank you Jason Gran!
Great totorial;, very concise and authoritative! You listed the famous Toccata & fugue in D minor (often attributed to J S Bach) as BWV 568 rather than BWV 565 as is usual. probably oversight?
11:28 The changing of the whole step to leap of a third is in Bach F major Fugue book 1 856! (many other examples but this one was the first I could think of)
Thank you! Those two terms have a long history, and I exaggerated a little bit how unfounded they are. In my experience many musicians get these terms confused because the difference between a "real" and "tonal" answer is not that one is more real or more tonal than the other. So I exaggerated in order to impress the technical meanings.
Another exceptionally fine video - so lucid and informative. What is a 'tenor cadence'? Not a term I've come across before - possibly a US/UK difference? Also in the Bach E-flat Fugue (12:36): labelling that A-natural seems problematic because it doesn't belong to any of the three hexachords in E-flat. And it can't be 'ti' because that's not included in the hexachord system? Is it 'mi', as a dominant of the dominant?
Early Music Sources has a video on the 16th and 17th c. melodic cadential formulas (ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-jaCRUdxTRSM.html&ab_channel=EarlyMusicSources). These names were common in older Italian and French treatises, but only quite recently have I heard these terms thrown around in English (often "Clausula" instead of "Cadence") The tenor clausula is ^2 - ^1, The soprano clausula is ^7 - ^1, and the bass clausula is ^5 - ^1 (ascending a fourth or descending a fifth). But the cadence formulas do not have to take place in the voice parts after which they are named. From what I understand in the hexachordal system, any chromatically raised note that would resolve upward by semitone in the manner of a leading tone would be considered a Mi-note, and chromatically lowered notes were called fa-notes. That way, all pairs of notes separated by a semitone (which are often tendency tones) are sung with the same syllables.
If you want to torture your students you could make them go through the entire WTC and state whether the answer is real and, if not, the reason why Bach chose to have a tonal answer. After watching this video that's my plan for the weekend lol.
Just a few curious questions: How did you grow from four thousand to 24 thousand in a few months and when? When did your first species counterpoint video become viral and how?
It happened at the end of September, perhaps when theory courses get to counterpoint in the fall semester? I wish I knew what happened, I really didn’t do anything to cause it.
On the section in the video describing tonal answers, I don't understand why a direct transposition(real answer) wouldn't work for subject that starts on the dominant degree and moves through the tonic hexachord. The video said a real answer of that kind of subject would result on the first note being the 2nd degree of the tonic hexachord which is bad. But why is that bad? I understand why a tonic degree through dominant hexachord would have a bad real answer because the answer would resolve on the second scale degree(dominant degree in the dominant hexachord) which would leave us further from the tonic up the circle of fifths. I am also curious how the hexachord theory even was conceived in the first place. Like why hexachords? (in my head)It could probably just work fine if it was just major scales transposed on their respective degrees(which would make accidentals but I don't see how that is a problem considering that the subdominant hexachord already has a accidental in it). Thanks (for reading this long question)
Very good question. An answer starting on scale degree 2 is not necessarily bad per se; Beethoven does exactly what you are describing in the fugue "Preiset ihn, ihr Engelschore" from Christ on the Mount of Olives, and honestly I didn't notice anything was unusual about it until I saw the score. Beethoven's choice doesn't sound bad at all, but it's not idiomatic to traditional fugal style. Most textbooks of the 18th and 19th centuries that taught fugue were written with the "strict style" of Palestrina and conservative church music in mind, where modality was still the normal vocabulary. So a fugue subject that began on ^5 and is answered by ^2 (let's say in C major) could be confused for ^1 answered by ^5 in the Mixolydian mode, which would create confusion in defining the mode of the piece. This ties in to the last section of the video where I talk about how certain tonal adjustments are made to help establish the mode-defining notes of the scale. As a general rule, you are going to see subjects that begin on ^1 paired with answers on ^5 and vice versa because those are the "pillars" of the key, even in tonal repertoire.
@@JacobGran Thanks for the explanation. I understood your answer, but I have a very limited knowledge about modal theory. I did a google search on modal theory and spent some time reading, but I still don't exactly understand modal theory. Perhaps could you references some introductory books/articles/links to me so that I can understand this theory? Also will you be making a video anytime in the future on modal theory, (since your video throws terms like hypodorian which I do not know)? Thanks again
@@JacobGran No no no :) Alberch. is here but in Smurfs I heard something like this and I can't add here link with notes so...I can't give you example :(
Very good. You made it crystal clear WHY these sort of tedious rules were necessary to maintain the tonal and modal integrity. This is what "music" was for people on the continent of Europe, or Christendom at that time. As in the Romanesque design of Cathedrals, tradition was valued greatly. I sometimes wonder how a composer like J.S. Bach would fare if he were dropped into our present time frame. That might be an interesting topic to explore in a paper or magazine article.
@@sstuddert - I think on one hand we would find it very restricting snd bristle at the demands to conform to an authoritarian religious based society. On the other hand we might find it pleasant for our minds to be unfettered by countless anxieties and distractions, and experience a lifestyle where music making was integral to everything. I think men in the 17th and 18th centuries were smarter and more productive. Men like Newton, Euler, J.S. Bach. I can't think of any modern equivalents.
@@robbes7rh I doubt that genius of that kind can be attributed to the _religiosity_ of their age. I see no reason to make that move at all: the death of God isn't the only development since then, after all (for my part, I think the cultural decline of Europe has more to do with it being utterly destroyed during the former half of the last century).
@@sstuddert -- I'm not lamenting the onset of the modern age or suggesting that we should become a religious society like things were in the Middle Ages. I'm just pointing out there was something about that time in history that was conducive to such monumental achievements in the music, mathematics, and science. We just don't see men like Bach, Euler, or Newton in our own time.
What is barring a fugue answer from being on another degree than 4 or 5, specifically, what's there to lose in an answer being an octave transposition of the subject?
This is a fantastic question! Fugues at the octave were once considered quite normal; even as late as 1753 Marpurg described them as a type of "ordinary fugue" along with fugues at the fifth and fourth. The first four of J.S. Bach's two-part inventions are great examples of octave fugues. Imitation at the perfect intervals involving scale degrees 1 and 5 were believed to be important for establishing the mode and key of the exposition of a fugue (hence Marpurg's naming them "ordinary"), but imitation at other intervals are often reserved for later entry groups.
@11:58, why isn't Fa (C) the same as Do(Lower C)? and then why isnt the last Do Sol instead? I guess I dont grasp how to accurately convert notation to solfege perfectly everytime. Is there a guide in english that simply explains this.
Here is a great primer by Early Music Sources ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-IRDDT1uSrd0.html He is explaining the syllables according to the 16th century, but the principles remained the same all the way into the early 19th century. There is an entire book written on the topic of the continuation of the tradition through the 18th century by Nicholas Baragwanath. He gives a summary in an interviewed here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Ti5HCqGLWcw.html
A little bit of understanding of language would do no harm in understanding the fuge. "real" as a word originally also held me meaning of "unaffected" too, hence being a good way of describing the Answer as being unaffected by the notes it stands on.
I agree that the names for those terms were appropriately used in their original contexts, and they are not as confusing as I made out in the video. In my earliest draft, I was considering getting into the origins of all of the names and terms, but I decided that might be too much of a dry digression. I am happy if the viewer comes away understanding the practical meaning.
Great video Jacob, this is the best explanation i ever read or watched on the relationship beetwen subject and answer, you seem to always give a historical perspective that makes the subjects you cover feel more natural and less rigid. I just finished a fugue semester at university and i wondering now whats the point in writing a scholastic fugue if this kind of subjects are not covered. Anyway, looking forward to the next video in the series!
Mr. Gran: *What books do you recommend to go deeper into the amazing world of counterpoint and voice leading, apart from the classic Gradus ad Parnassum?* 📚 I'd love to study more deeply about counterpoint, but sadly I don't have the time nor the money to study it formally at a music conservatory. Btw: Thank you again for upload and share with us such quality material (for free!). You're a truly inspiration! ✌
Thank you for the kind words. There are probably too many good books on the topic to name without knowing specifically what you aim to get out of it. If you are interested in the Fux species approach with a focus in modal counterpoint, I would recommend Knud Jeppesen's Counterpoint (which is frankly even better for this purpose than Gradus). If you are more interested in the species approach applied to the tonal common practice, then Salzer and Schachter's Counterpoint in Composition is kind of the gold standard, but rather expensive and extremely dense. Peter Schubert's books on renaissance and Baroque counterpoint are both excellent if you are more interested in a repertoire-based approach rather than the species.
@@aangtonio5570 Please call me Jacob! Prout's books are great, and if you don't mind reading them online I believe most of them are available for free by searching his name in Google Books. This is a great way to read older theory books and treatises that are out of copyright.
Can you please help me? I don't understand why the subdominant hexachord has B-flat instead of B-natural, since B-flat doesn't exist in the key of C major.
The transposable hexachord always follows a pattern of whole-whole-half-whole-whole in terms of whole and half steps of the scale. We need the flat in order to make the Mi note (A) a half step away from the Fa note (B-flat).
This is definitely helpful for me whenever I go and try to do a fugue variation upon a given melody. There's one melody that I have tried twice now to do a fugal variation on and that's the first 10 bars of Beethoven's Fifth Symphony. You see, I had this idea a couple years ago to do a Theme and Variations upon the first theme of Beethoven's Fifth, ending each at the Bb chord that ends the first theme. That is, until the final fugue variation where I end the subject at bar 10, the first move to a C minor chord. And I just can't seem to get the countersubject to sound right and I wonder if it's because of the answer. Right now, it sounds too much like C major because of me adjusting the Ab's to A naturals to avoid a false relation with the answer. Based on the hexachord theory you described, if C is to be the minor tonic(which it is), that means the tonic hexachord would be this: Bb, C, D, Eb, F, G A Bb major hexachord. And thus the dominant hexachord would be F, G, A, Bb, C, D and the subdominant hexachord would be Eb, F, G, A, Bb, C, giving me this solfege: G, G, G, Eb, F, F, F, D, G, G, G, Eb, Ab, Ab, Ab, G, Eb, Eb, Eb, C La, La, La, Fa, Sol, Sol, Sol, Mi, La, La, La, Fa, ?, ?, ?, La, Fa, Fa, Fa, Re Those question marks are the Ab moving to G in the subject, notes outside of the hexachord. The rest are all within the Bb tonic hexachord. But, if I go to the dominant, I have to either alter all the Bb's to B naturals as well as the Ab to an A natural that you get by staying in the hexachord, thus changing modes, or stick with the Bb's and only change the Ab to an A natural to stay in the minor mode. With the subdominant, this would mean a change of mode of the subdominant hexachord from Lydian to Major. So basically, it's impossible to have a real answer for this subject, right?
That's an interesting question. The note that lies outside of a minor hexachord a semitone above La, Ab in this case, would be considered a "Fa Note," just like the B-flat we saw at around 4:08 in the video. We can interpret the entire theme in terms of the syllables from the tonic hexachord other than that one note, like this (ignoring note repetitions): G Eb Ab G Eb C La Fa ( Fa ) La Fa Re Because the theme begins on scale degree 5 and moves through the tonic hexachord, the only way to create a real answer would be to allow the theme to modulate into F minor, like this: C Ab Db C Ab F La Fa ( Fa ) La Fa Re where the solfege syllables come from the subdominant hexachord, like the Toccata and Fugue at 9:54. The other option would be to use the dominant hexachord with a shifted first tone, which would result in a tonal answer: C Bb Eb D Bb G Sol Fa ( Fa ) La Fa Re But, then we would have changed the fate motive into a descending second instead of a third!
@@JacobGran So I can create a real answer for the Beethoven's Fifth subject, but it has to be on the subdominant. Otherwise, it has to be a tonal answer. And because the fate motive becomes a second with a tonal answer on the dominant, it's probably better if I go with the subdominant real answer, like Bach does in his Toccata and Fugue in D minor.
Let's say a fugue subject moves through a tonic hexachord, but leaps up an octave on the tonic note. Does it mean the subject exceeded the tonic hexachord?
Great question. Are you thinking of a particular example? The short answer is yes, octave leaps will usually involve a mutation (switch) between hexachords like the example at 11:30. Albrechtsberger here is describing strict vocal style fugues, but it is also possible in instrumental music (like Contrapunctus 9 from the Art of Fugue and the finale of Brahms's Cello Sonata no. 1) for an octave leap to behave more like a repetition of the same note in a different register than a melodic leap requiring a mutation. After all there are no singers who will need to find the correct syllables, so I imagine that is one reason why things are looser in instrumental music.
16:18 another example of this is ricercar a 6. The answer is tonal because what should be the fifth actually is transformed into a fourth instead. Otherwise it's the same subject
No you are right, the leading tone is a chromatic note outside the hexachord. Raised notes like that were given the syllable "mi" and chromatically lowered notes were "fa."
I don't see why not. Counter melodies used to happen a lot in the popular music of the 1960s (like the ending of Eleanor Rigby by the Beatles), but I can't think of very many recent examples, except maybe some Lady Gaga songs.
Yeah, I tried to give a primer in the video, but it is probably necessary to learn the hexachord stuff elsewhere. Here is a good interview where Nicholas Baragwanath explains the hexachords starting around the 4:20 mark: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-GRxKZDXEqSs.html&ab_channel=NikhilHoganShow
Each video I understand a little less, but hopefully still learn something. The approach to solfeggio is particularly confusing to me as I learned that the solfeggio syllables corresponds to scale degrees.
They certainly do nowadays, but the modern solfege syllables that line up with scale degrees did not gain widespread usage until the 18th and 19th centuries, and fugal theory was developed long before that using hexachords. Even well into the 19th century, we still find some authors explaining fugue in terms of the old solfeggio. And in my opinion, that is the only way to completely understand how tonal answers work. Think of the transposable hexachords as simply the first six scale degrees in any particular key, so everything except the leading tone. The most important is the tonic hexachord, which is just ^1 through ^6. Next most important is the hexachord from the key of the dominant, which are scale degrees ^1 through ^6 in that key, but also happen to be ^5 through ^3 in the tonic key. So thinking in terms of hexachords has the advantage that we can see the same notes as being part of this special area of the scale that overlaps between the two keys. The subdominant hexachord is just ^1 through ^6 in the key of the subdominant, which also happens to have five out of six notes that overlap with the key of the tonic.
