This is about the best presentation of tonnetz I've ever seen. I started taking notes on tonnetz transformations and didn't really follow through but this lesson really reawakened the fire of revisiting tonnetz. I was feeling like it would be useful but I had yet to see a direct representation of how it could be applied and digested until now. And the app just compounds on the helpful visual representation. So glad I found you and this fantastic video. Now I have tons of study to do. 😂 Can't thank you enough for creating this!
This is one of the best videos on music theory I've ever watched - I want to thank you for how articulately you've managed to make this concept 'click' for me after I've been confounded by a lot of other explanations!
This was an interesting take! Thanks for putting it together. I think even when speaking English it would make sense to pronounce the "Ton-" in Tonnetz as "Tone."
The part A#m7 D#7/A G#, if you consider the top melody, it sounds like implying vim7 II7 V(7) in C# melodic minor, the vim7 was somehow borrowed from C#major which has a major third, but the top melody has minor third. I think this is a technique which appears a lot in romantic classical/latin Jazz. So the tonality of this section was not in G# major but C# minor for me, it interrupted the major II V cadence by temporarily setting a half cadence to its relative minor. So it is originally like vi II7 ii V7 in E, now it is something like: vi (vi/vi II7/vi V/vi) ii V7 The most important thing I heard from Kapustin was that he could make all other “keys” sound like spinning around the center one.
It seems to me that the Tonnetz should be front and center in teaching music, and introduced early. Many of the concepts you present are beyond my theory but become immediately intuitive when cast on the net visually. What seems arbitrary on the page becomes obvious on the lattice. Thank you.
I think there’s some interesting stuff to be said about how chord changes can subvert this idea you present about lum7, specifically in the case you bring up around 19:40 where that flip happens on some, but it kind of leaves a “ghost note” resolution to the other notes. That assumes of course that subconsciously you expect all the notes to follow the same flip, but they don’t. In this case, they are just left out.
Good observation. This would be akin to what some theorists call "fuzzy" transformations, which are very close to a full transformation but missing some notes, or some notes don't go where they ought.
There is a video with a live Nikolai Kapustin, in which he talks about his concept of tonal arrangement in his cycle ‘24 Preludes and Fugues’ > For example, C major and A-flat minor, then F major and C-sharp minor, then B-flat major and G minor, and so on.... Notice that the major tonalities are exactly a fourth apart. So it is also a circle of fifths, in essence! I understand that Nikolai Kapustin is NOT talking about relative tonalities here, but about something else. About some ‘native’ tonalities. But, it has to be said, it sounds very beautiful! ----------------------------------------------- And one more interesting observation. Here is how the principle of tonalities arrangement looks like in the cycle ‘24 Preludes and Fugues’ by N. G. Kapustin maj | min 0# | | 5# 1b | | 4# 2b | | 3# 3b | | 2# 4b | | 1# 5b | | 0# 6b = 6# | 1b 5# | | 2b 4# | | 3b 3# | | 4b 2# | | 5b 1# | | 6b I found this in the comments under the video ‘Nikolai Kapustin - 24 Preludes and Fugues, Op 82’. ----------------------------------------------- And to summarise what you said in your video. There is another principle of arranging tonalities in twos, but the distance between these twos will not be a minor third, but a major third. That is, it's the other way round. Here's an example: 1) Let's just take a pair of relative tonalities - E major and C-sharp minor. The distance between them = minor third, since they are relative tonalities. 2) Then we step the major third down - we get A major and F-sharp minor relative to it. And then by analogy - D major and B minor, G major and E minor, and so on.... And you know what the 2 main features of this arrangement are? 1) Major tonalities are a circle of fifths. E, A, D, G, C, F.... 2) The lower/left (depending on how one represents the circle) one moves, the darker the tonality becomes. Correspondingly, the higher/right you move, the lighter it becomes. The brightest tonalities are B major, E major, A major. The darkest tonalities are B-flat minor, E-flat minor, A-flat minor.
I still don't know why Kapustin used that order. Actually we can also let some other intervals downwards, for example, C Major - B Minor, F Major - E Minor, Bb Major - A Minor, then the left remains but the right becomes 2#, 1#, 0#......
I want to let you know that you have a rare gift - forensic mind and a wonderfully engaging way of sharing what your forensic explorations have discovered, arranged - in order to illuminate some of the underlying patterns embedded within western tonal music. As a professional composer and performing musician - I salute you and thank you for this excellent illustration of our art.
Everything you told is very similar to matrices and their linear transformations (hi, 1blue3brown), especially when it came to ‘Lum7’. It turns out that Lum1 and Lum7 are nothing but basis vectors in 2D space.
The Tonnetz is actually 3d as a torus (donut) shape. Apparently, they've done brain scans that show this same shape as people listen to music modulating through all the keys.
is harmonic "luminosity" as described here another word for pitch? seems in the examples you give a brighter sound is always higher, a darker sound is always lower
@liamp.9201 one example is a C major chord moving down to an A major chord. While the pitches move down, it is a Lum6 and sounds brighter because of the C#
What do you mean C maj is darker than G maj? 4:23 they both have the exact same intervals, so unless you have perfect pitch you wouldn’t be able to tell which one is which. Do you mean that when playing a song in C maj and switching keys would make G maj sound brighter?
That's exactly it. Though they are the exact same chord type, we would hear the shift from one to another by they're relation to each other. We would hear G as brighter in that shift.
All this unfortunately doesn't prevent Kapustin's pieces from sounding harmonically somewhat "amorphous". There is always happening too much, so that the ear can't single out passages with really outstanding, characteristic harmonic color. Compare his music to, say, Wagners Tristan ouverture, to Scriabin's works, to Ravel's "Forlane" or to Kenny Kirkland's "Dienda". All pieces with much harmony going on, AND the ear can really discern harmonic "colors" typical for these pieces. Not in Kapustin's pieces - there is a sameness, it sounds arbitrary. So much theoretical thought, so little actual impact on the listener.
The harmonic process is much different than those composers, and I think how you hear Kapustin's music depends a bit on how much you have been primed in jazz styles and improvisation. His chord progressions are tonally directed, using mostly standard jazz syntax like 2-5-1s, with quick chromatic substitutions and elaborations that emulate the unexpected turns and kaleidoscopic effect of improvisation. More of a roller coaster than harmonic scenery.
@@willdaniels1608 I am a professional jazz pianist, so... I love bebop and Oscar Peterson and Bill Evans and all these jazz giants, but Kapustin leaves me totally cold.
@@willdaniels1608 can u share with us a download link? :D really intresting stuff in this vid bro ! stumbled across it from knowhere keep up the good work and thank you for your knowledge shared with us here :)
Hi, I watched your video, is the "harmonic luminosity" your original idea? What do you think of the F#? It could be seen either 6 steps clockwise from C or counterclockwise. And what about the difference between F# and Gb, etc.?
@yikunqi9353 Yes I coined the term "harmonic luminosity" and the definition laid out in the video. But its based on what I believe to be a common way of describing harmony, in terms of brightness and darkness. I set out to offer a technical way to describe that common experience. The relationship of C to F# is certainly a pole that is maximally distant and therefore difficult to discern, without further context, whether one hears it as brighter or darker. But that's just an individual note. Usually we hear these tones in a larger context of a key, or a collection of pitches, and then it may be more clear which side of that collection F# is closer to.
Но, все это можно проще, логичнее и математически более обоснованно, представить в пространстве кратностей, где по одной оси отложениы терции, а по другой квинты. Реально не существует малой терции, минорный аккорд это трезвучие, в котором тональной нотой является последняя - наиболее высокая, и внет точно так же отложен интервал терции (большой терции) от последней ноты, именно этот интервал соответсует резонансу по 5-й гармонике, что и составляет звучание аккорда. Малые терции не являются онтервалом образующим аккорды, ито всего лишь эмпирическое правило игры, предполагающее что следующим интервалом будет квинта, и терцию мы получим автоматически
This was amazingly helpful. I've been working on some of Kapustin's Op.59 Ten Bagatelles and found them really hard to commit to memory because of all the crazy harmony and ambiguous cadences. I have some pretty crazy looking pencil scrawl trying to notate the chords. A harmonic language which actually makes sense for his music will be so so useful. Thank you!
Holy heck this is fantastic. I was a little disoriented since this is the first time I've seen the tonnetz with the fifths axis in a vertical direction, but my brain was able to adjust pretty quick. Anyway, this is a great framework for comparing brightness. I'll be re-watching this several times for sure. Did you write your own Tonnetz program to respond to MIDI input?
I was disoriented too!! It's a program I found online called hexachord (I think, going off memory right now). I couldn't orient it any other way, and it was the best tool I could find.
@@willdaniels1608 Thank you. I'm curious: if you became emperor and could choose a standard way of drawing the Tonnetz that everyone would adhere to, what would you choose? My default has been to draw upward 5ths to the east, up minor 3rds to the northeast, and up major 3rds to the northwest, but now I'm wondering if that's optimal if one wants directions of increasing brightness to to be intuitive.
@@PohlLongsine If I were emperor I think I would agree with that. I would prefer what I think is pretty standard, exactly what you described. Then bright-shifting modulations would take you upwards, and dark would be downwards, while small diatonic changes would be sideways. Bright would be that vertical association with the heavens.
I'm sorry to say this, but it's entirely unfounded. While your experimental spirit is commendable, this content is quite erroneous and misleading. Firstly, it's fundamentally wrong from the beginning. Discussing the brightness and darkness, the essence of chords is not something that can be oversimplified. The connotation of chords is related to the context of the musical progression, and auditory contrast is relative. Your method can be considered an overinterpretation of the circle of fifths. For instance, when you claim that moving from C major to F major darkens the mood, it's merely because the F major you played is in a lower octave than C. If you were to play an F major in a higher register (fourth up), it would sound significantly brighter. The fact is, F major isn't inherently darker compare to C major; anyone can verify this. Therefore, from the outset, this is somewhat of an exaggerated interpretation. Please stop propagating these misconceptions.
I actually agree with you. This is generally how music theory works, that depending on how you look at it, we can draw out different associations. Defining a system within which to understand these associations, like brightness and darkness, is not an attempt to say this is the absolute essence of the chords, but an attempt to reflect a common experience, but yes it's purely relative, just measured relativity.
I've always felt this way too about the terms "brightness" and "darkness". I'm only starting to get it now after watching this video and arguing with myself about it (lol) - why would someone (whoever chose to use these terms originally) pick words that are sight-oriented to describe the way a harmonic shift sounds? It's never made sense to me until I realized they're just words we use to describe the direction, clockwise or counterclockwise, the shift is going on the circle of fifths. Obviously, from C major, both F major and G major are the same harmonic distance. To me, they sound the same distance but one doesn't sound "brighter" or "darker" than the other. As you said, the effect of shifting to one of those keys on the mood of a piece can be brighter or darker depending on the register. So it's really just a way to say "we're moving down/up in the circle of fifths", which, I guess if your ear is good enough (mine isn't yet lol) you should be able to discern the direction just from hearing the harmonic shift. So I agree that it's important to have words to describe the direction of motion but I will never agree that "bright' and "dark" are reasonable words to use lol. But that's not on Will, he didn't come up with those terms
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This video is a revelation. I’ve always loved Kapustin! 🥰
This is about the best presentation of tonnetz I've ever seen. I started taking notes on tonnetz transformations and didn't really follow through but this lesson really reawakened the fire of revisiting tonnetz. I was feeling like it would be useful but I had yet to see a direct representation of how it could be applied and digested until now. And the app just compounds on the helpful visual representation. So glad I found you and this fantastic video. Now I have tons of study to do. 😂 Can't thank you enough for creating this!
Glad it was helpful!
О мой Бог! Музыкальный анализ Капустина 🤩🥺😍
This is one of the best videos on music theory I've ever watched - I want to thank you for how articulately you've managed to make this concept 'click' for me after I've been confounded by a lot of other explanations!
Glad to hear it was helpful!
Never heard of tonnetz before, I really appreciate your effort to explain rare music theory concepts for us
that was actually insightful and understandable, thank you!
such a work you have done, impressive
This was an interesting take! Thanks for putting it together.
I think even when speaking English it would make sense to pronounce the "Ton-" in Tonnetz as "Tone."
The part A#m7 D#7/A G#, if you consider the top melody, it sounds like implying vim7 II7 V(7) in C# melodic minor, the vim7 was somehow borrowed from C#major which has a major third, but the top melody has minor third. I think this is a technique which appears a lot in romantic classical/latin Jazz. So the tonality of this section was not in G# major but C# minor for me, it interrupted the major II V cadence by temporarily setting a half cadence to its relative minor.
So it is originally like vi II7 ii V7 in E, now it is something like: vi (vi/vi II7/vi V/vi) ii V7
The most important thing I heard from Kapustin was that he could make all other “keys” sound like spinning around the center one.
Brilliant presentation! ♥
Thank you!
Really interesting and very easy to understand! Thanks for this video!
It seems to me that the Tonnetz should be front and center in teaching music, and introduced early. Many of the concepts you present are beyond my theory but become immediately intuitive when cast on the net visually. What seems arbitrary on the page becomes obvious on the lattice. Thank you.
@@intevolver 100%!
You are awesome
This is cool! I've never heard of the Tonnetz before
I think there’s some interesting stuff to be said about how chord changes can subvert this idea you present about lum7, specifically in the case you bring up around 19:40 where that flip happens on some, but it kind of leaves a “ghost note” resolution to the other notes. That assumes of course that subconsciously you expect all the notes to follow the same flip, but they don’t. In this case, they are just left out.
Good observation. This would be akin to what some theorists call "fuzzy" transformations, which are very close to a full transformation but missing some notes, or some notes don't go where they ought.
There is a video with a live Nikolai Kapustin, in which he talks about his concept of tonal arrangement in his cycle ‘24 Preludes and Fugues’
>
For example, C major and A-flat minor, then F major and C-sharp minor, then B-flat major and G minor, and so on....
Notice that the major tonalities are exactly a fourth apart. So it is also a circle of fifths, in essence!
I understand that Nikolai Kapustin is NOT talking about relative tonalities here, but about something else. About some ‘native’ tonalities.
But, it has to be said, it sounds very beautiful!
-----------------------------------------------
And one more interesting observation.
Here is how the principle of tonalities arrangement looks like in the cycle ‘24 Preludes and Fugues’ by N. G. Kapustin
maj | min
0# |
| 5#
1b |
| 4#
2b |
| 3#
3b |
| 2#
4b |
| 1#
5b |
| 0#
6b = 6#
| 1b
5# |
| 2b
4# |
| 3b
3# |
| 4b
2# |
| 5b
1# |
| 6b
I found this in the comments under the video ‘Nikolai Kapustin - 24 Preludes and Fugues, Op 82’.
-----------------------------------------------
And to summarise what you said in your video.
There is another principle of arranging tonalities in twos, but the distance between these twos will not be a minor third, but a major third. That is, it's the other way round.
Here's an example:
1) Let's just take a pair of relative tonalities - E major and C-sharp minor. The distance between them = minor third, since they are relative tonalities.
2) Then we step the major third down - we get A major and F-sharp minor relative to it.
And then by analogy - D major and B minor, G major and E minor, and so on....
And you know what the 2 main features of this arrangement are?
1) Major tonalities are a circle of fifths.
E, A, D, G, C, F....
2) The lower/left (depending on how one represents the circle) one moves, the darker the tonality becomes. Correspondingly, the higher/right you move, the lighter it becomes.
The brightest tonalities are B major, E major, A major.
The darkest tonalities are B-flat minor, E-flat minor, A-flat minor.
I still don't know why Kapustin used that order. Actually we can also let some other intervals downwards, for example, C Major - B Minor, F Major - E Minor, Bb Major - A Minor, then the left remains but the right becomes 2#, 1#, 0#......
@@yikunqi9353 beautiful sequence
It’s weird as an engineer to see someone brilliant approach music with an engineer-like imagination.
great video!
Indeed
I feel like I watched the music version of a playlist about linear algebra by 3blue1brown
incredible video
The lum system is really cool
Great video, thanks!!!
Great lesson!
awesome video!!!
I want to let you know that you have a rare gift - forensic mind and a wonderfully engaging way of sharing what your forensic explorations have discovered, arranged - in order to illuminate some of the underlying patterns embedded within western tonal music. As a professional composer and performing musician - I salute you and thank you for this excellent illustration of our art.
Wow thank you so much
@@willdaniels1608 Kapustin is russian! Proud of our homelander
Everything you told is very similar to matrices and their linear transformations (hi, 1blue3brown), especially when it came to ‘Lum7’.
It turns out that Lum1 and Lum7 are nothing but basis vectors in 2D space.
*3blue1brown
It looks a lot like the DNA of music, the fundamental blueprint of music, the sefirot tree of life applied to music… it should be 3-dimensional
The Tonnetz is actually 3d as a torus (donut) shape. Apparently, they've done brain scans that show this same shape as people listen to music modulating through all the keys.
The sonata seems to have glimpses of chopins Polonaise in A
is harmonic "luminosity" as described here another word for pitch? seems in the examples you give a brighter sound is always higher, a darker sound is always lower
@liamp.9201 not quite, it's the relative quality of pitches, though it often corresponds with higher and lower pitches
@@willdaniels1608 I don't really follow, can you provide an example where luminosity and pitch move in different directions?
@liamp.9201 one example is a C major chord moving down to an A major chord. While the pitches move down, it is a Lum6 and sounds brighter because of the C#
What software do you use for the Circle of Fifths?
louisbigo.com/hexachord
I already have DL Hexachord, I didn't know there was an interactive Circle of 5ths too lol
Yup !
1:40 Sounds like a little bit of an "Iron Man" motif in there in a few spots, at least to my uncultured ears lol
What do you mean C maj is darker than G maj? 4:23 they both have the exact same intervals, so unless you have perfect pitch you wouldn’t be able to tell which one is which. Do you mean that when playing a song in C maj and switching keys would make G maj sound brighter?
Great video though, I'm just confused about what you meant
That's exactly it. Though they are the exact same chord type, we would hear the shift from one to another by they're relation to each other. We would hear G as brighter in that shift.
Ahh I see thanks for elaborating, is that because we can view the shift as a modal shift from C maj to C Lydian?
@@NationDixon yes!
@@NationDixon but i think the view of their relative position around the circle of fifths is more fundamental and general
All this unfortunately doesn't prevent Kapustin's pieces from sounding harmonically somewhat "amorphous". There is always happening too much, so that the ear can't single out passages with really outstanding, characteristic harmonic color. Compare his music to, say, Wagners Tristan ouverture, to Scriabin's works, to Ravel's "Forlane" or to Kenny Kirkland's "Dienda". All pieces with much harmony going on, AND the ear can really discern harmonic "colors" typical for these pieces. Not in Kapustin's pieces - there is a sameness, it sounds arbitrary. So much theoretical thought, so little actual impact on the listener.
The harmonic process is much different than those composers, and I think how you hear Kapustin's music depends a bit on how much you have been primed in jazz styles and improvisation. His chord progressions are tonally directed, using mostly standard jazz syntax like 2-5-1s, with quick chromatic substitutions and elaborations that emulate the unexpected turns and kaleidoscopic effect of improvisation. More of a roller coaster than harmonic scenery.
Appreciate your thoughts!
@@willdaniels1608 I am a professional jazz pianist, so... I love bebop and Oscar Peterson and Bill Evans and all these jazz giants, but Kapustin leaves me totally cold.
@@christophmunch4796 I had a fear I may offend a jazz pianist 😅 . I'll just say I can sympathize with your take.
Really interesting. So well explained. Thanks very much!
Thank you Will. This is an enlightenment for me. Subscribed instantly.
What program do you use for showing the tonnetz?
Glad you found it enlightening. The program is Hexachord, I used it for the circle of fifths also.
@@willdaniels1608 can u share with us a download link? :D
really intresting stuff in this vid bro !
stumbled across it from knowhere
keep up the good work and thank you for your knowledge shared with us here :)
@@benrijkmans8983 Sure thing! louisbigo.com/hexachord
Hi, I watched your video, is the "harmonic luminosity" your original idea? What do you think of the F#? It could be seen either 6 steps clockwise from C or counterclockwise. And what about the difference between F# and Gb, etc.?
@yikunqi9353 Yes I coined the term "harmonic luminosity" and the definition laid out in the video. But its based on what I believe to be a common way of describing harmony, in terms of brightness and darkness. I set out to offer a technical way to describe that common experience. The relationship of C to F# is certainly a pole that is maximally distant and therefore difficult to discern, without further context, whether one hears it as brighter or darker. But that's just an individual note. Usually we hear these tones in a larger context of a key, or a collection of pitches, and then it may be more clear which side of that collection F# is closer to.
I really loved this. That software is genius. I'm a visual learner and this was amazing!
Но, все это можно проще, логичнее и математически более обоснованно, представить в пространстве кратностей, где по одной оси отложениы терции, а по другой квинты. Реально не существует малой терции, минорный аккорд это трезвучие, в котором тональной нотой является последняя - наиболее высокая, и внет точно так же отложен интервал терции (большой терции) от последней ноты, именно этот интервал соответсует резонансу по 5-й гармонике, что и составляет звучание аккорда. Малые терции не являются онтервалом образующим аккорды, ито всего лишь эмпирическое правило игры, предполагающее что следующим интервалом будет квинта, и терцию мы получим автоматически
Really love kapustins works and this explained and helped soo much!
This was amazingly helpful. I've been working on some of Kapustin's Op.59 Ten Bagatelles and found them really hard to commit to memory because of all the crazy harmony and ambiguous cadences. I have some pretty crazy looking pencil scrawl trying to notate the chords.
A harmonic language which actually makes sense for his music will be so so useful. Thank you!
Holy heck this is fantastic. I was a little disoriented since this is the first time I've seen the tonnetz with the fifths axis in a vertical direction, but my brain was able to adjust pretty quick. Anyway, this is a great framework for comparing brightness. I'll be re-watching this several times for sure. Did you write your own Tonnetz program to respond to MIDI input?
I was disoriented too!! It's a program I found online called hexachord (I think, going off memory right now). I couldn't orient it any other way, and it was the best tool I could find.
@@willdaniels1608 Thank you. I'm curious: if you became emperor and could choose a standard way of drawing the Tonnetz that everyone would adhere to, what would you choose? My default has been to draw upward 5ths to the east, up minor 3rds to the northeast, and up major 3rds to the northwest, but now I'm wondering if that's optimal if one wants directions of increasing brightness to to be intuitive.
@@PohlLongsine If I were emperor I think I would agree with that. I would prefer what I think is pretty standard, exactly what you described. Then bright-shifting modulations would take you upwards, and dark would be downwards, while small diatonic changes would be sideways. Bright would be that vertical association with the heavens.
Bravo
I'm sorry to say this, but it's entirely unfounded. While your experimental spirit is commendable, this content is quite erroneous and misleading. Firstly, it's fundamentally wrong from the beginning. Discussing the brightness and darkness, the essence of chords is not something that can be oversimplified. The connotation of chords is related to the context of the musical progression, and auditory contrast is relative. Your method can be considered an overinterpretation of the circle of fifths.
For instance, when you claim that moving from C major to F major darkens the mood, it's merely because the F major you played is in a lower octave than C. If you were to play an F major in a higher register (fourth up), it would sound significantly brighter. The fact is, F major isn't inherently darker compare to C major; anyone can verify this. Therefore, from the outset, this is somewhat of an exaggerated interpretation. Please stop propagating these misconceptions.
I actually agree with you. This is generally how music theory works, that depending on how you look at it, we can draw out different associations. Defining a system within which to understand these associations, like brightness and darkness, is not an attempt to say this is the absolute essence of the chords, but an attempt to reflect a common experience, but yes it's purely relative, just measured relativity.
I've always felt this way too about the terms "brightness" and "darkness". I'm only starting to get it now after watching this video and arguing with myself about it (lol) - why would someone (whoever chose to use these terms originally) pick words that are sight-oriented to describe the way a harmonic shift sounds? It's never made sense to me until I realized they're just words we use to describe the direction, clockwise or counterclockwise, the shift is going on the circle of fifths.
Obviously, from C major, both F major and G major are the same harmonic distance. To me, they sound the same distance but one doesn't sound "brighter" or "darker" than the other. As you said, the effect of shifting to one of those keys on the mood of a piece can be brighter or darker depending on the register. So it's really just a way to say "we're moving down/up in the circle of fifths", which, I guess if your ear is good enough (mine isn't yet lol) you should be able to discern the direction just from hearing the harmonic shift. So I agree that it's important to have words to describe the direction of motion but I will never agree that "bright' and "dark" are reasonable words to use lol. But that's not on Will, he didn't come up with those terms