*NOTES:* Big things coming this academic year for the channel! Patreon will soon fill up with a series of exclusive interviews as part of a truly massive project. Until then, shorter videos. 6:17 By “failure” here, I mean *the establishment of a musical grammar to supplant the tonal system.* This wasn’t Debussy’s (or Stravinsky’s) goal. They didn’t leave theoretical treatises; they inspired many other composers, but left no “school” in the same way that Schoenberg did (who was, by this metric, “successful.”)
If I understood correctly, the interval vector for the harmonic minor scale is 335442. For example, for ABCDEF and G# we have: m2: BC, EF, AG# M2: AB, CD, DE m3: FG#, AC, BD, DF, BG# M3: CE, EG#, CG#, AF P4: AD, BE, CF, AE TT: DG#, BF This would be easy to count on a piano but I don't have one of those at hand. Please let me know if I miscounted. [For n notes, the sum of the vector is always n(n-1)/2. For example, for n=7 we get 7*6/2 = 7*3 = 21 = 3+3+5+4+4+2.]
just a fun fact, the six-note collection made up of the diatonic scale without the leading tone also contains a unique number of each interval class vector, but with no tritone and one fewer of each of the other possible vectors
awesome... a similar fact about the diatonic scale and PC set theory explained to me by a professor: The near-transpositional symmetry of the diatonic collection (how it transposes by T5/T7 and only changes one pitch at a time) combined with successive transpositions of T5/T7 covering all 12 chromatic pitches using a single member of the set class as a starting point - take set class (0148) as a counterexample here - is incredibly rare... only 19 set classes have this property necessary for complete tonal modulation. Of these 19, only 5 have cardinality of at least seven (for uninterrupted stepwise melodic motion), and the diatonic scale has the lowest ratio of maintained pitches:new pitches, which maximizes variety among members of the set class!
Very interesting and informative. Although I think the way Stravinsky transposed pentatonic scales, diatonic scales and also larger ones like octatonic scales together was part of what made the rite of spring so incredibly interesting and influential. It is interesting as well that it’s harder for theorists to analyse polytonal works using these scales for that reason, but it’s also interesting to think of three other levels to this, which I would love to see you do a video on: 1. What exactly is the music theory behind how composers like Scriabin created their later works, besides the use of the ‘mystic’ scale and octatonic and augmented scales; 2. How about the music theory of neoclassical composers, such as Poulenc or Prokofiev, or mid period Stravinsky; and 3. Messiaen arguably did find a way to create lots of transpositions between more exotic scales, what is your opinion on this? It is also cool to see think with polytonality, one can indeed use diatonic/major and minor scales in different keys simultaneously and create huge harmonies, without even needing to use other scales. Many thanks again for this video.
Maybe the anomally regarding the tritone transposition of the diatonic set (2 common tones instead of the expected 1) is explained by the use of enharmonic notes. If we transpose the C maj diatonic set an aug 4th up, to get F# maj, for example, the F goes to B (the 1 common tone) while the B goes to E#, and while in 12 TET E# and F are enharmonic, if we used a different temperament system then B would really be the only common tone between these 2 sets
Great!!! It's nice to see a video that's actually about the interval class vector, it's useful but gets so little attention. Out of curiosity, I plugged in some simple microtonal scales into Jeremiah Goyette's set calculator to see if other similar-size scales had an interval class vector with a smooth variety of numbers. Here, I denote a larger consistent step size with "L," and a smaller step size with "s." So these scales each only have two types of adjacent step sizes, like the diatonic 5L 2s. Of course, there are more interval possibilities with higher 'TET's, and odd-numbered TET's don't have a half-octave and thus their last number doesn't have to be doubled/halved. 4L 3s in 15-TET: 4L 3s in 18-TET: 5L 4s in 19-TET: 5L 3s in 21-TET: 7L 1s in 15-TET: 7L 1s in 22-TET:
This reminds me of how I was taught to analyze Monteverdi as using the “Mollis” and “Durus” in his operas, like Orfeo, to represent ascent and descent (in and out of the underworld) of the various characters. Not entirely related, but an interesting rabbit hole lol
> Mollis and Durus I'm curious where those words come from-note that "dur" and "mol(l)" are the Danish (German) words for major and minor. [I _think_ there's one more ell in German compared to Danish.] The sound similar so maybe there's a connection.
Here's my explanation: Even pitches ascend, odd pitches descend, in Lydian mode, like so, in (unambigous) interval classes: -5 -3 -1 0 2 4 6. In Locrian, it's opposite. Dorian ics: 022020 (symmetric) Alternating formula: (-1)^(i+m), i (interval class ) in 1...6, m (mode) in 0...1. The contingent ic's comes from this alteration. Direction flips with odd/even, so polarity is a better word. Another way to obtain this set is by repeated division. 12 = 6+6. 6 = 3+3 or 2+2+2. 3 = 2+1 or 1+2.
this really gives a new perspective to the scale, thanks for sharing! by the way, can you please make a video one day about the books you have on the shelves, those seem interesting ones, and curious what are some books that you can maybe recommend :))
Tonality may not have yet been exhausted, but I do believe it's being used in increasingly less interesting ways. As with all styles, I feel it best fits as a technique to be used sparingly in conjunction with newer burgeoing styles. Similar to how spectralism and serialism are techniques and were mixed in both purely instrumental and electronic contexts (electronics the burgeoning style of their day), and how forms have been used throughout music history to call back to previous styles and comment on the present through them. Imo, tonality can only exist as a shade of color or a phrase to fit into a cohesive whole
First of all, I don't agree with the claim that tonality has been exhausted at all, but I don't think the comparison is apt. It's impossible to write literature without words but it is possible to write atonal music (serialist music or gamelan for example).
@@Tscott2279it’s not that at all. It’s actually moreso that all of the non interesting tonal music gets forgotten and lost to time, and so over time our canon is built off of what seems to be the cream of the crop. In our time, we see a lot of the new stuff and don’t really experience what gets filtered out, so you might be apt to say tonality is used in increasingly less interesting ways. There’s tons of comepltely boring and uninteresting tonal music over the last 400 years if you do digging into contemporary composers of the ones we consider great, or study into their less successful students.
@@apolace7242 In principle I agree that we cannot have an exact idea of which contemporary authors will have historical significance in the future. However, an avid listener will surely come across what will be considered geniuses in the future. And I personally have never heard a tonal piece from the last few decades that isn't essentially a recycling of already used ideas, formulas or structures. If you know about it, I'd be very curious to hear about it.
@@designstudio8013 the scales are for a large part based on the overtone series, they are not just made up and even if they were they would still contain comparable proportions.
PC set theory is entirely enharmonic. See my video on the theory (I think it's linked in the endscreen of this one) for a more substantive explanation.
Well, that escalated quickly... how did you go from the number of intervals of a given kind to that number being the same as the quantity of common tones after a transposition? It would have been helpful to see more clear examples. Where can I learn more? What is this? Schoenberg?
This is pitch-class set theory. It was invented after Schoenberg (developed on Babbitt, Forte, and Hanson’s work), but is very useful for analyzing his stuff. You can mess around on a keyboard with transposing by various intervals and seeing what is maintained. For example, playing a major chord and then transposing it by major third, your original third will become your new root (or vice versa) and the other two pitches will change. Or, if you transpose by minor third, your original 5th will become your new 3rd, or vice versa. Any movement by an interval you have will sort of “pivot” two of the notes. You can increase the complexity by trying pitch class sets with 4 notes each, or with multiple of the same type of interval, or get to the “why” behind the question by trying it with just 2-note sets (that is, with just a bare interval - I’m sure you can see there why transposition by the interval you have 1 of will maintain one pitch, and transposition by anything else won’t!)
Western music scales of merely a convoluted weird way to try to make things fit in fact they never can fit mathematically. The b&e notes are called whole notes but mathematically they are not.
You are very confused about music theory. There is no such thing as "whole notes", there are whole tones, these are intervals, meaning distances between 2 notes. B and E are two notes but their distance is not the same as a whole tone. Your first sentence also sounds like you are jumping to conclusions. In short, nothing of what you said makes any sense. Perhaps you need more formal study of music theory.
![PASULOL รามเกียรติ์ ตอนที่ 4 นนทกพบรัก [Ramakien Ep.4: Nonthok in love]](http://i.ytimg.com/vi/tMUQgmZzf_A/mqdefault.jpg)
*NOTES:* Big things coming this academic year for the channel! Patreon will soon fill up with a series of exclusive interviews as part of a truly massive project. Until then, shorter videos.
6:17 By “failure” here, I mean *the establishment of a musical grammar to supplant the tonal system.* This wasn’t Debussy’s (or Stravinsky’s) goal. They didn’t leave theoretical treatises; they inspired many other composers, but left no “school” in the same way that Schoenberg did (who was, by this metric, “successful.”)
If I understood correctly, the interval vector for the harmonic minor scale is 335442.
For example, for ABCDEF and G# we have:
m2: BC, EF, AG#
M2: AB, CD, DE
m3: FG#, AC, BD, DF, BG#
M3: CE, EG#, CG#, AF
P4: AD, BE, CF, AE
TT: DG#, BF
This would be easy to count on a piano but I don't have one of those at hand. Please let me know if I miscounted.
[For n notes, the sum of the vector is always n(n-1)/2. For example, for n=7 we get 7*6/2 = 7*3 = 21 = 3+3+5+4+4+2.]
just a fun fact, the six-note collection made up of the diatonic scale without the leading tone also contains a unique number of each interval class vector, but with no tritone and one fewer of each of the other possible vectors
awesome... a similar fact about the diatonic scale and PC set theory explained to me by a professor:
The near-transpositional symmetry of the diatonic collection (how it transposes by T5/T7 and only changes one pitch at a time) combined with successive transpositions of T5/T7 covering all 12 chromatic pitches using a single member of the set class as a starting point - take set class (0148) as a counterexample here - is incredibly rare... only 19 set classes have this property necessary for complete tonal modulation. Of these 19, only 5 have cardinality of at least seven (for uninterrupted stepwise melodic motion), and the diatonic scale has the lowest ratio of maintained pitches:new pitches, which maximizes variety among members of the set class!
hmm... so, if scale form analogus "circle of fifth", then we may see only a small amount of such scales with useful interval vector among other?
Just found this channel, it’s gold.
Thank you I enjoyed👌
super interesting approach!
Very interesting and informative. Although I think the way Stravinsky transposed pentatonic scales, diatonic scales and also larger ones like octatonic scales together was part of what made the rite of spring so incredibly interesting and influential. It is interesting as well that it’s harder for theorists to analyse polytonal works using these scales for that reason, but it’s also interesting to think of three other levels to this, which I would love to see you do a video on: 1. What exactly is the music theory behind how composers like Scriabin created their later works, besides the use of the ‘mystic’ scale and octatonic and augmented scales; 2. How about the music theory of neoclassical composers, such as Poulenc or Prokofiev, or mid period Stravinsky; and 3. Messiaen arguably did find a way to create lots of transpositions between more exotic scales, what is your opinion on this?
It is also cool to see think with polytonality, one can indeed use diatonic/major and minor scales in different keys simultaneously and create huge harmonies, without even needing to use other scales. Many thanks again for this video.
Maybe the anomally regarding the tritone transposition of the diatonic set (2 common tones instead of the expected 1) is explained by the use of enharmonic notes. If we transpose the C maj diatonic set an aug 4th up, to get F# maj, for example, the F goes to B (the 1 common tone) while the B goes to E#, and while in 12 TET E# and F are enharmonic, if we used a different temperament system then B would really be the only common tone between these 2 sets
Great!!! It's nice to see a video that's actually about the interval class vector, it's useful but gets so little attention. Out of curiosity, I plugged in some simple microtonal scales into Jeremiah Goyette's set calculator to see if other similar-size scales had an interval class vector with a smooth variety of numbers. Here, I denote a larger consistent step size with "L," and a smaller step size with "s." So these scales each only have two types of adjacent step sizes, like the diatonic 5L 2s. Of course, there are more interval possibilities with higher 'TET's, and odd-numbered TET's don't have a half-octave and thus their last number doesn't have to be doubled/halved.
4L 3s in 15-TET:
4L 3s in 18-TET:
5L 4s in 19-TET:
5L 3s in 21-TET:
7L 1s in 15-TET:
7L 1s in 22-TET:
This reminds me of how I was taught to analyze Monteverdi as using the “Mollis” and “Durus” in his operas, like Orfeo, to represent ascent and descent (in and out of the underworld) of the various characters.
Not entirely related, but an interesting rabbit hole lol
> Mollis and Durus
I'm curious where those words come from-note that "dur" and "mol(l)" are the Danish (German) words for major and minor. [I _think_ there's one more ell in German compared to Danish.] The sound similar so maybe there's a connection.
Here's my explanation: Even pitches ascend, odd pitches descend, in Lydian mode, like so, in (unambigous) interval classes: -5 -3 -1 0 2 4 6. In Locrian, it's opposite. Dorian ics: 022020 (symmetric)
Alternating formula: (-1)^(i+m), i (interval class ) in 1...6, m (mode) in 0...1.
The contingent ic's comes from this alteration. Direction flips with odd/even, so polarity is a better word.
Another way to obtain this set is by repeated division. 12 = 6+6. 6 = 3+3 or 2+2+2. 3 = 2+1 or 1+2.
this really gives a new perspective to the scale, thanks for sharing! by the way, can you please make a video one day about the books you have on the shelves, those seem interesting ones, and curious what are some books that you can maybe recommend :))
I did a huge livestream going through the library a while back :)
Honestly, the claim that tonality has exhausted itself is like saying that using color in painting exhausted itself (or words in literature).
No, the equivalent to that would be that using notes (or even sound) has exhausted itself
Tonality may not have yet been exhausted, but I do believe it's being used in increasingly less interesting ways.
As with all styles, I feel it best fits as a technique to be used sparingly in conjunction with newer burgeoing styles. Similar to how spectralism and serialism are techniques and were mixed in both purely instrumental and electronic contexts (electronics the burgeoning style of their day), and how forms have been used throughout music history to call back to previous styles and comment on the present through them.
Imo, tonality can only exist as a shade of color or a phrase to fit into a cohesive whole
First of all, I don't agree with the claim that tonality has been exhausted at all, but I don't think the comparison is apt. It's impossible to write literature without words but it is possible to write atonal music (serialist music or gamelan for example).
@@Tscott2279it’s not that at all. It’s actually moreso that all of the non interesting tonal music gets forgotten and lost to time, and so over time our canon is built off of what seems to be the cream of the crop. In our time, we see a lot of the new stuff and don’t really experience what gets filtered out, so you might be apt to say tonality is used in increasingly less interesting ways. There’s tons of comepltely boring and uninteresting tonal music over the last 400 years if you do digging into contemporary composers of the ones we consider great, or study into their less successful students.
@@apolace7242 In principle I agree that we cannot have an exact idea of which contemporary authors will have historical significance in the future. However, an avid listener will surely come across what will be considered geniuses in the future. And I personally have never heard a tonal piece from the last few decades that isn't essentially a recycling of already used ideas, formulas or structures. If you know about it, I'd be very curious to hear about it.
Hi,i love your videos, can you make one on kapustin please?
I wonder if you might be able to do a video on the octatonic scale.
Cool. Now you need to make a video about Thomas Noll's "Ionian theorem"!
I've found that pitch class set theory is very useful when analyzing music by Leonard Bernstein.
Could you make a video on Samuil Feinberg, please?
Can we say that diatonic has this useful interval vector, becose it form a "circle of fifth" structure?
So where did the "major scale" come from in the first place?
The Ionian tetrachord has a spiral like structure of decreasing intervals, 204 182 112, similar to the volute in the capital of the Ionian pillar.
I seriously doubt anything in music can be compared to engineering. Music is completely made up as you can tell by the many many different scales.
@@designstudio8013 the scales are for a large part based on the overtone series, they are not just made up and even if they were they would still contain comparable proportions.
@@designstudio8013 What does “made up” mean, in this context?
What does “204 182 112” mean?
@@unoaotroa they refer to cents where a 100 would equal a semitone in the equal temperament system
how would you analyze scales with augmented tones? would you count it as a m3 or have a different form of vector?
PC set theory is entirely enharmonic. See my video on the theory (I think it's linked in the endscreen of this one) for a more substantive explanation.
Well, that escalated quickly... how did you go from the number of intervals of a given kind to that number being the same as the quantity of common tones after a transposition?
It would have been helpful to see more clear examples. Where can I learn more? What is this? Schoenberg?
This is pitch-class set theory. It was invented after Schoenberg (developed on Babbitt, Forte, and Hanson’s work), but is very useful for analyzing his stuff.
You can mess around on a keyboard with transposing by various intervals and seeing what is maintained. For example, playing a major chord and then transposing it by major third, your original third will become your new root (or vice versa) and the other two pitches will change. Or, if you transpose by minor third, your original 5th will become your new 3rd, or vice versa. Any movement by an interval you have will sort of “pivot” two of the notes. You can increase the complexity by trying pitch class sets with 4 notes each, or with multiple of the same type of interval, or get to the “why” behind the question by trying it with just 2-note sets (that is, with just a bare interval - I’m sure you can see there why transposition by the interval you have 1 of will maintain one pitch, and transposition by anything else won’t!)
Well I've listened to this video twice now and I have not got a clue what you are talking about
Western music scales of merely a convoluted weird way to try to make things fit in fact they never can fit mathematically. The b&e notes are called whole notes but mathematically they are not.
You are very confused about music theory. There is no such thing as "whole notes", there are whole tones, these are intervals, meaning distances between 2 notes. B and E are two notes but their distance is not the same as a whole tone. Your first sentence also sounds like you are jumping to conclusions. In short, nothing of what you said makes any sense. Perhaps you need more formal study of music theory.