
English: 
hey, welcome to 12tone! if you're the sort
of person who watches my videos, you probably
already know who Jacob Collier is, but just
in case, he's a Grammy-winning singer, songwriter,
arranger, and multi-instrumentalist who rose
to fame for his intricate one-man arrangements
of popular songs on YouTube. but he's also
a really talented theorist who's done some
awesome videos about the ideas behind his
work. plus he's five years younger than me
and I'm totally not jealous. anyway, today
I want to take a closer look at one idea that
seems to be pretty central to his conceptions
about harmony: the duality between fifths
and fourths. that is, his idea that perfect
5ths are effectively major, whereas perfect
4ths are minor.
Jacob often justifies this by comparison to
the circle of 5ths. we've covered this before,
but basically, it's a mnemonic device for
remembering how key signatures work. we start
at the top with C major, whose key signature
is empty, then we go clockwise around the
circle, moving up a perfect fifth every time,
so G, D, A, and so on, until we get back around
to Bb, F, and finally C, completing the circle.
we can also move counter-clockwise, going

Chinese: 
嗨，這裡是12tone(頻道名)
如果你是此頻道的觀眾
你大概早就知道誰是Jacob Collier
但還是簡單做個介紹
Jacob是葛萊梅獲獎歌手、作曲家、編曲家和樂手
他在youtube發布自己重編的流行歌，使他聲名大噪
此外，他同時對"理論"也相當有一套
他有幾支影片就在討論他創作的想法
雖然他比我小五歲，但我完全不忌妒他
言歸正傳
今天我想仔細探討Jacob和聲使用的中心思想
五度和四度的"二元性"
五度有大調的效果
而四度則是小調的感覺
Jacob經常使用五度圈來說明此事
之前的影片有介紹過五度圈
基本上，它是個能幫助記憶調號的工具
從最上面C大調開始，沒有升降
然後順時針，相隔完全五度數下來
所以是G、D、A...等等
直到輪完一圈，經由Bb、F，最後由回到C
也能逆時針，向下完全五度地數

Chinese: 
也就是向上完全四度的意思
向下完五和向上完四是一樣的，只是方向不同
所以這能怎樣?
我們觀察各個調之間的關係發現
每當順時針往數一個調，就表示要升一個音
所以從A大調到E大調，要將原本的D，改為#D
換言之，將主音上移五度會有明亮的感覺
因為將整組音略上移了點
逆時針看五度圈也有類似的規律
從E到A表示要降一個音(#D->D)
聲響上會有暗沉的感覺
當然，並沒有絕對"明亮"的調
因為這是一個循環，你可以不斷升高，或是降低。
但Jacob對這個概念的理解，遠超過單單轉調那麼簡單。
他將這個概念用來組建每個和絃
例如這個和弦，聽起來比較明亮且有大調聲響
和這個和弦相比之下
盡管我使用的都是同一組音
我想稍微討論一下這個點
因為Jacob並不是憑空創造出這些的

English: 
down a perfect 5th at a time, which is equivalent
to going up a perfect 4th. they're the same
thing, just different directions.
so what does this give us? well, if we look
at the various major scales associated with
these notes, we see that going one step clockwise
around the circle means raising one note in
the scale, so going from A major (bang) to
E major (bang) means taking this D natural
and making it sharp. in a sense, then, moving
to a key whose root is a perfect fifth higher
does result in a brighter tonality, because
the set of notes we can use has shifted slightly
upwards. and going the other way round the
circle works similarly: going from E major
back to A major means lowering the D# to D
natural, darkening the overall pitch collection.
of course, all of this is relative: there
is no objectively brightest key. that's why
it's a circle: you can keep rising or falling
forever, depending on which way you want to
go.
but Jacob expands this idea far past simple
key changes. he also applies it to the construction
of individual chords, so that this (bang)
would sound more bright and major than this
(bang) even though I used the same notes both
times. and here is where I want to focus,
because Jacob isn't just pulling this out
of thin air. you may have noticed at the beginning

Chinese: 
你應該有注意到我前面用"二元論"稱呼這個理論
這個稱呼，非常恰當。
我從沒聽過Jacob使用"二元論"這種說法
但他的想法可以追朔到19世紀
一個相當具爭議性的論調，"和聲二元論
好的，或許也不是那麼有爭議性
但假使我們回到1860至1870這段時間
你會看到理論家們激烈的鬥爭，
發生於"一元論"與"二元論"的支持者之間。
因為我們這些理論學者是相當嚴肅的
總會為了"重要"的事情奮鬥
簡單說一下他們的爭議
一元論者認為，大三和弦是"王"
而小三和弦卻是微不足道
二元論者則認為兩者同樣重要
這只是對這長達十多年的爭論的略述
當然多少有漏一點細節
不過在此，"一元論"並不太重要
所以，我們繼續看下去吧
首先要提及的重要人物是德國學者，莫里茨.赫普曼
他1853年發布了"和聲的本質"和"音韻學"
在裡頭，他把三和弦拆成有三個不同功能的音。
並將其稱作"moment"
每個和弦都是由「unity」、「duality」
及「union」所組成
這是用彼此之間的關係來定義的
如果同時彈奏「unity」和「duality」，
可以得到一個完全五度

English: 
that I described his theory as "duality",
and that was a very deliberate choice. I've
never heard Jacob use that specific word to
describe it, but his ideas trace their roots
back to one of the most controversial ideas
of the 19th century: Harmonic dualism.
ok, maybe not the most controversial, but
if we traveled back in time to around the
1860s and 70s, you'd see a battle raging among
the theorist crowd between monism and dualism,
because we theorists are very serious people
who only fight about very important things.
to quickly summarize the dispute, monism is
the idea that the major triad is king, and
the minor triad is just a corruption or shadow
of it, whereas dualism puts both on equal
footing. that's a one-sentence explanation
of a decades-long argument, so it's admittedly
missing a little of the nuance, but monism
isn't really relevant here so let's dive in.
probably the first important name is the German
theorist Moritz Hauptmann, who published The
Nature of Harmony and Metre in 1853. in it,
he broke the notes of a triad up into three
different functions which he called "moments":
each chord consists of the unity, the duality,
and the union, which are defined by their
relations to each other. if you play the unity

English: 
and duality at the same time, you get a perfect
fifth, while the unity and union make a major
3rd. the unity is effectively the root of
the chord, in that it participates in both
these relationships, but it doesn't have to
actually be the root in the classic sense
of the word. in a major chord, it is: if we
take A as our unity, then we can have E as
the duality a perfect fifth up and C# as the
union in between. but if we have A minor,
then it's no longer really an A chord at all:
the only note that's part of both a perfect
fifth and a major third is E, so that becomes
our unity even though it's actually the fifth
of the chord. this is where the comparison
to Jacob's theory comes from: in a major chord,
the duality is a perfect 5th above the unity,
whereas in a minor one the duality is a perfect
5th down, or a perfect 4th up. in effect,
Hauptmann believed that the minor triad was
just like the major one, but upside-down.
this idea is at the heart of the dualist philosophy,
but it received some significant pushback.
perhaps one of the most famous critics was
Hermann von Helmholtz, who argued that the
minor triad couldn't be equivalent to the
major one, because its overtones were significantly
messier. you see, Hauptmann had been assuming
that going up and going down were the same

Chinese: 
「unity」和「union」則能組成大三度
「unity」的功能相當於和弦根音
因為它同時與另外兩個音有完五與大三的關係
但它與傳統意義上的"根音"，不盡相同。
以一個大三和弦來說
如果我們將A設為我們的unity
那E就會是duality，因為呈完五度，而#C則是union
但如果是A小三和弦
那它就不再是一個真正的A和弦
唯一有符合同時有大三與完五的音是E
因此E才是Am的unity，儘管它實際上是Am的五音
這正是Jacob所使用的概念出處
在大和弦中，duality是unity向上完五度
而小和弦的duality則是向下完五度
或是向上完四度
赫普曼實際上認為大三和弦與小三和弦是相似的
只是相反而已
這個想法是二元論的核心
而它卻受到一股非同小可的阻礙
Hermann von Helmholtz或許是最有名的批評者
它對大小和弦的同質性提出質疑
因為兩者造成的諧波有顯著的不同
如你所見，赫普曼假設無論音程的方向為何，
都是一樣的

Chinese: 
但在實際聲響上，它們並不相同
當你彈奏一個音，你會創造出具特定頻率的聲波
此外，你也同時創造出了泛音
泛音的頻率是基頻的整數倍
假設我彈奏一個頻率為220hz的A
它會伴隨著440hz、660hz、880hz...等音
這些泛音是真實會發生的現象
而會造就大和絃色彩明亮的原因之一
正是因為和弦音與根音產生的泛音列相同
但並不存在向下泛音，至少不真實存在
許多人，包括Helmholtz都有類似關於向下泛音列的理論
也就是220hz會創造出110、73、55hz等頻率
但實際上當你在彈奏樂器時，並不會有這音產生
這只是個想法，雖然想法很好，但卻違背實際聲響
因此Helmholtz才會批評赫普曼的"和聲對稱說"
雖然很可愛，但卻經不起嚴酷的現實考驗
但二元論尚未結束
下一個二元論的擁護者是 Arthur von Oettingen
他是一位受過訓練的物理學家
Oettingen透過兩個概念來解釋這個原理
"phonicity" 和"tonicity"

English: 
thing, but in a real, acoustic sense, they're
not. when you play a note, you're creating
a soundwave with a specific frequency, but
you're also creating what are called overtones,
which are just all the multiples of that frequency.
so the A I just played was 220 hertz, or cycles
per second, but there were also shades of
440 hertz, 660, 880, and so on, forever. these
overtones are real acoustic phenomena, and
one of the reasons a major triad sounds so
pleasant is that the base frequencies of the
notes line up to create a fairly simple overtone
series.
but there's no downward equivalent, at least
not a real one. plenty of people, including
Helmholtz, have theorized about a sort of
undertone series, where 220 hertz would create
110, 73, 55, and so on, but it's not an actual
thing that actually happens when you play
the note on an actual instrument. it's just
an idea, and while ideas are great, they're
not the same as sounds, so Helmholtz argued
that Hauptmann's symmetries, while lovely
in concept, fell apart in the harsh realities
of practice.
but that wasn't the end for dualism. its next
advocate was Arthur von Oettingen, who was
actually a physicist by training. Oettingen's
approach was based on two ideas he called

English: 
"phonicity" and "tonicity". the phonicity
of an interval or chord has to do with which
overtones all the notes share, so if we have,
say, 220 and 330 hertz, the lowest note they
both create is 660 hertz, which is called
their phonic overtone. tonicity, on the other
hand, is about which fundamental note generates
all the others as overtones, so if we again
take 220 and 330, we find they're both overtones
of 110 hertz, which we call the tonic fundamental.
in a major triad, the tonic fundamental is
consonant: that is, it's a note that's already
part of the chord. with A major, for instance,
it's a low A. however, the phonic overtone
isn't: it's a G#, which is a whole new note.
A minor, on the other hand, has a phonic overtone
of E, which is in the chord, re-establishing
a real symmetry between the two and reinforcing
Hauptmann's idea that the actual root of a
minor triad is its fifth. it's worth noting
that this is based, in part, on Helmholtz's
own work, combining it with Hauptmann's models
in order to rescue them from Helmholtz's attack,
because sometimes music theory is just wonderfully
petty.
Oettingen also developed the idea of laying
notes out on a grid, like this, in order to
build chords. here, the rows represent movement
by perfect fifth, and the columns go by major

Chinese: 
phonicity的音程或和弦，
會有那麼一個音，同時是所有音的泛音
所以，假如我們有220hz和330hz，
它們最低的共同泛音頻率便是660hz
660hz就會被稱作為這兩個音的phonic泛音
另一方面，tonicity表示，所有音同時是某個音的泛音
假如再拿220hz與330hz為例
我們發現它們同時是110hz的第一與第二諧音
110hz就會被稱作為220與330hz的共同基頻
大三和弦的共同基頻是和諧的。
也就是說這個音，是該和弦的組成音
比方說一個A大三和弦，它們的共同基頻就是低音A
然而A大三和弦的共同泛音就不和諧了
它們的共同泛音是#G，完全不再和弦裡頭
然而A小三和弦的共同泛音是E，卻是和弦內音
重新建立兩者之間的對稱關係，以及加強赫普曼的觀點
意即A小三和弦真正的根音其實是它的五音(E)
值得注意的是，這個理論是以Helmholtz的研究為基礎
並結合了赫普曼的概念原型
為了從Helmholtz的打擊下逃過一劫
因為有時候，樂理的容忍度實在很奇妙
Oettingen同時發展出將音填入矩陣的概念，如圖。
並以此來建立和弦
將音以完全五度的間隔填入列，
並以大三度的間隔填入行

English: 
third. he claimed that all proper triads could
be found along diagonals like this one, connecting
notes a minor third apart. the only difference
is that major triads also include the note
above the line, while minor triads use the
one below instead. if you've heard me talk
about Neo-Riemannian theory before, this may
look familiar: it's a tonnetz. normally those
are rotated a bit so the triangles are equilateral,
but structurally it's the exact same thing.
if you don't know what that is, don't worry,
it's not that important right now, I just
needed a segue to talk about our last and
most famous dualist, Hugo Riemann.
Riemann further formalized these dualist relationships
by developing a set of transformations that
could take you from one chord to another.
the first of these are Schritte, or step transformations,
which slide the chord up or down to a new
root without changing the quality. of particular
note is the Quintschritt transformation, which
moves to a chord whose root is your starting
chord's duality, to borrow Hauptmann's terms,
so like from A major to E major or A minor
to D minor. the second transformation is the
Wechsel, or change transformation, which flips
the chord across its unity, changing it from
major to minor or vice versa. most notable

Chinese: 
他聲稱任何一個三和弦都能在對角線上找到，例如這個，兩個音相連成和弦中的小三度
其中的差別是，大和絃要包含斜線上的音，
小和弦則是斜線下的音。
如果你聽過我之前關於新里曼理論的討論，
你會發現其中的相似
這其實就是"里曼調性網絡"
只是里曼調性網絡將圖畫成正三角形，
但實際上它們是同樣的東西
如果你不曉得里曼調性網絡是啥，別擔心
這目前並不重要
這只是需要一個橋段，來銜接到我們最後一位，也是最有名的一位二元論學者，胡戈·里曼。
里曼透過發展出一套轉換的方式，
使二元關係更一步的系統化
這套轉換方式可以讓你將和弦，轉移到另一個和弦
第一種方式是平移轉換，它可以將和弦在不改變大小的情況下，轉換到另一個和弦
這種轉換式將起始和弦的duality作為目標和弦的unity
比如A大三和弦可以轉移到E大三和弦
或是A小三和弦轉移到D小三和弦
第二種轉換方式是Wechsel，或稱映射轉換。
它將和弦以unity為軸，翻到另一側。
大變小，小變大
最顯著的轉換是Terzwechsel

English: 
of these is the Terzwechsel, which moves it
to the chord's union, then flips it, so like
A major to F# minor or vice versa. these two
relationships formed the basis of Riemann's
theories of functional harmony, but we've
talked a lot about that elsewhere and this
video is getting long enough already, so let's
get back to Jacob Collier.
what does all of this say about Jacob's theory
that fifths are major and fourths are minor?
is he right? well, that's where this all kinda
falls apart, because in music theory, the
ideas of right and wrong are sorta meaningless.
it's all just expert interpretation, and Jacob's
an expert, so in a sense he's inherently right,
even if his ideas don't really match mine,
or Helmholtz's, or anyone else's. thinking
about harmony the way he does has allowed
him to make some awesome songs, and if his
models help you write or understand music
better, that's what matters. that's all any
of us are trying to do, and I'd say Jacob's
doing it really well.
anyway, thanks for watching, and thanks to
our Patreon patrons for supporting us and
making these videos possible. if you want
to help out, and get some sweet perks like
sneak peeks of upcoming episodes, there's
a link to our Patreon on screen now. you can
also join our mailing list to find out about
new episodes, like, share, comment, subscribe,
and above all, keep on rockin'.

Chinese: 
先將和弦根音移到union，再翻轉。
例如A大三和弦可以轉換到F#小和弦
這兩個關係正是里曼和聲功能論的基礎
但我們好像有點扯遠了
而且這部影片也有點太長了
所以，讓我們回來談談Jacob Collier吧
這一切都說明Jacob Collier理論了些什麼？
五度是大調，四度是小調？
他是對的嗎
這或許會有點令人沮喪
因為在討論樂理時，評斷創作概念的對錯是沒有意義的。
這全都是大師們的解釋
而Jacob是大師，意義上他無論如何都是正確的
即使他與我的觀念互相違背，
或是與Helmholtz地想法背道而馳，他都是正確的
他思考和聲的方式，造就他創造出美妙的音樂
假使他的方式使你創作出更好的音樂，
或是更加理接音樂
對錯有什麼關係呢?
這正是我們一直以來都在做的事情，而我必須要說，Jacob做得相當好。
 
 
 
 
 
