# Why are keys a fifth away most closely related?



## millionrainbows

You know, the circle of fifths. Why does it go in fifths, adding sharps? Furthermore, why are there 12 keys?


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## tdc

The circle of fifths goes that way I would assume, because that is the order sharps accumulate as occurring in the major scales. The interval of a fifth is a perfect consonance, the V chord in a given key contains the leading tone as its 3rd, which creates a kind of a magnetic attraction between the chords V and I.

Our modern day system of tuning has at its roots the mathematics of Pythagoras, it has been altered slightly to allow for modulation to other keys while still sounding relatively in tune in all keys. I think it is interesting that these intervals of sound are based on naturally occurring mathematical acoustical concepts. It just so happens that the number of notes in a diatonic scale (7) and the number of notes in the chromatic scale (12), are also significant astrological numbers relating to lunar and solar cycles of time.


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## EdwardBast

These questions are like asking "Why are there ducks?" There is the obvious and immediate answer: Because pairs of ducks mated and had more ducks, and there is the answer that actually explains: beginning with the Big Bang, the distribution of heavy elements due to super novae, and the history of the evolution of life on earth. In that spirit:

The obvious . . . :

12 keys because 12 tones. 12 tones because of Pythagoras et alia. Keys a 5th apart more closely related because they have more common tones. Adding sharps/subtracting flats because that was the solution (historically) for notating a 12 tone system using seven letter names.

. . . and the actual explanation:

The whole history of western music and western music theory.


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## Larkenfield

Because each fifth is the dominant fifth of the next key in the cycle. It's a cycle or circle that jazz musicians often practice because it takes them through the 12 keys or scales. The bridge to I've Got Rhythm is based on the circle of fifths as just one example, though in general the entire circle is usually not utilized. It's an often used chord progression worth learning in every key and it’s often been found in classical music.


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## JeffD

I might be wrong, but I don't think you are looking for a music theory answer. There are tons of websites and videos that explain all that, anyway, and you would have found them if you were looking for them, I am sure.

I think you are looking for the answer to a more fundamental question, like: Did these "standards" get picked arbitrarily, or is their something truly fundamental about them, something physical, and not arbitrary.

The interval of the fifth is a pair of notes with a frequency ratio of 3:2. This can be shown on an oscilloscope, a taught string, etc. The ear drum responds this way as well. It is the first natural overtone, depending on how you count things.

Emotionally all folks of all cultures can recognize the ratio of the fifth, and identify a fifth above a give note accurately. And the two together sound pleasing to most all people. Aside from the unison and the octave, the fifth is considered the most consonant of harmonies.

Other harmonies are considered less consonant by most listeners, and different cultures start to rank the different intervals as to pleasantness etc., differently depending on what they are used to.

Dividing the octave into 12 tones is somewhat arbitrary, since the piano was well tempered.

From that basic stuff we get all the keys. There is a video of Leonard Bernstein showing this on the piano, and it is fascinating. All the keys and their progression follow mathematically from these fundamentals.

Bernstein explains it much better than I can. But the interval of the fifth leads directly to all the keys and all the 12 division of the scale: 




Watch the whole thing, its only five minutes but it is really great. 3 minutes in is the most relevant to your question.


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## BabyGiraffe

millionrainbows said:


> Furthermore, why are there 12 keys?


Are there really only 12 keys?

19 equal is even better, traditional harmony would sound slightly more consonant (would be worse for power chords and extended jazz chords, based on stacking fifths/fourth, but this is true for any meantone tuning). The best small, non-meantone equal temperaments in 5-limit are 34 and 53 equal. 53 is basically 5-limit just intonation (the only notable comma, tempered in it is the schisma).
If you want to modulate the diatonic scale in non-meantone tuning or just intonation, you get one additional chroma that is not mentioned in simple school books, so when modulating a fifth up - let's say from C major to G major - you have to notate two accidentals, because of the 81/80 difference appearing:

0: 1/1 0.000000 unison, perfect prime 
1: 9/8 203.910002 major whole tone 
2: 5/4 386.313714 major third 
3: 4/3 498.044999 perfect fourth 
4: 3/2 701.955001 perfect fifth 
5: 5/3  884.358713 major sixth, BP sixth 
6: 15/8 1088.268715 classic major seventh
7: 2/1 1200.000000 octave

x 3/2 =

0: 1/1 0.000000 unison, perfect prime
1: 9/8 203.910002 major whole tone 
2: 5/4 386.313714 major third
3: 45/32 590.223716 diatonic tritone
4: 3/2 701.955001 perfect fifth 
5: 27/16  905.865003 Pythagorean major sixth 
6: 15/8 1088.268715 classic major seventh 
7: 2/1 1200.000000 octave

27/16:5/3 = 81/80 - 21.506290 syntonic comma, Didymus comma, it's tempered in all meantone systems, so all theoretical Western music, but not in traditional Indian or Arabic, or Chinese music.


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## ECraigR

EdwardBast said:


> These questions are like asking "Why are there ducks?" There is the obvious and immediate answer: Because pairs of ducks mated and had more ducks, and there is the answer that actually explains: beginning with the Big Bang, the distribution of heavy elements due to super novae, and the history of the evolution of life on earth. In that spirit:
> 
> The obvious . . . :
> 
> 12 keys because 12 tones. 12 tones because of Pythagoras et alia. Keys a 5th apart more closely related because they have more common tones. Adding sharps/subtracting flats because that was the solution (historically) for notating a 12 tone system using seven letter names.
> 
> . . . and the actual explanation:
> 
> The whole history of western music and western music theory.


But WHY are there ducks!?


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## mbhaub

Keys are "close" because they have a lot in common. Take C and G: all the notes are the same except one: F becomes F#. In the same way C and a-minor are close (parallel is the term) since in the ascending natural minor mode they have exactly the same notes - just different starting and ending points. C and G-flat have little in common (just two notes) and are distant.


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## millionrainbows

tdc said:


> The circle of fifths goes that way I would assume, because that is the order sharps accumulate as occurring in the major scales. The interval of a fifth is a perfect consonance, the V chord in a given key contains the leading tone as its 3rd, which creates a kind of a magnetic attraction between the chords V and I.
> 
> *Our modern day system of tuning has at its roots the mathematics of Pythagoras,* it has been altered slightly to allow for modulation to other keys while still sounding relatively in tune in all keys. I think it is interesting that these intervals of sound are based on naturally occurring mathematical acoustical concepts. It just so happens that the number of notes in a diatonic scale (7) and the number of notes in the chromatic scale (12), are also significant astrological numbers relating to lunar and solar cycles of time.





EdwardBast said:


> These questions are like asking "Why are there ducks?" There is the obvious and immediate answer: Because pairs of ducks mated and had more ducks, and there is the answer that actually explains: beginning with the Big Bang, the distribution of heavy elements due to super novae, and the history of the evolution of life on earth. In that spirit:
> 
> The obvious . . . :
> 
> 12 keys because 12 tones. *12 tones because of Pythagoras* et alia. Keys a 5th apart more closely related because they have more common tones. Adding sharps/subtracting flats because that was the solution (historically) for notating a 12 tone system using seven letter names.
> 
> . . . and the actual explanation:
> 
> The whole history of western music and western music theory.


Edwardbast, you don't seem to understsnd that I ask these questions so that I can see for myself the answers that are given. This is the internet, and it's cut-throat.

I've had the Pythagoran connection questioned before:



> (Pythagoras) didn't do anything like closing the circle at 12, there is no good reason to believe he did. It would mean nothing to a Greek music theorist to talk about such things. If you want to believe legends are literally true then how about Heracles? Or Beowulf?


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## EdwardBast

millionrainbows said:


> Edwardbast, you don't seem to understsnd that I ask these questions so that I can see for myself the answers that are given. This is the internet, and it's cut-throat.
> 
> I've had the Pythagoran connection questioned before:


Pythagoras _et alia_.


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## Minor Sixthist

ECraigR said:


> But WHY are there ducks!?


Where do the lousy ducks go in winter?


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## jegreenwood

ECraigR said:


> But WHY are there ducks!?


Ask Groucho.






(Best I could find. 36 minute mark.)


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## millionrainbows

EdwardBast said:


> Pythagoras _et alia_.


Yeah, I know what "et al." means; I do crossword puzzles, too.

Explain yourself, then. Where did "12 notes" come from, if not Pythagoran principles? You already said it: Pythagoras. it's too late to go back on that. Exactly what "hair" are you splitting?


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## millionrainbows

Before you can use "notes in common" and "closely related keys," you have to explain where the notes came from. Scales are "generated," they don't just appear. Academic theorists rarely think about how scales are "made."

So where did the 12-note division of the octave come from? What procedure, idea, or concept?


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## millionrainbows

Before you can use "notes in common" and "closely related keys," you have to explain where the notes came from. Scales are "generated," they don't just appear. Academic theorists rarely think about how scales are "made."

So where did the 12-note division of the octave come from? What procedure, idea, or concept?

Also, this shows how people take things as "givens" and can't explain them when asked.


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## millionrainbows

EdwardBast said:


> These questions are like asking "Why are there ducks?" There is the obvious and immediate answer: Because pairs of ducks mated and had more ducks, and there is the answer that actually explains: beginning with the Big Bang, the distribution of heavy elements due to super novae, and the history of the evolution of life on earth. In that spirit:
> 
> The obvious . . . :
> 
> 12 keys because 12 tones. 12 tones because of Pythagoras et alia. Keys a 5th apart more closely related because they have more common tones. Adding sharps/subtracting flats because that was the solution (historically) for notating a 12 tone system using seven letter names.
> 
> . . . and the actual explanation:
> 
> The whole history of western music and western music theory.


This "cute" reply shows how people take things as "givens" and _can't explain them when asked._


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## BabyGiraffe

12 tones in just intonation (simplest and most consonant 5-limit interpretation) and equal temperaments, plus deviations in cents and scale pattern in equal temperaments:

0: 1/1 0.000000 unison, perfect prime
1: 16/15 111.731285 minor diatonic semitone
2: 9/8 203.910002 major whole tone
3: 6/5 315.641287 minor third
4: 5/4 386.313714 major third
5: 4/3 498.044999 perfect fourth
6: 45/32 590.223716 diatonic tritone
7: 3/2 701.955001 perfect fifth
8: 8/5 813.686286 minor sixth
9: 5/3 884.358713 major sixth
10: 9/5 1017.596288 just minor seventh
11: 15/8 1088.268715 classic major seventh
12: 2/1 1200.000000 octave
|
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 11.4284 c. M: 17.5963 c. Twelve-tone Chromatic (1/11-comma)
19: 2 1 2 1 2 1 2 2 1 2 1 2 P M ME SD: 10.6924 c. M: 21.8027 c. Genus diatonico-chromaticum (if we use minor whole tone and augmented fourth, 19 comes way more accurate)
22: 2 2 2 1 2 2 2 2 1 3 1 2 P D3 SD: 9.4792 c. M:-18.7673 c. 
29: 3 2 3 1 3 2 3 3 1 4 1 3 P SD: 11.4957 c. M:-16.8865 c. 
31: 3 2 3 2 3 2 3 3 2 3 2 3 SP M ME SD: 6.3773 c. M: 11.1447 c. Genus diatonico-chromaticum
34: 3 3 3 2 3 3 3 3 2 4 2 3 SP D3 SD: 5.0594 c. M:-9.7763 c. 
41: 4 3 4 2 4 3 4 4 2 5 2 4 SP SD: 4.7936 c. M:-6.7940 c. 
53: 5 4 5 3 5 4 5 5 3 6 3 5 SP SD: 1.1527 c. M: 1.5445 c. Genus diatonico-chromaticum
118: 11 9 11 7 11 9 11 11 7 13 7 11 SP SD: 0.3350 c. M: 0.6471 c.

Arabs have theories about 17, 22 and 24 tones musical systems. It is more interesting how did they came with them.
Indians have a theory about 22-"shrutis".
Babylonians are the oldest culture, developing 12 tone system, way before Greeks and Pythagoras. It was probably related to their calendar and numerical system.
Anyway, it's easy to see that lower overtones of simple ratios will form 12 tone system, if we reduce them to octave. Stacking fifths is another way (3/2 = 5/4 x 6/5, so we will always have some approximations to just major and minor sixths and thirds in this scale). Another way: transposing 7-note diatonic scale and tempering any two of these commas: syntonic, major diesis (known as the difference between 4 minor thirds and octave) , minor diesis (difference between 3 major thirds and octave), diaschisma (difference between the two diatonic tritones; these diatonic tritones are different from augmented fourths and diminished fifths, and septimal tritones or 11th harmonic and its inverse, so there is a whole sea of intervals that sound roughly like "tritones"). (These commas are not musically irrelevant, if you want to notate what actually - let's say- a real violin player performs.)

Another approach using statistical physics and lattice structures:
https://advances.sciencemag.org/con...BnNc56HpzVbrXtqzebf5pd4pHWcixcM-kISaq6bKNWxqo


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## millionrainbows

BabyGiraffe said:


> 12 tones in just intonation (simplest and most consonant 5-limit interpretation) and equal temperaments, plus deviations in cents and scale pattern in equal temperaments:
> 
> 0: 1/1 0.000000 unison, perfect prime
> 1: 16/15 111.731285 minor diatonic semitone
> 2: 9/8 203.910002 major whole tone
> 3: 6/5 315.641287 minor third
> 4: 5/4 386.313714 major third
> 5: 4/3 498.044999 perfect fourth
> 6: 45/32 590.223716 diatonic tritone
> 7: 3/2 701.955001 perfect fifth
> 8: 8/5 813.686286 minor sixth
> 9: 5/3 884.358713 major sixth
> 10: 9/5 1017.596288 just minor seventh
> 11: 15/8 1088.268715 classic major seventh
> 12: 2/1 1200.000000 octave
> |
> 12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 11.4284 c. M: 17.5963 c. Twelve-tone Chromatic (1/11-comma)
> 19: 2 1 2 1 2 1 2 2 1 2 1 2 P M ME SD: 10.6924 c. M: 21.8027 c. Genus diatonico-chromaticum (if we use minor whole tone and augmented fourth, 19 comes way more accurate)
> 22: 2 2 2 1 2 2 2 2 1 3 1 2 P D3 SD: 9.4792 c. M:-18.7673 c.
> 29: 3 2 3 1 3 2 3 3 1 4 1 3 P SD: 11.4957 c. M:-16.8865 c.
> 31: 3 2 3 2 3 2 3 3 2 3 2 3 SP M ME SD: 6.3773 c. M: 11.1447 c. Genus diatonico-chromaticum
> 34: 3 3 3 2 3 3 3 3 2 4 2 3 SP D3 SD: 5.0594 c. M:-9.7763 c.
> 41: 4 3 4 2 4 3 4 4 2 5 2 4 SP SD: 4.7936 c. M:-6.7940 c.
> 53: 5 4 5 3 5 4 5 5 3 6 3 5 SP SD: 1.1527 c. M: 1.5445 c. Genus diatonico-chromaticum
> 118: 11 9 11 7 11 9 11 11 7 13 7 11 SP SD: 0.3350 c. M: 0.6471 c.
> 
> Arabs have theories about 17, 22 and 24 tones musical systems. It is more interesting how did they came with them.
> Indians have a theory about 22-"shrutis".
> Babylonians are the oldest culture, developing 12 tone system, way before Greeks and Pythagoras. It was probably related to their calendar and numerical system.
> Anyway, it's easy to see that lower overtones of simple ratios will form 12 tone system, if we reduce them to octave. *Stacking fifths is another way (3/2 = 5/4 x 6/5, so we will always have some approximations to just major and minor sixths and thirds in this scale). *Another way: transposing 7-note diatonic scale and tempering any two of these commas: syntonic, major diesis (known as the difference between 4 minor thirds and octave) , minor diesis (difference between 3 major thirds and octave), diaschisma (difference between the two diatonic tritones; these diatonic tritones are different from augmented fourths and diminished fifths, and septimal tritones or 11th harmonic and its inverse, so there is a whole sea of intervals that sound roughly like "tritones"). (These commas are not musically irrelevant, if you want to notate what actually - let's say- a real violin player performs.)
> 
> Another approach using statistical physics and lattice structures:
> https://advances.sciencemag.org/con...BnNc56HpzVbrXtqzebf5pd4pHWcixcM-kISaq6bKNWxqo


That's nice little exposition on five-limit systems. But what do you mean by "stacking fifths?" What can this method of generating an octave-division otherwise be described as? Has it anything to do with Pythagoras or Pythagorian-derived principles?

Are scales and octave divisions "created" and derived from principles and/or procedures, or do scales "just exist?" Should we question why there are 12 notes? Should we try to find this out, and why? or Why not?

Did scales "evolve historically," and are they inextricably interwoven with the practices of music throughout history, and so, "this is just the way it turned out?"

Where do we draw the line between accepting things as "given," or looking at things in more detail, which seems to be frustrating for many academic thinkers?


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## BabyGiraffe

"What can this method of generating an octave-division otherwise be described as? "

I'm pretty sure that we can find some academically correct definition in professional music theory journals or sites like Xenwiki (in the mathematical section - they use abstract algebra for their definitions, while some guys like D.Lewin or G.Mazzola would probably use category theoretical language).


"Are scales and octave divisions "created" and derived from principles and/or procedures, or do scales "just exist?" -

You are going into philosophical territory. For mathematicians or platonists such constructions probably "exist".

"Did scales "evolve historically," and are they inextricably interwoven with the practices of music throughout history, and so, "this is just the way it turned out?"

- I can see current popularity of 12 equal coming from mass commercial production of pianos in 19th century (I doubt piano manufacturers wanted customers to have out of tune intervals, making some keys unplayable), it is well known that meantone was the dominant musical tuning for several centuries in Western world.

Today: anyone can buy isomorphic keyboard or guitar, designed to play in 19, 22, 24, 31 (or fretless).
Anyway, 12 equal has more compositional resources than 12 notes meantone - we will need more than 12 keys to reach all enharmonics in meantone, leading to systems like 19 or 31 microtone, or in the case of some stuff by Liszt or Stravinsky use non-meantone 12 note tunings (there are various such options that will give more in-tune augmented, diminished or Messiaen scales, but regular diatonic will be worse than in meantone). Or septimal scales for blues and barbershop music. 

What we will never get with only 12 tones - arabic music - the closest thing to is 12 notes out of 17 equal and only one of the permutations of this scale is close enough to oriental intervals to sound "exotic".

The minimum size of closed tonal system for oriental music with neutral thirds is 17 tones. 17 equal, 17 notes out of 22 (has neutral seconds, but not neutral thirds, so only some arabic/turkish scales work), 17 out of 24/26/27/29/31 equal are good enough.


"Where do we draw the line between accepting things as "given," or looking at things in more detail?" - 

Understanding how sound works may give us new compositional resources, so we better not accept elementary theory as given.

Timbres with distorted harmonic series like metal bars, based on frequency or phase modulation, or waveshaping synthesis techniques, or detuned partials (additive synthesis) can sound better in unusual music systems than in more harmonic equal temperaments like 12, 19 etc. (Still, very inharmonic timbres will never work well (or at all, depending on the sound) for complex chords and counterpoint and this is because of our cognitive system and hearing apparatus, not because of some ideological limitations).

Gamelan and certain African musical tribal traditions are good example for original musical cultures, not based on harmonic series (Arabic music can be explained as using higher harmonics like 11th and 13th), but on the timbres of their gongs and similar instruments. 
Chopi people in Africa use the anti-diatonic scale where major and minor are interchanged and we have augmented triad, not diminished on "B" (the scale can be generated by a flat "fifth" - 675 cents) , so any work composed with diatonic scales has a dual equivalent there. Of course, it will sound terrible and out of tune on typical Western instruments with harmonic overtones; we have to use some instruments that work in this tuning - like Chopi marimbas, xylophones (I'm not too sure about what other instruments they use, wikipedia mentions musical bows and pipes/flutes).


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## millionrainbows

BabyGiraffe said:


> "Are scales and octave divisions "created" and derived from principles and/or procedures, or do scales "just exist?" -
> 
> You are going into philosophical territory. For mathematicians or platonists such constructions probably "exist".


Howard Hanson's book "Harmonic Materials of Modern Music" describes ways of creating scales by using "interval projection."

How do scales arise? are they made, or do they form out of relationships of notes?


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## BabyGiraffe

How do scales arise? -



Compositional resources are abstractions and this has nothing to do with music theory, it's a question from the field of philosophy of music.


are they made, or do they form out of relationships of notes? - 

It doesn't matter; they may exist in some kind in some non-physical space or they could be a construction, invented by human mind.
You can call any pitch collection used as a scale a "scale" - even these without any structural relationships between the tones (as long as it is used in a musical piece). Wikipedia gives us this definition: " sequence of ordered musical notes". Or: "In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Some scales contain different pitches when ascending than when descending, for example, the melodic minor scale. " - It is interesting that last sentence better describes Indian ragas or Medieval modes - these are something like proto-melodic patterns, not something like our current scales or pitch class sets, which are more abstract. ( I wonder who is the genius behind the ascending/descending minor - certainly it's not hard to find counterexamples, even pieces where Dorian or Harmonic minor are used along natural and melodic minor in all kinds of motion.
The modern, vague concept of minor tonality (with "flexible" notes or "alterations", and different ascending and descending patterns) is almost as bad in describing actual music as the theory behind "blues" scale in rock music.)


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## Phil loves classical

millionrainbows said:


> Howard Hanson's book "Harmonic Materials of Modern Music" describes ways of creating scales by using "interval projection."
> 
> How do scales arise? are they made, or do they form out of relationships of notes?


The basis of musical foundations is all in the math. Scales were constructed from the symmetry of those relationships. (tetrachords). The human brain is like a wave analyser, it can detect consonance and dissonance. So it is possible for us to perceive a posteriori without actually understanding the math.

Here is a good overview

https://golem.ph.utexas.edu/category/2010/02/a_look_at_the_mathematical_ori.html

It's because of this, that tonality is natural, and even if we sent a race of humans to another planet that never heard a note of music, over time (a few thousand years or so) they would still develop the same scales  The only thing that is arbitrary or may be different is another mode could be more popular than major/minor


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## millionrainbows

BabyGiraffe said:


> How do scales arise? -
> 
> Compositional resources are abstractions and this has nothing to do with music theory, it's a question from the field of philosophy of music.


You mean all those scales you talk about, like the Pythagoran scale with its 9/8 seconds, are not based on relationships?

And you're always talking about "just" intonation. Any "just" scale has to have a starting point, from which interval relationships proliferate.

Take just one interval, the 3:2 fifth. When placed in an octave, we immediately get its reciprocal, the just fourth 3:4.


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