# Just representation of dominant 7th chord



## BrahmsWasAGreatMelodist

So here's the thing. A lot of people talk about how the 7-limit Just Intonation (JI) representation of a dominant (major-minor) 7th chord is
1/1
5/4
3/2
7/4

Now here's what I don't get. The ratio 7:4 is commonly regarded as (and sounds like to my ears) a relatively "consonant" interval, with no tendency towards stepwise downwards resolution. If we're trying to produce a dominant chord, then aren't we supposed to inherit from the diatonic scale of the key we're resolving to? I.e. in C major, the dominant 7th chord is:

G (3/2)
B (15 / 8)
D (9 / 4)
F (8 / 3)

Normalizing to the root of the chord yields ratios of:

1/1
5/4
3/2
16 / 9 (the Pythagorean minor seventh, a bit sharper than 7:4)

for the dominant 7th chord. The 5-limit JI and 7-limit JI 7th chords should be equivalent; there is no need for a 7-limit interval.


Put another way: I think the diatonic scale is clearly (at most) a 5-limit scale, and so all diatonic chords (diatonic in any key) are 5-limit chords.

That said, 7:4 obviously does have its uses. The 7th in a blues scale is obviously a 7:4. Barbershop quartets often sing dominant chords with (close to) a 7:4. I just think composers like Mozart rarely, if ever, employed 7-limit harmonies or melodies.


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## BrahmsWasAGreatMelodist

For a more concrete example, here's Beethoven's Moonlight Sonata Mvt. 3 with the dominant chords as 4:5:6:7. Sounds off IMO, I don't think Beethoven would have internalized this as the sound he was looking for.


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## BrahmsWasAGreatMelodist

If anyone else has a justification for the dominant 7th in common practice era music ever being any other ratio besides 16/9 (9:5, 7:4, etc) I'd love to hear it.


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## Bwv 1080

7/4 is the partial from the harmonic series, 16/9 is just intonation and Pythagorean

meantone and ET are less than a half-cent different on a minor 7th, so essentially the same

So ET minor 7ths are essentially all of CP


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## BrahmsWasAGreatMelodist

Of course the 7ths are literally meantone, when sounded. I am talking about their just representations. For tonal music, this isn't necessarily a matter of "closeness" so much as deducing the ratios from the rational functions of the chords. Does 7-limit harmony ever have any justification in a tonal context? Seeing as the diatonic and chromatic scales are both clearly 5-limit scales in JI, I don't think so. But how about in extended harmony? Maybe someone like Wagner or Debussy or Scriabin wrote chords whose just approximation is (arguably) 7-limit, but I'm not sure. I'd like to know. I've actually read somewhere that the Wagner's harmony in _Parsifal_, while seemingly defying standards of functional harmony, becomes "functional" if you allow for higher limit intervals. But there was no follow-up to this claim, so who knows its origin.

To be clear, I am operating under the assumption that just intonation forms the theoretical basis for tonality and functional harmony, even if that isn't the way the music ends up sounding. A strong assumption, but I'd ask you to suspend disbelief for a moment.


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## BabyGiraffe

"Diatonic scale" is better thought as 3-limit scale in historical context - stack of fifths or fourths.
There are many possible possible detemperaments to 5-limit scale, the most famous are just major and just natural minor - two different scales that can be constructed by stacking 3 major or 3 minor triads. In meantone temperament there is no distinction between them.
We can also detemper the 3-limit scale to 2.3.7 subgroup and the consonant chord will be a root+fifth+septimal seventh/septimal sixth - something like inverse of common practice. 
12 equal supports both - the first temperament is with better thirds and flattens the perfect fifth, while the second sharpens it - so in 12 equal the 3/2 is almost perfectly in tune while the 5-limit thirds and septimal seventh and sixth are out of tune.

" Does 7-limit harmony ever have any justification in a tonal context? " - not in pre-romantic music unless as dissonances. Still, in a Renaissance or Classical era with gamut of only 12 pitches on the keyboard you only had access to 8 major and 8 minor triads and 6 copies of the diatonic scale, so you sometimes had to unintentionally use approximations to septimal intervals unless your music was mostly trivial.
Added 9th chords are even more complex form of septimal harmony and the most compact equal divisions that support them in tune are 41, 46, 53 - the last one is almost perfect as Pythagorean system.

I personally think that septimal harmony implies some kind of chromatic system (most likely based on "superdiatonic" scales with anything from 9 up to 19 pitches per octave) with its own functional harmony that has nothing to do with heptatonic scale degrees or 5-limit sequences based on major and minor thirds. 
It will be based on sequences of septimal seventh chords. Or sequences of septimal added 9th chords.

In common practice the seventh is both 16/9 or 9/5. If it is also 7/4, it is called "dominant" temperament and this one is somewhat in tune only in 12 and 29 equal, so it is not a good septimal temperament. Maybe it is useful to think about it only 12 equal since 29 equal has better approximations to harmonics. 
domnant 29 equal 12 equal 29 equal with better approximation to septimal tetrad 
0 cents 0 cents 0
413.793 cents 400. cents 372.413793
703.448 cents 700 cents 703.448
993.103 cents 1000 cents 951.724138


Music that makes a lot of usage of augmented triads or augmented hexatonic/nonatonic scales can be retuned to 27 equal which is a equal division of the octave that supports septimal harmony.


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## BrahmsWasAGreatMelodist

Hi BG,

Thanks for responding, I knew this would be up your alley. Can I ask you to elaborate on a few points please?



> "Diatonic scale" is better thought as 3-limit scale in historical context - stack of fifths or fourths.
> There are many possible possible detemperaments to 5-limit scale, the most famous are just major and just natural minor - two different scales that can be constructed by stacking 3 major or 3 minor triads


The way I understand it, Ptolemy's Intense Diatonic Scale was widely understood to represent the "diatonic" (major) scale by the 1500s (Zarlino, e.g.). However as I understand it, many medieval theorists thought in terms of Pythagorean tuning. *When did this transition - from a 3-limit diatonic scale to 5-limit diatonic scale - occur?*



> . " Does 7-limit harmony ever have any justification in a tonal context? " - not in pre-romantic music unless as dissonances. Still, in a Renaissance or Classical era with gamut of only 12 pitches on the keyboard you only had access to 8 major and 8 minor triads and 6 copies of the diatonic scale, so you sometimes had to unintentionally use approximations to septimal intervals unless your music was mostly trivial.


Could you elaborate on this a bit? Is there an example of 7-limit harmony, or an _intentional_ approximation thereof, in pre-Romantic music used as a dissonance? Or in Romantic music, used as either a consonance or a dissonance? I'm interested in the earliest theoretical and practical applications of septimal harmony, particularly in CPT. I've heard some Renaissance composers/theorists considered even 11-limit to be consonant; would you happen to know anything about this?



> .In common practice the seventh is both 16/9 or 9/5.


Could you please give an example of a context in CPT where 9:5 is more appropriate than 16:9? I'm sure there are many, just can't think of one off the top of my head.


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## BrahmsWasAGreatMelodist

> .I personally think that septimal harmony implies some kind of chromatic system (most likely based on "superdiatonic" scales with anything from 9 up to 19 pitches per octave) with its own functional harmony that has nothing to do with heptatonic scale degrees or 5-limit sequences based on major and minor thirds.
> It will be based on sequences of septimal seventh chords. Or sequences of septimal added 9th chords.


I'm also interested in hearing more of your thoughts about this, if you have any. Fascinating idea!


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## BabyGiraffe

"When did this transition - from a 3-limit diatonic scale to 5-limit diatonic scale - occur?" 
My guess is - during 1300s in England and in the next two centuries - the rest of Western Europe.

"Is there an example of 7-limit harmony, or an intentional approximation thereof, in pre-Romantic music used as a dissonance?"
Well, there is no way to know what was the "mode" of the 12 keys (where the sharp wolf fifth is placed) used by Mozart or anyone. I think at some point it was G#-Eb as standard (but I am not sure). 
Anything chromatic will unintentionally use septimal intervals. Someone had a post here on Chopin's letters and it is interesting that he used meantone tuning, so some of his music was probably somewhat septimal.

" I'm interested in the earliest theoretical and practical applications of septimal harmony, particularly in CPT. I've heard some Renaissance composers/theorists considered even 11-limit to be consonant; would you happen to know anything about this?"
You will find some chords from any limit to be consonant - it will mainly depend on the voicing and timbre. I think hammers are placed in a such way on the pianos to reduce the volume of 7th harmonic, because it is out of tune with 12 equal.
For example: try 2:5:8:11 chord = C-E - C - F^ = the best chords seem to based on sequence with skipping of harmonics
0: 1/1 0.000000 unison, perfect prime
1: 5/2 1586.313714 major 10th
2: 4/1 2400.000000 2 octaves
3: 11/2 2951.317942 undecimal semi-augmented fourth +2 octaves

Or added eleventh chord=> C - E - G - Bb (7/4) - D - F^

0: 1/1 0.000000 unison, perfect prime
1: 5/4 386.313714 major third
2: 3/2 701.955001 perfect fifth
3: 7/4 968.825906 harmonic seventh
4: 9/4 1403.910002 major ninth
5: 11/4 1751.317942 undecimal semi-augmented fourth +1 octave

D to F^ is a neutral third, so maybe this chord has some kind of future in oriental jazz?

"Could you please give an example of a context in CPT where 9:5 is more appropriate than 16:9? " 
If you place 16/9 in minor mode, you get 40/27 instead of 3/2 on the mediant degree => 6/5 x 3/2 = 9/5.In standard theory it is both 16/9 and 9/5 and this equivalence is not reflected in notation system unlike enharmonic equivalences.

"I'm also interested in hearing more of your thoughts about this, if you have any. Fascinating idea! "

Well, I can recommend you reading:
https://plato.stanford.edu/entries/symmetry-breaking/ and
https://www.oxfordhandbooks.com/vie...9935321.001.0001/oxfordhb-9780199935321-e-003

Equal divisions are examples of closed scales with perfect symmetry. We are discarding most of the symmetries in regular temperaments of a given rank.
If we break all the symmetries, we get something like the structures that are equal on linear scale (instead of logarithmic scale). They are not generated by some kind of specific interval - all the intervals are generators.
For example: harmonics 12 -> 24 give the 12 equal linear scale: 12:13:14:15:16:17:18:19:20:21:22:23:24 
Intervals get progressively smaller - 13/12 is 138 cents, 23/12 to octave is 74 cents. 
0: 1/1 0.000000 unison, perfect prime
1: 13/12 138.572661 tridecimal 2/3-tone
2: 7/6 266.870906 septimal minor third
3: 5/4 386.313714 major third
4: 4/3 498.044999 perfect fourth
5: 17/12 603.000409 2nd septendecimal tritone
6: 3/2 701.955001 perfect fifth
7: 19/12 795.558015 undevicesimal minor sixth
8: 5/3 884.358713 major sixth, BP sixth
9: 7/4 968.825906 harmonic seventh
10: 11/6 1049.362941 undecimal neutral seventh, 21/4-tone
11: 23/12 1126.319346 vicesimotertial major seventh
12: 2/1 1200.000000 octave

In classical diatonic music only the meantone symmetry is not broken, so we mainly use such sequences. And we are also analyzing everything like it is based on 7 notes scale.
We can take another symmetry (even from 7 equal) from some equal division and create new functional theory.
Let's take a scale that is generated by neutral third in 24 equal: here are all the modes ( I hope the formatting won't be horrible) 
1 2 3 4 5 6 7 
1/1 : 150.0 350.0 500.0 700.0 850.0 1050.0 2/1 
150.0 : 200.0 350.0 550.0 700.0 900.0 1050.0 1200.0
350.0 : 150.0 350.0 500.0 700.0 850.0 1000.0 1200.0
500.0 : 200.0 350.0 550.0 700.0 850.0 1050.0 1200.0
700.0 : 150.0 350.0 500.0 650.0 850.0 1000.0 1200.0
850.0 : 200.0 350.0 500.0 700.0 850.0 1050.0 1200.0
1050.0: 150.0 300.0 500.0 650.0 850.0 1000.0 1200.0
2/1

We can see that we have a neutral triad (0 -350 - 700 cents) on scale degree I, II, III, IV, VI - maybe these will be the main chords in some kind dissonant style? The meantone symmetry is broken - we have two wolf fifths but we have another type of symmetry - neutral thirds instead of major or minor, the 300 cents neutral third is some kind of "false" third like the tritone in Locrian mode is a false fifth.

I am thinking that septimal music is chromatic, because if we want more than a 1 tetrads or pentads in a scale or chord progression, we end up with something similar.
There is also a mathematical way to generate them. For example we want a scale that has major or minor thirds and it is also in meantone temperament. There is only one such scale and it is 7 equal. In other equal divisions major and minor thirds are not found on the same scale degree or it is not meantone. We break the major/minor symmetry and end up with diatonic scale in meantone temperament and its related scales (like melodic minor etc).
If we want a scale where septimal intervals 12/7 and 7/4 are on the same scale degree and it is in septimal meantone, there is only one such scale - 19 equal. We can use 19 unequal meantone as septimal superdiatonic. 
12 equal as septimal superdiatonic is = septimal meantone + 7/6 and 6/5 on the same degree.

Septimal meantone is the tuning where 126/125 x 225/224 = 81/80, we get these simple equivalences in it:
septimal tritone (7/5) becomes augmented fourth, diminished fourth becomes a septimal major third (9/7), augmented second becomes 7/6 (septimal minor third), diminished third becomes 8/7 (septimal whole tone).

Maybe 12 tone music can become a thing one day based on a septimal scale, but I doubt it will be in meantone temperament - it offers no advantage over just intonation with only 12 notes per octave in terms of number of septimal tetrads (only 4) and it has worse tuning than just intonation.
There are regular temperaments in 22 and 26, and 27 equal that are better than that in number of tetrads with only 12 notes. 
If we are after pentads, it seems that 19 notes scales are also good choice - there are two good ones in 41 and 46 equal.
I would not recommend going over 22 notes per octave in unequal tuning (especially on a guitar, it becomes kind of unplayable).


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## BabyGiraffe

So, if we want to extend common practice theory by modulating by a fifth or fourth, but this time using septimal tetrads instead of 5-limit triads, we run into these commas = I have added the error of optimal tuning that tempers them.

rank-3 temperaments (there is no single optimal scale in this temperament - often times they are enantiomorphs ((a familiar example would be harmonic minor and harmonic major)), - construction is also more complex, needs at least two different generating intervals)
error in cents and comma
1.348 = 225/224 =
3.847 = 81/80 = This is the familiar 5-limit comma, I guess the rank-3 scale is a modification of the scales, generated by a chain of fifths - for example: dorian or mixolydian mode with 7/4 etc.
5.211 = 64/63 
5.733 = 525/512
6.675 = 50/49
7.084 = 49/48
10.624= 256/245 
13.097= 36/35
14.745= 135/128 
15.048= 28/27

some rank 2 temperaments
3.879 = Septimal Meantone (81/80, 225/224) Good tuning, not efficient in terms of scale size (at least 19 pitches to get something over just tuning)
5.944 = (525/512, 81/80) Flattone = Flatter meantone = 26 and 45 equal = the difference between this and septimal meantone is that 7/4 is augmented sixth in septimal meantone and kind of diminished seventh in flattone
7.222 = (225/224, 64/63, 50/49) = Sharp P5, flat M3, sharp 7/4 = 22 equal = the child of 10 and 12 equal
7.230 = (225/224, 525/512, 49/48) = 8/7 and 7/6 (7/4 and 12/7) are tempered together; simple tunings are 19 and 29 equal
8.809 = (81/80, 50/49) = Two parallel chains of meantone fifths = 26 and 38 equal
8.983 = (525/512, 50/49) = 10 + 16 equal = 26 equal seems optimal
13.237 = (64/63, 36/35, 81/80) Dominant temperament, 12 equal and another interpretation of 29 equal
13.288 = (36/35, 225/224) 7/6 and minor third are tempered = 12 equal and 21 equal


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## pianozach

BrahmsWasAGreatMelodist said:


> For a more concrete example, here's Beethoven's Moonlight Sonata Mvt. 3 with the dominant chords as 4:5:6:7. Sounds off IMO, I don't think Beethoven would have internalized this as the sound he was looking for.


This becomes painful at times.


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