Just Chord Tunings &dflat;

Major Triad
Minor Triad
Augmented Triad
Diminished Triad
Major 7th Chord
Minor 7th Chord
Major-Diminished 7th Chord
Minor-Augmented 6th Chord
Dominant 7th Chord
German 6th Chord
Swiss 6th Chord
Half-Diminished 7th Chord
Tristan 6th Chord
Eulenspiegel 6th Chord
Diminished 7th Chord
Minor-Major 7th Chord
Augmented-Major 7th Chord
Major 6th Chord
Minor 6th Chord
French 6th Chord
Cuckoo Chord
Augmented-minor 7th Chord
Major 9th Chord
Minor 9th Chord
Dominant 9th Chord
Flatted 9th Chord
Undecimal 9th Chord
Major 11th Chord
Minor 11th Chord
Dominant 11th Chord
Harmonic 11th Chord
Split 11th Chord
Diminished 11th Chord
Major 13th Chord
Dominant 13th Chord
Harmonic 13th Chord
Flatted 13th Chord
5th Inversion 13th Chord

The following is an overview of various chord tunings, with tables giving their intervals above a root of C. The table shows the name of each note, its harmonic value, its frequency ratio with the root, and the MIDI pitch bend adjustment. To hear a chord, hold the mouse over the colored button near the chord. Red is for equal temperament tunings. Yellow is for Pythagorean (3-limit) tunings. Green is for 5-limit tunings. Blue is for septimal (7-limit) tunings and beyond. In some chords, the choice between the just or 5-Limit tunings of a chord is open to debate depending on the musical context. The official “just” tuning is given first, taking into account low harmonic values (consonance) and its musical applicability. In the most consonant tunings, if you listen carefully the fundamental can be heard far below the chord even though it isn’t really being played. You will also note that the fundamental is sometimes on a different note than the root (e.g. the minor triad), as indicated when the root falls on a harmonic that is not a power of 2, such as in the minor triad.

Major Triad

a.k.a. harmonic triad.

This is the simplest chord, containing the first three notes in the harmonic series not counting octaves. It is rooted on the the fundamental, i.e. of the same pitch class as the fundamental, which is itself two octaves down. Notice how out of tune the Pythagorean (3-limit) tuning sounds, due to the sharp third alone.

The septimal tuning, also known as “supermajor triad", rooted a harmonic 7th above the fundamental, is really the upper three notes of a dominant 11th chord, although it sounds out of tune by itself. It is named after its supermajor 3rd interval 9/7.

The tredecimal median tuning, rooted a 9th above the fundamental, is really the upper three notes of a harmonic 13th chord. Note how by itself it sounds more like an out-of-tune diminished triad.

The Wolf tuning is the one heard on the 7th degree of the just-minor scale, named after the sound of its narrow 40/27 fifth.

If you look at the “just” tuning, you can see that the notes of a C major triad are C, E and G. It is first shown appearing at harmonics 4, 5 and 6. The intervals above the root are 1/1 (the root itself), 5/4 (the major third) and 3/2 (the perfect fifth). The values 0, -561 and 80 are the pitchbend values you would add to the notes in a MIDI sequence to “just tune&rdquo ;the triad (4096 is an equal temperament halfstep).

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 4 1/1 0
E 5 5/4 -561
G 6 3/2 80
Septimal
Pitch Harmonic Interval Pitchbend
C 14 1/1 0
E 18 9/7 1437
G 21 3/2 80
Tredecimal Median
Pitch Harmonic Interval Pitchbend
C 9 1/1 0
E 11 11/9 -2154
G 13 13/9 -2596
Pythagorean
Pitch Harmonic Interval Pitchbend
C 64 1/1 0
E 81 81/64 320
G 96 3/2 80
Wolf
Pitch Harmonic Interval Pitchbend
C 108 1/1 0
E 135 5/4 320
G 160 40/27 -801

Minor Triad

a.k.a. sub-harmonic triad.

This chord, rooted a major 3rd above the fundamental, is really the upper three notes of a major 7th chord. It can also be thought of as the sub-harmonic dual of the major triad.

The septimal, also known as a “subminor” triad, rooted a perfect 5th above the fundamental, is really the upper three notes of a dominant 9th chord. It is named after its subminor 3rd interval 7/6. This is also the chord to which a major 6th is added to create a just minor 6th chord, which is really a half-diminished 7th chord in first inversion.

The Pythagorean tuning, based on its harmonics, is really rooted rooted a Pythagorean major 6th above the fundamental, i.e. its 3rd is on the fundamental.

The overtone tuning is so-called because all its notes are intervals above the fundamental pitch class. It is the only minor triad tuning rooted on the fundamental.

The wolf tuning, included only for illustration, is the tuning of the ii chord in the just major scale, which is named after the howling sound of its narrow 40/27 fifth.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭   1.1892 0
G   1.4983 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 10 1/1 0
E♭ 12 6/5 641
G 15 3/2 80
Septimal
Pitch Harmonic Interval Pitchbend
C 6 1/1 0
E♭ 7 7/6 -1357
G 9 3/2 80
"Pythagorean”
Pitch Harmonic Interval Pitchbend
C 54 1/1 0
E♭ 64 32/27 -240
G 81 3/2 80
"Overtone”
Pitch Harmonic Interval Pitchbend
C 16 1/1 0
E♭ 19 19/16 -102
G 24 3/2 80
"Wolf”
Pitch Harmonic Interval Pitchbend
C 27 1/1 0
E♭ 32 32/27 -240
G 40 40/27 -801

Augmented Triad

This chord, rooted a harmonic 7th above the fundamental, is actually the upper three notes of a harmonic 11th chord. None of the intervals appear in the just major scale, since they have a limit higher than 5. The 9/7 interval is called a septimal major 3rd, also known as a supermajor 3rd. The upper third (11/9) when heard by itself, actually sounds like it’s midway between major and minor, and is thus often referred to as an undecimal median 3rd. The 11/7 interval is the undecimal augmented 5th. The 5-limit tuning consists of two just major thirds, based on the III+ triad in the harmonic minor scale. The resulting outer interval (25/16) is the just augmented 5th (25/16). It is rooted on the fundamental. Since the 11th harmonic is introduced here, this is the first of the chords that pose a dilemma in tuning choices. The 5-limit tuning is harmonically unstable due to the high harmonic values, whereas the undecimal tuning is melodically unstable due to the large pitch bends present.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G♯   1.5874 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 7 1/1 0
E 9 9/7 1437
G♯ 11 11/7 -717
5-Limit
Pitch Harmonic Interval Pitchbend
C 16 1/1 0
E 20 5/4 -561
G♯ 25 25/16 -1121
Pythagorean
Pitch Harmonic Interval Pitchbend
C 4096 1/1 0
E 5184 81/64 320
G♯ 6561 6561/4096 641

Diminished Triad

This chord, rooted a major 3rd above the fundamental, is really the upper 3 notes of a dominant 7th chord. While the lower interval of the chord is a regular 6/5 minor 3rd, the upper interval is actually a subminor third, also known as a septimal minor third due to its ratio of 7/6. These two thirds are reversed in the sub-harmonic tuning, whose actual fundamental would now be on the Bb near the top of the piano range! The 5-Limit tuning is the one heard on the 7th degree of the just major scale, and is actually out of tune due to the dissonant diminished 5th interval (64/45). The stacked version is built from two just minor thirds, and due to the lower harmonic values, is actually more in tune than the 5-Limit version.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭   1.1892 0
G♭   1.4142 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 5 1/1 0
E♭ 6 6/5 641
G♭ 7 7/5 -716
Sub-harmonic
Pitch Harmonic Interval Pitchbend
C 1/7 1/1 0
E♭ 1/6 7/6 -1357
G♭ 1/5 7/5 -716
Stacked
Pitch Harmonic Interval Pitchbend
C 25 1/1 0
E♭ 30 6/5 641
G♭ 36 36/25 1281
5-Limit
Pitch Harmonic Interval Pitchbend
C 45 1/1 0
E♭ 54 6/5 641
G♭ 64 64/45 400
Pythagorean
Pitch Harmonic Interval Pitchbend
C 729 1/1 0
E♭ 864 32/27 -240
G♭ 1024 1024/729 -480

Major 7th Chord

Add the just major 7th to a major triad and you get this chord. It is the chord you would get by sharping the 7th of the dominant 7th chord (4:5:6:7) to the 15th harmonic (which, as you see, puts it an octave higher in the harmonic series). It is rooted on the fundamental. This chord could also be used as a “5-Limit” minor 9th chord with the root missing. The septimal tuning, also known as a “supermajor” 7th chord, rooted a harmonic 7th above the fundamental, is really the upper 4 notes of a dominant 13th chord, although it sounds out of tune by itself.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
B   1.8877 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 8 1/1 0
E 10 5/4 -561
G 12 6/4 80
B 15 15/8 -481
Septimal
Pitch Harmonic Interval Pitchbend
C 14 1/1 0
E 18 9/7 1437
G 21 3/2 80
B 27 27/14 1517
Pythagorean
Pitch Harmonic Interval Pitchbend
C 128 1/1 0
E 162 81/64 320
G 192 3/2 80
B 243 243/128 400

Minor 7th Chord

Add the just minor 7th to a minor triad and you get this chord. Rooted a major 3rd above the fundamental, it is really the upper four notes of a major 9th chord. Musically speaking, this chord could be thought of as having three roots. It can be used: as a major 9th chord in which the root is missing, that root being on the fundamental; as a true minor 7th chord, rooted a major 3rd above the fundamental; or as a major 6th, rooted a perfect 5th above the fundamental. The septimal tuning, also known as a “subminor” 7th chord, rooted a perfect 5th above the fundamental, is really the upper 4 notes of a dominant 11th chord. This tuning is useful for resolving to the dominant a fifth down (half-cadence), allowing the root and 3rd to become the 5th and 7th of the dominant without bending.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭   1.1892 0
G   1.4983 0
B♭   1.7818 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 10 1/1 0
E♭ 12 6/5 641
G 15 3/2 80
B♭ 18 9/5 721
Septimal
Pitch Harmonic Interval Pitchbend
C 12 1/1 0
E♭ 14 7/6 -1357
G 18 3/2 80
B♭ 21 7/4 -1277
Pythagorean
Pitch Harmonic Interval Pitchbend
C 54 1/1 0
E♭ 64 32/27 -240
G 81 3/2 80
B♭ 96 16/9 -160

Major-Diminished 7th Chord

This enharmonic of the minor 7th chord can be found on the 7th degree of the Swiss scale, where it sounds like a major 6th chord. It is made of the minor 6th scale degree, the major 7th degree, the raised 2nd degree and the raised 4th degree. It is akin to the minor-diminished 7th (Tristan 6th ) but with a major triad in place of the minor one. It is really a minor-augmented 6th chord standing on its head. It isn’t very powerful in its tonal function, but is included because it is a 7th chord in the Swiss scale.

5-Limit
Pitch Harmonic Interval Pitchbend
C 256 1/1 0
D♯ 300 75/64 -1041
Fx 375 375/256 -1602
A♯ 450 225/128 -961
Pythagorean
Pitch Harmonic Interval Pitchbend
C 1048576 1/1 0
D♯ 1259712 19683/16384 721
Fx; 1594323 1594323/1048576 1041
A♯ 1889568 59049/32768 801

Minor-Augmented 6th Chord

This enharmonic of the minor 7th chord can be found on the 2nd degree of the Eulenspiegel scale, where it sounds like a minor 7th chord in 3rd inversion. It is made of the 4th scale degree, the minor 6th scale degree, the tonic and the raised 2nd degree. It is akin to the German 6th chord but with a minor triad in place of the major one. It is really a major-diminished 7th chord standing on its head. It isn’t very powerful in its tonal function, but is included because it is a 7th chord in the Eulenspiegel scale. An barbershop example of it can be heard in the last “West Virginia” in the tag of “Country Roads", where the baritone sharps the 2nd degree.

5-Limit
Pitch Harmonic Interval Pitchbend
C 640 1/1 0
E♭ 768 6/5 641
G 960 3/2 80
A♯ 1125 225/128 -961
Pythagorean
Pitch Harmonic Interval Pitchbend
C 884736 1/1 0
E♭ 1048576 32/27 -240
G 1327104 3/2 80
A♯ 1594323 59049/32768 801

Dominant 7th Chord

a.k.a. major-minor 7th, harmonic 7th, barbershop 7th.

The harmonic tuning, hence the Harmonic 7th, is the simplest 4-part chord, containing the first four notes in the harmonic series. It is the septimal extension of a major triad. It is rooted on the fundamental.

The 5-Limit tuning is based on its position on the dominant note of the just major scale. It is rooted on the 9th harmonic. This regular tuning of the chord has a strong desire to resolve to the tonic. The septimal comma is applied to this tuning to produce the harmonic version.

What I called the involute dominant 7th chord is a 17-limit tuning rooted on the 11th harmonic, making the chord a tritone removed from the just dominant 7th chord on the same fundamental. This tuning is really only useful as part of a larger chord such as the diminished 11th chord. In actual practice, dominant 7th chords used in these musical contexts should still use the harmonic tuning for better sound.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
B♭   1.7818 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 4 1/1 0
E 5 5/4 -561
G 6 3/2 80
B♭ 7 7/4 -1277
5-Limit
Pitch Harmonic Interval Pitchbend
C 36 1/1 0
E 45 5/4 -561
G 54 3/2 80
B♭ 64 16/9 -160
Involute
Pitch Harmonic Interval Pitchbend
C 11 1/1 0
E 14 14/11 717
G 17 17/11 2197
B♭ 20 20/11 1433
Pythagorean
Pitch Harmonic Interval Pitchbend
C 576 1/1 0
E 729 81/64 320
G 864 3/2 80
B♭ 1024 16/9 -160

German 6th Chord

This enharmonic of the dominant 7th chord can be found on in its true root position on the 4th degree of the Hungarian-minor scale, where it sounds like a a dominant 7th chord in 3rd inversion. It consists of the minor 6th scale degree, the tonic, the minor 3rd degree and the raised 4th degree. This chord is distinctive in the way it resolves. What sounds like its 7th resolves upward, revealing its nature as an augmented 6th. The chord is fully resolved by moving its major triad down a half-step and its augmented 6th up a half-step to a double root. It’s name, whose origin is uncertain, may have been coined by someone who thinks it has a very German sound, probably because it was used by the likes of Beethoven and Schubert. It is rooted on the fundamental.

5-Limit
Pitch Harmonic Interval Pitchbend
C 128 1/1 0
E 160 5/4 -561
G 192 3/2 80
A♯ 225 225/128 -961
Pythagorean
Pitch Harmonic Interval Pitchbend
C 32768 1/1 0
E 41472 81/64 320
G 49152 3/2 80
A♯ 59049 59049/32768 801

Swiss 6th Chord

This enharmonic of the dominant 7th chord is also called the English 6th, or the Alsatian 6th. It is made of the minor 6th scale degree, the tonic, the raised 2nd degree and the raised 4th degree. I created a “Swiss scale” as a container for this chord, located on the 2nd degree, where it sounds like a dominant 7th chord in 2nd inversion. It has also been called a misspelled German 6th chord, and is mainly used as an alternative for a German 6th when resolving to a major tonic. It differs from the German in that its 5th is spelled as a doubly-augmented 4th. The chord is fully resolved by moving its lower major 3rd down a half-step and its upper two notes up a half-step, forming a minor triad in 2nd inversion with a doubled 3rd. This resolution is actually used in some barbershop passages. Among songs I can remember are I Can’t Give You Anything but Love on “baby", Pretty Baby right on the title words, and Red Head on “prettiest gal". 1927 Kansas City arranged by Brian Beck uses these chords in the tag on “love won’t fade".

5-Limit
Pitch Harmonic Interval Pitchbend
C ? 1/1 0
E ? 5/4 -561
Fx ? 375/256 -1602
A♯ ? 225/128 -961
Pythagorean
Pitch Harmonic Interval Pitchbend
C 1048576 1/1 0
E 1327104 81/64 320
Fx 1594323 1594323/1048576 1041
A♯ 1889568 59049/32768 801

Half-Diminished 7th Chord

a.k.a. diminished-minor 7th, sub-harmonic 7th.

Musically speaking, this chord can be thought of as having three roots. It can be used: 1) as a dominant 9th chord with the root missing, which puts the root on the fundamental; 2) as a minor 6th chord, which puts the root at the bottom of a perfect fifth; or 3) as a true half-diminished 7th chord, mainly by resolving it down a fifth. The third option could, however, be heard as resolving a minor 6th with an ascending 2nd.

The harmonic tuning, rooted on the 5th harmonic, is really the upper four notes of a dominant 9th chord.

The septendecimal tuning is made by flatting the fifth of the sub-minor 7th chord to a septendecimal diminished 5th (17/12). It is rooted on the 3rd harmonic. This tuning is useful for resolving to the dominant a 5th down (Spanish half-cadence), allowing the root and 3rd to become the 5th and 7th of the dominant without bending, since both chords have a common fundamental.

The subharmonic tuning, hence the sub-harmonic 7th, is really a harmonic 7th chord “standing on its head", i.e. built from subharmonics 4, 5, 6 and 7. This means that the 7th is now on the fundamental, which is now 2 octaves up! In fact, when you listen to this chord, you should be able to hear this high tone two octaves above the top of the chord. This can easily be explained in that this tone is the lowest common harmonic of all four chord notes (the 4th, 5th, 6th and 7th harmonic of the four chord tones respectively to be exact), and as such is enforced fourfold.

The 5-Limit tuning is based on its position on the leading tone of the just major scale. It is out of tune due to the dissonant diminished 5th (64/45) and minor 7th (16/9) intervals.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭   1.1892 0
G♭   1.4142 0
B♭   1.7818 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 5 1/1 0
E♭ 6 6/5 641
G♭ 7 7/5 -716
B♭ 9 9/5 721
5-Limit
Pitch Harmonic Interval Pitchbend
C 45 1/1 0
E♭ 54 6/5 641
G♭ 64 64/45 400
B♭ 80 16/9 -160
Pythagorean
Pitch Harmonic Interval Pitchbend
C 729 1/1 0
E♭ 864 32/27 -240
G♭ 1024 1024/729 -480
B♭ 1296 16/9 -160
Septendecimal
Pitch Harmonic Interval Pitchbend
C 12 1/1 0
E♭ 14 7/6 -1357
G♭ 17 17/12 123
B♭ 21 7/4 -1277
Sub-harmonic
Pitch Harmonic Interval Pitchbend
C 1/7 1/1 0
E♭ 1/6 7/6 -1357
G♭ 1/5 7/5 -716
B♭ 1/4 7/4 -1277
Overtone
Pitch Harmonic Interval Pitchbend
C 16 1/1 0
E♭ 19 19/16 -102
G♭ 22 11/8 -1994
B♭ 28 7/4 -1277

Tristan 6th Chord

The three well-known “national” augmented 6th chords have all one thing in common: they have the minor 6th scale degree, the tonic degree, and the raised 4th degree. That much is the Italian 6th chord. The German 6th adds to this the minor 3rd degree. The French 6th adds the 2nd degree. But here we introduce what is really the “lost chord” of the augmented 6th chords that can be found within the notes of the Hungarian-minor scale. It is different from the three known +6 chords in that the tonic degree is gone, replaced by the major 7th degree.

Named after its appearance as the curious first chord in Tristan und Isolde, This enharmonic of the half-diminished 7th chord can be found in its true root position on the 7th degree of the Hungarian-minor scale as a minor-diminished 7th chord, and it sounds like a half-diminished 7th chord in 1st inversion. It is made of the minor 6th scale degree, the major 7th degree, the 2nd degree and the raised 4th degree. It is really a German 6th chord standing on its head.

It hasn’t earned itself a “country” name like three other augmented 6th chords, mainly because its eccentric tonality makes it sound unearthly, period, and its use in classical music is fairly uncommon. And so appropriately I have referred to it as the Neptune 6th chord, after its use in the final fadeout in The Planets by Holst. The four notes forming this chord in Neptune fully resolved, but there was an fifth note a 3rd below it, forming an enharmonic of the harmonic 9th chord that resolved like a hybrid of the Tristan and the Swiss 6th chords. Another name I’d like to give it is the Swedish 6th chord, since Hugo Alfven, one of the few composers to make a rather hysterical use of it in a composition, is Swedish.

Referring to the chord as a half-diminished 7th, as at least one internet source did in perusing the Hungarian Minor scale, is an error tantamount to referring to the German 6th as a dominant 7th, and was probably done in haste. People hear the German 6th chord resolve enough times to be used to what makes it a different beast than the Dominant 7th chord. But the Tristan 6th and its distinguishing resolution is so seldom heard that most musical ears haven’t caught onto what makes it different from a half-diminished 7th.

The chord is fully resolved by moving the minor triad on top of the chord up a half-step and the bass note down a half-step, making a minor triad in 2nd inversion and doubled fifth. The Original Wagnerian setting didn’t fully resolve it. However, a handful of composers such as Hugo Alfvén and Michael Haydn, did.

The septendecimal tuning is the result of adding the 17th harmonic to a just minor triad (10:12:15). It is shown inverted here to sound like a root position half-diminished. The full resolution would bring the 17/15 interval (septimal diminished 3rd) into unison at 16.

5-Limit
Pitch Harmonic Interval Pitchbend
C 128 1/1 0
D♯ 150 75/64 -1041
F♯ 180 45/32 -400
A♯ 225 225/128 -961
Pythagorean
Pitch Harmonic Interval Pitchbend
C 32768 1/1 0
D♯ 39366 19683/16384 721
F♯ 46656 729/512 480
A♯ 59049 59049/32768 801
17 20 24 30
Septendecimal
Pitch Harmonic Interval Pitchbend
C 17 1/1 0
D♯ 20 20/17 -764
F♯ 24 24/17 -123
A♯ 30 30/17 -683

Eulenspiegel 6th Chord

This enharmonic of the half-diminished 7th chord is named after its prominent use in Till Eulenspiegel’s Merry Pranks, and could even be called the Till 6th. One music scholar called it the Australian 6th. It is made of the 4th scale degree, the minor 6th, the major 7th and the raised 2nd degree. I created an “Eulenspiegel scale” as a container for this chord, located on the 7nd degree, where in its true root position it sounds like a half-diminished 7th chord in 2nd inversion. It could also be called a misspelled Tristan 6th, and is mainly used as a tritone substitute for the leading-tone Half-Diminished 7th chord when resolving to a major tonic. It differs from the Tristan in that its 5th is spelled as a doubly-augmented 4th. The chord is fully resolved by moving its lower major 3rd down a half-step and its upper two notes up a half-step, forming a major triad in 2nd inversion with a doubled 3rd. It is much more commonly used than the Tristan, and actually functions as an “altered” jazz chord.

5-Limit
Pitch Harmonic Interval Pitchbend
C 640 1/1 0
E♭ 768 6/5 641
F♯ 900 45/32 -400
A♯ 1125 225/128 -961
Pythagorean
Pitch Harmonic Interval Pitchbend
C 640 1/1 0
E♭ 768 32/27 -240
F♯ 900 729/512 480
A♯ 1125 59049/32768 801

Diminished 7th Chord

This chord, rooted a major 3rd above the fundamental, is really the upper four notes of the flatted 9th chord. Musically speaking, this chord is rather unique in that none of its notes are functional as a root, making it a kind of floating chord used as a “fly-over” for connecting between more grounded chords. The 1st septimal tuning modifies the top note so that both tritones are septimal diminished 5ths. It is rooted an augmented 5th above the fundamental, resulting in a just minor, a sub-minor and a just minor 3rd. The 2nd septimal tuning is different in that the two tritones are offset by a sub-minor 3rd instead of a just minor 3rd, giving a sub-minor, a just minor and a sub-minor third. The latter tuning is particularly useful for the I-7o7/ii-V7 sequence common in barbershop.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭   1.1892 0
G♭   1.4142 0
B♭♭   1.6818 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 10 1/1 0
E♭ 12 6/5 641
G♭ 14 7/5 -716
Bbb 17 17/10 764
5-Limit
Pitch Harmonic Interval Pitchbend
C 675 1/1 0
E♭ 810 6/5 641
G♭ 960 64/45 400
Bbb 1152 128/75 1041
Pythagorean
Pitch Harmonic Interval Pitchbend
C 19683 1/1 0
E♭ 23328 32/27 -240
G♭ 27648 1024/729 -480
Bbb 32768 32768/19683 -721
Septimal I
Pitch Harmonic Interval Pitchbend
C 25 1/1 0
E♭ 30 6/5 641
G♭ 35 7/5 -716
Bbb 42 42/25 -76
Septimal II
Pitch Harmonic Interval Pitchbend
C 30 1/1 0
E♭ 35 7/6 -1357
G♭ 42 7/5 -716
Bbb 49 49/30 -2073

Minor-Major 7th Chord

Add the undecimal median 7th to a subminor triad and you get this chord. Also known as a harmonic-minor 7th chord, rooted a perfect 5th above the fundamental, it is really the upper 4 notes of a harmonic 11th chord. The 9/6 interval, which reduces to 3/2, is the only low-limit interval. It contains the sub-minor 3rd, the super-major 3rd and and the undecimal median 3rd. The 5-limit tuning consists of minor triad with a major 7th added, based on its appearance in the just harmonic minor scale in the tonic position. But due to the high harmonic value it’s out of tune. It is rooted a major 3rd above the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭   1.1892 0
G   1.4983 0
B   1.8877 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 6 1/1 0
E♭ 7 7/6 -1357
G 9 3/2 80
B 11 11/6 -2074
5-limit
Pitch Harmonic Interval Pitchbend
C 40 1/1 0
E♭ 48 6/5 641
G 60 3/2 80
B 75 15/8 -481
Pythagorean
Pitch Harmonic Interval Pitchbend
C 3456 1/1 0
E♭ 4096 32/27 -240
G 5184 3/2 80
B 6561 243/128 400

Augmented-Major 7th Chord

Add the tredecimal major 7th to an augmented triad and you get this chord. Rooted a harmonic 7th above the fundamental, it is really the upper 4 notes of a harmonic 13th chord. The 5-limit tuning tuning consists of two major 3rds and a minor 3rd, based on its appearance on the 3rd degree of the harmonic minor scale. It is rooted on the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G♯   1.5874 0
B   1.8877 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 7 1/1 0
E 9 9/7 1437
G♯ 11 11/7 -717
B 13 13/7 -1159
5-Limit
Pitch Harmonic Interval Pitchbend
C 16 1/1 0
E 20 5/4 -561
G♯ 25 25/16 -1121
B 30 15/8 -481
Pythagorean
Pitch Harmonic Interval Pitchbend
C 4096 1/1 0
E 5184 81/64 320
G♯ 6561 6561/4096 641
B 7776 243/128 400

Major 6th Chord

Add a just major 6th to a major triad and you get this chord. Rooted a perfect 5th above the fundamental, it is actually a minor 7th chord in 1st inversion. But due to its musical ambiguity this can be thought of as a separate chord in root position. The “overtone” tuning uses a Pythagorean major 6th shown, and is rooted on the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
A   1.6818 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 12 1/1 0
E 15 5/4 -561
G 18 3/2 80
A 20 5/3 -641
Septimal
Pitch Harmonic Interval Pitchbend
C 14 1/1 0
E 18 9/7 1437
G 21 3/2 80
A 24 12/7 1357
Pythagorean
Pitch Harmonic Interval Pitchbend
C 64 1/1 0
E 81 81/64 320
G 96 3/2 80
A 108 27/16 240

Minor 6th Chord

Add a just major 6th to a septimal minor triad and you get this chord. Rooted a perfect 5th above the fundamental, it is actually a half-diminished 7th chord in 1st inversion. But due to its musical ambiguity this can be thought of as a separate chord in root position. The 5-limit tuning is shown to illustrate the out of tune sound when a minor triad is substituted for the subminor.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭   1.1892 0
G   1.4983 0
A   1.6818 0
Septimal
Pitch Harmonic Interval Pitchbend
C 6 1/1 0
Eb 7 7/6 -1357
G 9 3/2 80
A 10 5/3 -641
5-limit
Pitch Harmonic Interval Pitchbend
C 30 1/1 0
Eb 36 6/5 641
G 45 3/2 80
A 50 5/3 -641
Pythagorean
Pitch Harmonic Interval Pitchbend
C 432 1/1 0
Eb 512 32/27 -240
G 648 3/2 80
A 729 27/16 240

French 6th Chord

This is the one augmented 6th that is unique - not enharmonic to a more consonant chord. It can be found in its true root position on the 2nd degree of the Hungarian-minor scale. It is made of the minor 6th degree, the tonic, the 2nd degree and the raised 4th degree. The origin of the name is uncertain, but may have been coined by someone who associated the sound with the French, probably because it was used by Debussy or Ravel.

This harmonic tuning can be thought of as a harmonic 11th chord with the 11th in the middle. As such, it is the chord you would get by flatting the 5th of the dominant 7th chord (4:5:6:7) to the 11th harmonic. Its bass is on the fundamental. The altered tuning is the same one in 2nd inversion, and its bass is on the 11th harmonic.

The septimal tunings consist of a just major 3rd and another major 3rd a tritone higher. This tuning is basically formed by flatting the 5th of the dominant 7th chord by a septimal chroma (15/14). The two tunings are really inversions of each other, differing by whether the tritones are diminished 5ths (7/5) or augmented 4ths (10/7).

The 5-Limit and Pythagorean tunings are based on the notes found in the 5-limit and Pythagorean tunings of the Hungarian-minor scale.

Harmonic
Pitch Harmonic Interval Pitchbend
C 8 1/1 0
E 10 5/4 -561
G♭ 11 11/8 -1994
B♭ 14 7/4 -1277
Septimal (root position)
Pitch Harmonic Interval Pitchbend
C 20 1/1 0
E 25 5/4 -561
G♭ 28 7/5 -716
B♭ 35 7/4 -1277
5-Limit (root position)
Pitch Harmonic Interval Pitchbend
C 180 1/1 0
E 225 5/4 -561
G♭ 256 64/45 400
B♭ 320 16/9 -160
Pythagorean (root position)
Pitch Harmonic Interval Pitchbend
C 46656 1/1 0
E 59049 81/64 320
G♭ 65536 1024/729 -480
B♭ 82944 16/9 -160
Altered
Pitch Harmonic Interval Pitchbend
C 11 1/1 0
E 14 14/11 717
G♭ 16 16/11 1994
B♭ 20 20/11 1433
Septimal (2nd inversion)
Pitch Harmonic Interval Pitchbend
C 28 1/1 0
E 35 5/4 -561
F♯ 40 10/7 716
A♯ 50 25/14 155
5-Limit (2nd inversion)
Pitch Harmonic Interval Pitchbend
C 128 1/1 0
E 160 5/4 -561
F♯ 180 45/32 -400
A♯ 225 225/128 -961
Pythagorean (2nd inversion)
Pitch Harmonic Interval Pitchbend
C 32768 1/1 0
E 41472 81/64 320
F♯ 46656 729/512 480
A♯ 59049 59049/32768 801
Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G♭   1.4142 0
B♭   1.7818 0

Cuckoo chord

This is named after the movement in Carnival of the Animals where it is heard. It is a type of Augmented 6th chord and can be found on the 4th degree of the Swiss scale, and consists of the raised 4th scale degree, the minor 6th degree, the tonic and the major 3rd degree. Its full resolution is to the mediant. But in some cases such as in Carnival it is used as a neighboring chord to the tonic by simply contracting the diminished 3rd to a unison. This is akin to a German 6th chord resolving to a minor tonic 6/4.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭♭   1.1225 0
G♭   1.4142 0
B♭   1.7818 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 10 1/1 0
E♭♭ 11 11/10 -1433
G♭ 14 7/5 -716
B♭ 18 9/5 721
5-Limit
Pitch Harmonic Interval Pitchbend
C 225 1/1 0
E♭♭ 256 256/225 961
G♭ 320 64/45 400
B♭ 400 16/9 -160
Pythagorean
Pitch Harmonic Interval Pitchbend
C 59049 1/1 0
E♭♭ 65536 65536/59049 -801
G♭ 82944 1024/729 -480
B♭ 104976 16/9 -160

Augmented-Minor 7th Chord

This is a dominant 7th chord with a raised 5th. It is a type of Augmented 6th chord and can be found on the 5th degree of the Eulenspiegel scale. It consists of the 5th scale degree, the major 7th degree, the raised 2nd degree and the 4th degree. Its full resolution is to the tonic. It is really the cuckoo chord standing on its head. It isn’t very powerful in its tonal function, but is included because it is a 7th chord in the Swiss scale. But sometimes a non-tertian enharmonic is used where a minor 6th degree is used in lieu of the raised 5th, making it a flatted 13th chord, like the “chime” chord in Lida Rose.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G♯   1.5874 0
B♭   1.7818 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 144 1/1 0
E 180 5/4 -561
G♯ 225 25/16 -1121
B♭ 256 16/9 -160
Pythagorean
Pitch Harmonic Interval Pitchbend
C 36864 1/1 0
E 46656 81/64 320
G♯ 59049 6561/4096 641
B♭ 65536 16/9 -160

Major 9th Chord

Add the 9th harmonic to a major 7th Chord, and you get this chord. It is the largest just tuning consisting of alternating major and minor thirds. It is rooted on the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
B   1.8877 0
D   2.2449 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 8 1/1 0
E 10 5/4 -561
G 12 3/2 80
B 15 15/8 -481
D 18 9/4 160

Minor 9th Chord

This chord, rooted a major 3rd above the fundamental, is really the upper 5 notes of a major 11th chord. The 5-limit, 23-limit and harmonic tunings correspond to those given for the major 13th chord. The septimal tuning, also known as a “subminor” 9th chord, rooted a perfect 5th above the fundamental, is really the upper 5 notes of a dominant 13th chord.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭   1.1892 0
G   1.4983 0
B♭   1.7818 0
D   2.2449 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 20 1/1 0
E♭ 24 6/5 641
G 30 3/2 80
B♭ 36 9/5 721
D 45 9/4 160
Septimal
Pitch Harmonic Interval Pitchbend
C 12 1/1 0
E♭ 14 7/6 -1357
G 18 3/2 80
B♭ 21 7/4 -1277
D 27 9/4 160

Dominant 9th Chord

Add a 9th to a dominant 7th chord, and you get this chord. This is a the simplest 5-part chord, containing the first 5 different notes in the harmonic series. It is rooted on the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
B♭   1.7818 0
D   2.2449 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 4 1/1 0
E 5 5/4 -561
G 6 3/2 80
B♭ 7 7/4 -1277
D 9 9/4 160

Flatted 9th Chord

Add the septendecimal 9th to a dominant 7th chord and you get this chord. It is the chord you would get by flatting the 9th of the dominant 9th chord (4:5:6:7:9) to the 17th harmonic. It is rooted on the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
B♭   1.7818 0
D♭   2.1189 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 8 1/1 0
E 10 5/4 -561
G 12 3/2 80
B♭ 14 7/4 -1277
D♭ 17 17/8 203

Undecimal 9th Chord

Add an undecimal major 2nd to a half-diminished 7th chord and you get this rather beautiful chord. Rooted a major 3rd above the fundamental, it is really the upper 5 notes of a harmonic 11th chord. It’s inclusion is mainly out of an academic sense of adventure. But there is a chord in the Perry Mason theme that may be this one, making for a colorful cadence in the minor key.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭   1.1892 0
G♭   1.4142 0
B♭   1.7818 0
D   2.2449 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 5 1/1 0
E♭ 6 6/5 641
G♭ 7 7/5 -716
B♭ 9 9/5 -721
D 11 11/5 -1433

Major 11th Chord

These three tunings differ only in the tuning of the 11th degree added above a just major 9th chord. They are all rooted on the fundamental. The three tunings correspond to those given for the major 13th chord.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
B   1.8877 0
D   2.2449 0
F♯   2.8284 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 16 1/1 0
E 20 5/4 -561
G 24 3/2 80
B 30 15/8 -481
D 36 9/4 160
F♯ 45 45/16 -400

Minor 11th Chord

This chord, rooted a major 3rd above the fundamental, is really the upper 6 notes of a major 13th chord. The three tunings correspond to those given for the major 13th chord, and are all rooted a major 3rd above the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E♭   1.1892 0
G   1.4983 0
B♭   1.7818 0
D   2.2449 0
F   2.6697 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 20 1/1 0
E♭ 24 6/5 641
G 30 3/2 80
B♭ 36 9/5 721
D 45 9/4 160
F 54 27/10 801

Dominant 11th Chord

Take out the 3rd of a dominant 9th chord and add an 11th a perfect 5th above the 7th, and you get this chord. It is rooted on the fundamental. This chord is common in popular music and is prominantly used in Paul McCartney’s With a Little Luck.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
G   1.4983 0
B♭   1.7818 0
D   2.2449 0
F   2.6697 0
Septimal
Pitch Harmonic Interval Pitchbend
C 8 1/1 0
G 12 3/2 80
B♭ 14 7/4 -1277
D 18 9/4 160
F 21 21/8 -1197

Harmonic 11th Chord

This chord has one of the most glorious sounds from the golden age of music. It is the simplest 6-part chord, containing the first 6 notes in the harmonic series. It is the undecimal extension of a dominant 9th chord. It appears prominantly in the song Christmas Time is Here. It is rooted on the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
B♭   1.7818 0
D   2.2449 0
F♯   2.8284 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 4 1/1 0
E 5 5/4 -561
G 6 3/2 80
B♭ 7 7/4 -1277
D 9 9/4 160
F♯ 11 11/4 -1994

Split 11th Chord

I named this chord after the interval of a fourth heard between the highest two notes. This is the chord you would get by flatting the 9th of of the harmonic 11th chord to the 17th harmonic. It is rooted on the fundamental. The septimal and undecimal tunings are merely two just triads with roots separated by the septimal or undecimal tritones respectively.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
B♭   1.7818 0
D♭   2.1189 0
F♯   2.8284 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 8 1/1 0
E 10 5/4 -561
G 12 3/2 80
B♭ 14 7/4 -1277
D♭ 17 17/8 203
F♯ 22 11/4 -1994
Septimal
Pitch Harmonic Interval Pitchbend
C 20 1/1 0
E 25 5/4 -561
G 30 3/2 80
B♭ 35 7/4 -1277
D♭ 42 21/10 -636
F♯ 56 14/5 -716
Undecimal
Pitch Harmonic Interval Pitchbend
C 32 1/1 0
E 40 5/4 -561
G 48 3/2 80
B♭ 55 55/32 -2555
D♭ 66 33/16 -1914
F♯ 88 11/4 -1994

Diminished 11th Chord

This chord is so-called because it’s eleventh is flatted to sound like a 10th. The harmonic tuning is really an incomplete harmonic 11th chord in 5th inversion. It is rooted an undecimal tritone above the fundamental which is actually the Gb. The overtone tuning substitutes the involute dominant 7th chord in the upper 4 notes, and is rooted on the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
G♭   1.4142 0
B♭   1.7818 0
D♭   2.1189 0
F♭   2.5198 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 11 1/1 0
G♭ 16 16/11 1994
B♭ 20 20/11 1433
D♭ 24 24/11 2074
F♭ 28 28/11 717
Overtone
Pitch Harmonic Interval Pitchbend
C 8 1/1 0
G♭ 11 11/8 -1994
B♭ 14 7/4 -1277
D♭ 17 17/8 203
F♭ 20 5/2 -561

Major 13th Chord

I’ll start with the harmonic tuning, which is made by sharping the 7th of a harmonic 13th chord (4:5:6:7:9:11:13) to the 15th harmonic. You may find the harmonic tuning harsh, probably owing to the simultaneous sounding of the high-order 11th and 15th harmonics in inversion. The 23-limit tuning gives a lift to the 11th by tuning it to the nearby 23rd harmonic. The 13th, likewise, is given a lift by tuning it to the nearby 27th degree (which, by the way, forms a Pythagorean 6th with the root). The 5-limit version, which uses alternating major and minor 3rds, actually occurs in the harmonic series an octave higher, so as to use the 45th harmonic. These tunings are all rooted on the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
B   1.8877 0
D   2.2449 0
F♯   2.8284 0
A   3.3636 0
5-Limit
Pitch Harmonic Interval Pitchbend
C 16 1/1 0
E 20 5/4 -561
G 24 3/2 80
B 30 15/8 -481
D 36 9/4 160
F♯ 45 45/16 -400
A 54 27/8 240

Dominant 13th Chord

Add a 13th a perfect 5th above the 9th of a dominant 11th chord and you get this chord. It is rooted on the fundamental. It’s functionality is similar to the dominant 11th chord but has a little more color to it.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
G   1.4983 0
B♭   1.7818 0
D   2.2449 0
F   2.6697 0
A   3.3636 0
Septimal
Pitch Harmonic Interval Pitchbend
C 8 1/1 0
G 12 3/2 80
B♭ 14 7/4 -1277
D 18 9/4 160
F 21 21/8 -1197
A 27 27/8 240

Harmonic 13th Chord

This chord, used a lot in big band music as a closing chord, can be called the “accoustic 13th” because of its obvious basis in the harmonic series. It is the simplest 7-part chord, containing the first 7 notes in the harmonic series. It is the tredecimal extension of a harmonic 11th chord. A number of the chords mentioned above can be found inside this one. It is rooted on the fundamental. If you revoice this chord to 1:3:5:7:9:11:13, it should sound a lot like a clarinet playing the fundamental note, since it’s tone quality is based on the presence of the odd harmonics.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
E   1.2599 0
G   1.4983 0
B♭   1.7818 0
D   2.2449 0
F♯   2.8284 0
A   3.3636 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 4 1/1 0
E 5 5/4 -561
G 6 3/2 80
B♭ 7 7/4 -1277
D 9 9/4 160
F♯ 11 11/4 -1994
A 13 13/4 -2436

Flatted 13th Chord

Add the undecimal median 6th to the Diminished 11th Chord and you get this chord. Harmonic examination reveals however that it is really a harmonic 11th chord in 5th inversion. It is rooted an undecimal tritone above the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
G♭   1.4142 0
B♭   1.7818 0
D♭   2.1189 0
F♭   2.5198 0
A♭   3.1748 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 11 1/1 0
G♭ 16 16/11 1994
B♭ 20 20/11 1433
D♭ 24 24/11 2074
F♭ 28 28/11 717
A♭ 36 36/11 2154

5th Inversion 13th Chord

Add a tredecimal minor 3rd to the flatted 13th chord and you get this chord. Harmonic examination reveals however that it is really a harmonic 13th chord in 5th inversion. It is rooted an undecimal tritone above the fundamental.

Equal Temperament
Pitch Harmonic Interval Pitchbend
C   1 0
G♭   1.4142 0
B♭   1.7818 0
D♭   2.1189 0
F♭   2.5198 0
A♭   3.1748 0
E♭   4.7568 0
Harmonic
Pitch Harmonic Interval Pitchbend
C 11 1/1 0
G♭ 16 16/11 1994
B♭ 20 20/11 1433
D♭ 24 24/11 2074
F♭ 28 28/11 717
A♭ 36 36/11 2154
E♭ 52 52/11 -442

Scales
Index of Intervals
Chord Progressions
Glossary of Just Intonation
Tables of Pitch Bends
Fun with Vowel Formants
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