Table of Just Scales

Major Scale
Natural Minor Scale
Harmonic Minor Scale
Harmonic Major Scale
Hungarian Minor Scale
Neopolitan Minor Scale
Eulenspiegel Scale
Swiss Scale
Acoustic Scale
Jazz Minor Scale
Altered Scale
Romanian Major Scale
Hungarian Major Scale
Jeths’ Mode Scale
Chromatic Scale

The following are various scales and scale tunings. Listen to the notes by moving the mouse over the glass buttons on the right. You can peruse the 7th chords on each scale degree by holding the Alt key. You can also select the wavetype to use by selecting below.

Wavetype:

Major Scale

also known as: Ionian mode

The Pythagorean major scale (the oldest of the tunings) is generated by a series of tones a perfect 5th (3/2) apart. The notes F, C, G, D, A, E and B, if they are tuned to just 5ths (3/2), would be a Pythagorean C major scale.

The just major scale is constructed by lowering the three notes most remote from the tonic, A, E and B, by a syntonic comma. The result can be arranged as a “circle of thirds”, F, A, C, E, G, B and D, where the odd notes retain their Pythagorean tuning, while the even ones, lowered by a syntonic comma, form just thirds creating the three just major triads FAC, CEG and GBD. Two just minor triads ACE and EGB are also formed. Note that triads using the two end notes together are out of tune, which are BDF and DFA.

The septimal major scale alters the just scale by raising the 6th degree by a syntonic comma and lowering the 4th degree by a septimal comma. As a result, the tonic chord is just, and the four remaining notes form a just half-diminished 7th chord. Additionally, the ii chord is a septimal minor triad, the V chord and the vii diminished are both just, and the vi chord is a wolf-minor triad. This could appropriately be called the "overtone major scale" because its degrees are based on harmonics above the fundamental (note how the denominators of the ratios are all powers of 2). A justly tuned harmonica uses this tuning. You will also note that introducing the septimal factor allows a scale of even smaller ratio values than the just scale.

The quarter meantone scale is tuned by lowering each note in the circle of fifths by an additonal 1/4 syntonic comma, so that the note 4 links from the tonic is lowered a whole syntonic comma to form a just major third. This results in a scale with a lot of just major thirds, at the expense of narrowed fifths. This tuning is concurrent with the period when the triad was promoted to consonance, sometime around the period from the Renaissance to the Baroque.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Major
D
9/8
160
Minor
E
5/4
-561
Minor
F
4/3
-8
Major
G
3/2
80
Dominant
A
5/3
-641
Minor
B
15/8
-481
Half-Diminished
C
2
0
Major

Pythagorean

Note
Ratio
Pitchbend
7th Chord
C
1
0
Major
D
9/8
160
Minor
E
81/64
320
Minor
F
4/3
-8
Major
G
3/2
80
Dominant
A
27/16
240
Minor
B
243/128
400
Half-Diminished
C
2
0
Major

Septimal

Note
Ratio
Pitchbend
7th Chord
C
1
0
Major
D
9/8
160
Minor
E
5/4
-561
Minor
F
21/16
-1197
Major
G
3/2
80
Dominant
A
27/16
240
Minor
B
15/8
-481
Half-Diminished
C
2
0
Major

Quarter Meantone

Note
Ratio
Pitchbend
7th Chord
C
1
0
Major
D
1.1180
-280
Minor
E
5/4
-561
Minor
F
1.3375
140
Major
G
1.4953
-140
Dominant
A
1.6719
-420
Minor
B
1.8692
-701
Half-Diminished
C
2
0
Major

Natural Minor Scale

Also called: Aolean mode.

This is the 6th mode of the major scale. Proper tuning of the 4th degree requires flatting the 2nd of the relative major by a syntonic comma (81/80).

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Minor
D
9/8
160
Half-Diminished
Eb
6/5
641
Major
F
4/3
-80
Minor
G
3/2
80
Minor
Ab
8/5
561
Major
Bb
9/5
721
Dominant
C
2
0
Minor

Harmonic Minor Scale

This scale raises the 7th degree of the natural-minor, resulting in an augmented second between the minor 6th and the major 7th degrees. The harmonic minor is asymmetrical in that its mirror image is a different scale (based on modal equivalence). But all the chords in the scale include their mirror images. The septendecimal tuning tunes the minor 6th degree according to the 17th harmonic, resulting in the dominant flatted 9th chord being just.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Minor-Major
D
9/8
160
Half-Diminished
Eb
6/5
641
Augmented-Major
F
4/3
-80
Minor
G
3/2
80
Dominant
Ab
8/5
561
Major
B
15/8
-481
Diminished
C
2
0
Minor-Major

Septendecimal

Note
Ratio
Pitchbend
7th Chord
C
1
0
Minor-Major
D
9/8
160
Half-Diminished
Eb
6/5
641
Augmented-Major
F
21/16
-1197
Minor
G
3/2
80
Dominant
Ab
51/32
283
Major
B
15/8
-481
Diminished
C
2
0
Minor-Major

Harmonic Major Scale

The harmonic major scale is different from the harmonic minor only in its major 3rd degree. It is a mode of the harmonic minor scale’s mirror image.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Major
D
9/8
160
Half-Diminished
E
5/4
-561
Minor
F
4/3
-80
Minor-Major
G
3/2
80
Dominant
Ab
8/5
561
Augmented-Major
B
15/8
-481
Diminished
C
2
0
Major

Septendecimal

Note
Ratio
Pitchbend
7th Chord
C
1
0
Major
D
9/8
160
Half-Diminished
E
5/4
-561
Minor
F
21/16
-1197
Minor-Major
G
3/2
80
Dominant
Ab
51/32
283
Augmented-Major
B
15/8
-481
Diminished
C
2
0
Major

Hungarian Minor Scale

This scale results from raising the 4th and 7th degrees of the natural-minor, resulting in two augmented seconds in the scale. This scale is also home to well-known augmented sixth chords: the Italian 6th, with a root, 3rd and augmented 6th, often voiced with a doubled 3rd; the German 6th, which sounds like a dominant 7th but the 7th degree reveals its idently as an augmented 6th by its wanting to resolve upward; the French 6th, which is a whole-tone 7th consisting of a major 3rd, a major 2nd, and another major 3rd.

Then there is this very obscurely used chord on the scale’s 7th degree that sounds like a half-diminished 7th, but whose notes forming the minor 7th want to resolve outward to an octave, making it really a kind of augmented 6th chord. It is really a German 6th chord standing on its head. I dub it the Tristan 6th chord because its famous use in Tristan and Isolde spells it as F B D# G#, which would make it want to resolve to tonic E C E A chord in A minor. But Wagner doesn’t resolve it. He sort of half-cadences it to a regular dominant 7th chord with a connecting French 6th. The chord has been fully resolved in a few cases like Alfven's 3rd Symphony and Michael Haydn's C Symphony Perger 12. But its most common use by far is as a “Tristan mordent”.

The “gypsy scale” (aka the Arabic or Byzantine scale) is the 5th mode of the Hungarian minor. This mode is symmetrical in that it is its own mirror image.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Minor-Major
D
9/8
160
French 6th
Eb
6/5
641
Augmented-Major
F#
45/32
-400
German 6th
G
3/2
80
Major
Ab
8/5
561
Major
B
15/8
-481
Tristan 6th
C
2
0
Minor-Major

Neopolitan Minor Scale

This scale results from flatting the 2nd degree of the harmonic minor scale. I like to call it the Appalachian scale after how well it lends itself to Appalachian folk melodies. The main chorus of “When I Look In Your Eyes” makes use of this scale.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Minor-Major
Db
16/15
481
Major
Eb
6/5
641
Augmented-Minor
F
4/3
-80
Minor
G
3/2
80
French 6th
Ab
8/5
561
Major
B
15/8
-481
German 6th
C
2
0
Minor-Major

Eulenspiegel Scale

When you combine the notes of the Eulenspiegel 6th chord with those of the tonic doubled-3rd triad to which it fully resolves, the result is this “Eulenspiegel” scale (also known in South Indian classical music as the Gangeyabhushani scale) Note how the sharped 2nd degree and flatted 6th degree form a “doubly-diminished 5th”. A good mneumonic to consider is: the Eulenspiegel is a major scale with a sharped 2nd degree and a flatted 6th degree.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Major
D#
75/64
-1041
Minor-Augmented 6th
E
5/4
-561
Minor-Major
F
4/3
-80
Minor-Major
G
3/2
80
Augmented-Minor
Ab
8/5
561
Augmented-Major
B
15/8
-481
Eulenspiegel 6th
C
2
0
Major

Swiss Scale

When you combine the notes of the Swiss 6th chord with those of the mediant doubled-3rd triad to which it fully resolves, the result is this “Swiss” scale (also known in South Indian classical music as the Dhatuvardhani scale). This is (based on modal equivalence) a mirror-image of the Eulenspiegel scale, This chord was originally developed as an alternate spelling for the German 6th to resolve to the major 6/4 tonic. Note how the flatted 6th and sharped 2nd degrees form a “doubly-augmented 4th”. The Swiss can be made by sharping the 2nd and 3rd degrees of the Hungarian minor, putting the #2 where the b3 used to be. Here the German 6th is still where it used to be but spelled differently and hence the resolution.

The scales also introduce some additional chords to Tertian status such as some more whole-tone chords similar to the French 6th, and a couple enharmonics of the minor 7th chord which by its resolution turns out to be a kind of augmented 6th chord. Mneumonic: the Swiss differs from the Eulenspiegel only in its raised 4th degree.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Major
D#
75/64
-1041
Swiss 6th
E
5/4
-561
Minor-Major
F#
45/32
-400
"cuckoo"
G
3/2
80
Augmented-Major
Ab
8/5
561
Augmented-Major
B
15/8
-481
Major-Diminished
C
2
0
Major

Acoustic Scale

Also called: overtone scale, lydian-dominant scale.

This scale is based on the first 13 harmonics. It is a major scale with a sharped 4th degree and a flatted 7th degree. It it also the 4th mode of the melodic-minor scale. The Septimal tuning is merely two just dominant 7th chords offset by a Septimal 2nd.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Dominant
D
9/8
160
Dominant
E
5/4
-561
Half-Diminished
F#
45/32
-400
Half-Diminished
G
3/2
80
Minor-Major
A
5/3
-641
Minor
Bb
9/5
721
Augmented-Major
C
2
0
Dominant

Just

Note
Ratio
Pitchbend
7th Chord
C
1
0
Dominant
D
9/8
160
Dominant
E
5/4
-561
Half-Diminished
F#
11/8
-1994
Half-Diminished
G
3/2
80
Minor-Major
Ab
13/8
-2436
Minor
B
7/4
-1277
Augmented-Major
C
2
0
Dominant

Septimal

Note
Ratio
Pitchbend
7th Chord
C
1
0
Dominant
D
8/7
1277
Dominant
E
5/4
-561
Half-Diminished
F#
10/7
716
Half-Diminished
G
3/2
80
Minor-Major
Ab
12/7
1357
Minor
B
7/4
-1277
Augmented-Major
C
2
0
Dominant

Jazz Minor Scale

Also called: the ascending melodic minor scale.

This is the 5th mode of the acoustic scale. Named on the basis of a tradition of jazz to use a minor key that uses a major 6th degree.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Minor-Major
D
9/8
160
Minor
Eb
6/5
641
Augmented-Minor
F
4/3
-80
Dominant
G
3/2
80
Dominant
A
5/3
-641
Half-Diminished
B
15/8
-481
Half-Diminished
C
2
0
Minor-Major

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Minor-Major
D
13/12
160
Minor
Eb
7/6
-1357
Augmented-Minor
F
4/3
-80
Dominant
G
3/2
80
Dominant
A
5/3
-641
Half-Diminished
B
11/6
-2074
Half-Diminished
C
2
0
Minor-Major

Altered Scale

This is the 4th mode of the acoustic scale, and is associated with the creation of altered jazz chords, which have the 11th harmonic in the bass. It can also be described as a major scale with the interior notes all flatted.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Half-Diminished
Db
16/15
481
Minor-Major
Eb
6/5
641
Minor
Fb
32/25
1121
Augmented-Major
Gb
36/25
1281
Dominant
Ab
8/5
561
Dominant
Bb
9/5
721
Half-Diminished
C
2
0
Half-Diminished

Just

Note
Ratio
Pitchbend
7th Chord
C
1
0
Half-Diminished
Db
12/11
2074
Minor-Major
Eb
13/11
-442
Minor
Fb
14/11
717
Augmented-Major
Gb
16/11
1994
Dominant
Ab
18/11
2154
Dominant
Bb
20/11
1433
Half-Diminished
C
2
0
Hald-Diminished

Romanian Major Scale

The Romanian major scale (which I’ve referred to as the rural-major) is made by flatting the 2nd degree of the acoustic scale to a septendecimal minor 2nd. It is the also 4th mode of the Jeths&lsrq; Mode scale.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Dominant
Db
16/15
481
Flat-Rooted
E
5/4
-561
Diminished
F#
45/32
-400
Half-Diminished
G
3/2
80
Diminished-Major
A
5/3
-641
Minor
Bb
9/5
721
Split
C
2
0
Dominant

Just

Note
Ratio
Pitchbend
7th Chord
C
1
0
Dominant
Db
17/16
203
Flat-Rooted
E
5/4
-561
Diminished
F#
11/8
-1194
Half-Diminished
G
3/2
80
Diminished-Major
A
13/8
-2436
Minor
Bb
7/4
-1277
Split
C
2
0
Dominant

Hungarian Major Scale

This scale is made by sharping the 2nd degree of the acoustic scale to a 19-limit minor 3rd.

Just

Note
Ratio
Pitchbend
7th Chord
C
1
0
Dominant
D#
19/16
-102
Diminished
E
5/4
-561
Diminished-Major
F#
11/8
-1194
Half-Diminished
G
3/2
80
Split
A
13/8
-2436
Minor
Bb
7/4
-1277
Flat-Rooted
C
2
0
Dominant

Jeths’ Mode Scale

This scale (which I have referrd to as the rural-minor scale) is one I came up with to shoehorn some choice chords into "tertian" status. I found a guitar website that has the same scale and referenced it as Jeths” Mode. Mneumonic: The Jeths’ Mode is the melodic minor with a flatted 5th, or the diminished scale with the 6th degree missing.

5-Limit

Note
Ratio
Pitchbend
7th Chord
C
1
0
Diminished-Major
D
9/8
160
Minor
Eb
6/5
641
Split
F
4/3
-80
Dominant
Gb
36/25
1281
Flat-Rooted
A
5/3
-641
Diminished
B
15/8
-481
Half-Diminished
C
2
0
Diminished-Major

Just

Note
Ratio
Pitchbend
7th Chord
C
1
0
Diminished-Major
D
13/12
-2516
Minor
Eb
7/6
-1357
Split
F
4/3
-80
Dominant
Gb
17/12
123
Flat-Rooted
A
5/3
-641
Diminished
B
11/6
-2074
Half-Diminished
C
2
0
Diminished-Major

Chromatic Scale

This scale uses the range of fifths from Eb to G#. The interval between these two is really a diminished 6th, causing them to be out of tune and often referred to as a “wolf fifth”. The tunings are C-based, meaning that C has a pitchbend of 0 and other notes are tuned relative to C.

While the Pythagorean tuning had pure 5ths, its major 3rds (81/64) were too wide to be considered a consonant interval, hence the treatment of the 3rd as a dissonance back in the pre-common-practice period. Quarter-comma meantone sacrifices some of the purity of the 5ths by narrowing each them a quarter of a syntonic comma so that the ditones (made by 4 consecutive 5ths) are narrowed to pure major 3rds (5/4). This corresponded with the beginning of the emphasis of major thirds as a consonant interval in the Baroque. Indeed, baroque harmony considered the 3rd more vital to the completeness of a true chord than the 5th, since the meantone tuning had made the 3rd the pure interval.

Pythagorean

Note
Ratio
Pitchbend
C
1
0
C#
2187/2048
561
D
9/8
160
Eb
32/27
-240
E
81/64
320
F
4/3
-80
F#
729/512
480
G
3/2
80
G#
6561/4096
641
A
27/16
240
Bb
16/9
-2074
B
243/128
-160
C
2
400

5-Limit

Note
Ratio
Pitchbend
C
1
0
C#
25/24
-1201
D
9/8
160
Eb
6/5
641
E
5/4
-561
F
4/3
-80
F#
45/32
-400
G
3/2
80
G#
25/16
-1121
A
5/3
-641
Bb
9/5
721
B
15/8
-481
C
2
0

Quarter Meantone

Note
Ratio
Pitchbend
C
1
0
C#
1.0449
-981
D
1.1180
-280
Eb
1.1963
420
E
5/4
-561
F
1.3375
140
F#
1.3975
-841
G
1.4953
-140
G#
25/16
-1121
A
1.6719
-420
Bb
1.7889
280
B
1.8692
-701
C
2
0

Index of Intervals
Chord Tunings
Chord Progressions
Glossary of Just Intonation
Tables of Pitch Bends
Fun with Vowel Formants
Just Intonation home page
Billy”s home page