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Difference between revisions of "Ref: Intervals"

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An interval is the "distance" in terms of pitch between two notes.  This is usually defined in terms of the number of steps between them in the 12-tone chromatic scale.  The actual pitch difference depends on the tuning system used.  The two relative pitches also form a ratio that determine how harmonic they are.
 
An interval is the "distance" in terms of pitch between two notes.  This is usually defined in terms of the number of steps between them in the 12-tone chromatic scale.  The actual pitch difference depends on the tuning system used.  The two relative pitches also form a ratio that determine how harmonic they are.
 +
 +
== Pitches and Ratios ==
  
 
{| class="wikitable"
 
{| class="wikitable"
 
|-
 
|-
−
! Root<br>Note !! Other<br>Note !! Semitones<br>from<br />Root to<br>Other !! Interval<br>Short<br>Name !! Interval<br>Full<br>Name !! Just<br>Intonation<br>Ratio
+
! Root<br>Note !! Other<br>Note !! Semitones<br>from<br />Root to<br>Other !! Interval<br>Short<br>Name !! Interval<br>Full<br>Name !! Just<br>Intonation<br>Ratio !! Root<br>Cycles<br>before<br>resync
 
|-
 
|-
−
|  C  || C  || 0  || U  || Unison || 1:1
+
|  C  || C  || 0  || {{U|U}} || Unison || 1:1 || 1
 
|-
 
|-
−
| C  || D{{f}} || 1  || m2 || Minor 2nd || 16:15
+
| C  || D{{f}} || 1  || {{Mn2|m2}} || Minor 2nd || 16:15 || 16
 
|-
 
|-
−
| C  || D  || 2  || M2 || (Major) 2nd || 9:8
+
| C  || D  || 2  || {{M2|M2}} || (Major) 2nd || 9:8 || 9
 
|-
 
|-
−
| C  || E{{f}} || 3  || m3 || Minor 3rd || 6:5
+
| C  || E{{f}} || 3  || {{Mn3|m3}} || Minor 3rd || 6:5 || 6
 
|-
 
|-
−
| C  || E  || 4  || M3 || Major 3rd || 5:4
+
| C  || E  || 4  || {{M3|M3}} || Major 3rd || 5:4 || 5
 
|-
 
|-
−
| C  || F  || 5  || P4 || (Perfect) 4th ||4:3
+
| C  || F  || 5  || {{P4|P4}} || (Perfect) 4th ||4:3 || 4
 
|-
 
|-
−
| C  || G{{f}} || 6  || A4, d5, tt  || Augmented 4th/<br />Diminished 5th/<br />Tritone || 7:5
+
| C  || G{{f}} || 6  || {{tt|A4, d5, tt}} || Augmented 4th/<br />Diminished 5th/<br />Tritone || 45:32 || 45
 
|-
 
|-
−
| C  || G  || 7  || P5 || (Perfect) 5th ||3:2
+
| C  || G  || 7  || {{P5|P5}} || (Perfect) 5th || 3:2 || 3
 
|-
 
|-
−
| C  || A{{f}} || 8  || m6 || Minor 6th || 8:5
+
| C  || A{{f}} || 8  || {{Mn6|m6}} || Minor 6th || 8:5 || 8
 
|-
 
|-
−
| C  || A  || 9  || M6 || Major 6th || 5:3
+
| C  || A  || 9  || {{M6|M6}} || Major 6th || 5:3 || 5
 
|-
 
|-
−
| C  || B{{f}} || 10 || m7  || Minor 7th || 16:9
+
| C  || B{{f}} || 10 || {{Mn7|m7}} || Minor 7th || 9:5 || 9
 
|-
 
|-
−
| C  || B  || 11 || M7 || Major 7th || 15:8
+
| C  || B  || 11 || {{M7|M7}} || Major 7th || 15:8 || 15
 
|-
 
|-
−
| C  || C  || 12 || U  || Unison or Octave || 2:1
+
| C  || C  || 12 || {{U|U}} || Unison or Octave || 2:1 || 2
 
|}
 
|}
 +
 +
{| class="wikitable"
 +
|+ Ordered by Most to Least Consonant (Harmonic)
 +
|-
 +
! Root<br>Note !! Other<br>Note !! Semitones<br>from<br />Root to<br>Other !! Interval<br>Short<br>Name !! Interval<br>Full<br>Name !! Just<br>Intonation<br>Ratio !! Root<br>Cycles<br>before<br>resync
 +
|-
 +
|  C  || C  || 0  || {{U|U}}  || Unison || 1:1 || 1
 +
|-
 +
| C  || C  || 12 || {{U|U}}  || Unison or Octave || 2:1 || 2
 +
|-
 +
| C  || G  || 7  || {{P5|P5}} || (Perfect) 5th || 3:2 || 3
 +
|-
 +
| C  || F  || 5  || {{P4|P4}} || (Perfect) 4th ||4:3 || 4
 +
|-
 +
| C  || E  || 4  || {{M3|M3}} || Major 3rd || 5:4 || 5
 +
|-
 +
| C  || A  || 9  || {{M6|M6}} || Major 6th || 5:3 || 5
 +
|-
 +
| C  || E{{f}} || 3  || {{Mn3|m3}} || Minor 3rd || 6:5 || 6
 +
|-
 +
| C  || A{{f}} || 8  || {{Mn6|m6}} || Minor 6th || 8:5 || 8
 +
|-
 +
| C  || D  || 2  || {{M2|M2}} || (Major) 2nd || 9:8 || 9
 +
|-
 +
| C  || B{{f}} || 10 || {{Mn7|m7}}  || Minor 7th || 9:5 || 9
 +
|-
 +
| C  || B  || 11 || {{M7|M7}} || Major 7th || 15:8 || 15
 +
|-
 +
| C  || D{{f}} || 1  || {{Mn2|m2}} || Minor 2nd || 16:15 || 16
 +
|-
 +
| C  || G{{f}} || 6  || {{tt|A4, d5, tt}}  || Augmented 4th/<br />Diminished 5th/<br />Tritone || 45:32 || 45
 +
|}
 +
  
 
{|class="wikitable"
 
{|class="wikitable"
 +
|+ Comparing Just Intonation to Equal Temperament
 +
|-
 
! Interval<br>Name !! Just<br>Intonation<br>Ratio !!  Equal<br>Temperament !! %<br>Difference !! Just<br>Intonation<br>Cents !! 12-TET<br>Cents !! Cent<br>Difference
 
! Interval<br>Name !! Just<br>Intonation<br>Ratio !!  Equal<br>Temperament !! %<br>Difference !! Just<br>Intonation<br>Cents !! 12-TET<br>Cents !! Cent<br>Difference
 
|-
 
|-
−
| m2 || 16:15 = 1.0666... || 1.05946 || -0.67563 || 111.73  || 100  || -11.73
+
| {{Mn2|m2}} || 16:15 = 1.0666... || 1.05946 || -0.67563 || 111.73  || 100  || -11.73
 
|-                                       
 
|-                                       
−
| M2 || 9:8 = 1.125      || 1.12246 || -0.22618 || 203.91  || 200  || -3.91
+
| {{M2|M2}} || 9:8 = 1.125      || 1.12246 || -0.22560 || 203.91  || 200  || -3.91
 
|-                                               
 
|-                                               
−
| m3 || 6:5 = 1.2        || 1.18920 || -0.90028 || 315.64  || 300  || -15.64
+
| {{Mn3|m3}} || 6:5 = 1.2        || 1.18921 || -0.89941 || 315.64  || 300  || -15.64
 
|-                                               
 
|-                                               
−
| M3 || 5:4 = 1.25        || 1.25991 ||  0.79250 || 386.31  || 400  || +13.69
+
| {{M3|M3}} || 5:4 = 1.25        || 1.25992 ||  0.79368 || 386.31  || 400  || +13.69
 
|-                                               
 
|-                                               
−
| P4 || 4:3 = 1.333...    || 1.33482 ||  0.11153 || 498.04   || 500  || +1.96
+
| {{P4|P4}} || 4:3 = 1.333...    || 1.33484 ||  0.11299 || 498.05   || 500  || +1.96
 
|-                                               
 
|-                                               
−
| tt || 7:5 = 1.4        || 1.41419 ||  1.01349 || 582.51  || 600  || +17.49
+
| {{tt|tt}} || 45:32 = 1.40625  || 1.41421 ||  0.56630 || 590.22 || 600  || +9.78
 
|-                                               
 
|-                                               
−
| P5 || 3:2 = 1.5        || 1.49828 || -0.11491 || 701.96  || 700  || -1.96
+
| {{P5|P5}} || 3:2 = 1.5        || 1.49831 || -0.11286 || 701.96  || 700  || -1.96
 
|-                                               
 
|-                                               
−
| m6 || 8:5 = 1.6        || 1.58736 || -0.78975 || 813.69  || 800  || -13.69
+
| {{Mn6|m6}} || 8:5 = 1.6        || 1.58740 || -0.78743 || 813.69  || 800  || -13.69
 
|-                                               
 
|-                                               
−
| M6 || 5:3 = 1.666...    || 1.68175 ||  0.90492 || 884.36  || 900  || +15.64
+
| {{M6|M6}} || 5:3 = 1.666...    || 1.68179 ||  0.90757 || 884.36  || 900  || +15.64
 
|-                                       
 
|-                                       
−
| m7 || 16:9 = 1.777...   || 1.78175 ||  0.22318 || 996.09  || 1,000 || +3.91
+
| {{Mn7|m7}} || 9:5 = 1.8   || 1.78180 ||  -1.01125 || 1017.60 || 1,000 || -17.60
 
|-                                       
 
|-                                       
−
| M7 || 15:8 = 1.875      || 1.88769 ||  0.67669 || 1088.27  || 1,100 || +11.73
+
| {{M7|M7}} || 15:8 = 1.875      || 1.88775 ||  0.67993 || 1088.27  || 1,100 || +11.73
 
|}
 
|}
  
 
=== Minor 2nd ===
 
=== Minor 2nd ===
 
{{Crumbs|blue|1
 
{{Crumbs|blue|1
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of [[#Major 7th|M7]]
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 1 || [[#Major 7th|{{M7|M7}}]] || 16:15 = 1.0666... || 1.05946 || 111.73 || 100
 +
|}
 +
 
 
{|
 
{|
 
|-
 
|-
Line 82: Line 125:
 
* |i|-|-|-|-|
 
* |i|-|-|-|-|
 
* |-|-|-|-|p|
 
* |-|-|-|-|p|
 +
 
* all but G-B
 
* all but G-B
 
</pre>
 
</pre>
Line 88: Line 132:
 
B |i|-|-|-|
 
B |i|-|-|-|
 
G |-|-|-|p|
 
G |-|-|-|p|
 +
 
   
 
   
 
</pre>
 
</pre>
 +
|-
 
|
 
|
 
<pre>
 
<pre>
Line 95: Line 141:
 
A/D |-|-|-|-|
 
A/D |-|-|-|-|
 
E/A |i|-|-|-|
 
E/A |i|-|-|-|
 +
 
</pre>
 
</pre>
 
|
 
|
Line 101: Line 148:
 
G/B |-|-|-|-|-|
 
G/B |-|-|-|-|-|
 
D/G |i|-|-|-|-|
 
D/G |i|-|-|-|-|
 +
 +
</pre>
 +
|
 +
<pre>
 +
G |i|-|-|
 +
D |-|-|-|
 +
A |-|-|-|
 +
E |-|-|r|
 +
</pre>
 +
|
 +
<pre>
 +
B/E |i|-|
 +
G/B |-|-|
 +
D/G |-|-|
 +
A/D |-|m|
 
</pre>
 
</pre>
 
|}
 
|}
Line 115: Line 177:
 
=== Major 2nd ===
 
=== Major 2nd ===
 
{{Crumbs|blue|2
 
{{Crumbs|blue|2
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of [[#Minor 7th|m7]]
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 2 || [[#Minor 7th|{{Mn7|m7}}]] || 9:8 = 1.125 || 1.12246 || 203.91 || 200
 +
|}
  
 
{|
 
{|
Line 179: Line 246:
 
=== Minor 3rd ===
 
=== Minor 3rd ===
 
{{Crumbs|blue|3
 
{{Crumbs|blue|3
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of [[#Major 6th|M6]]
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 3 || [[#Major 6th|{{M6|M6}}]] || 6:5 = 1.2        || 1.18920 || 315.64 || 300
 +
|}
  
 
{|
 
{|
Line 236: Line 308:
 
=== Major 3rd ===
 
=== Major 3rd ===
 
{{Crumbs|blue|4
 
{{Crumbs|blue|4
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of [[#Minor 6th|m6]]
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 4 || [[#Minor 6th|{{Mn6|m6}}]] || 5:4 = 1.25        || 1.25991 || 386.31 || 400
 +
|}
  
 
{|
 
{|
Line 293: Line 370:
 
=== Perfect 4th ===
 
=== Perfect 4th ===
 
{{Crumbs|blue|5
 
{{Crumbs|blue|5
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of [[#Perfect 5th|P5]]
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 5 || [[#Perfect 5th|{{P5|P5}}]] || 4:3 = 1.333...    || 1.33484 || 498.05 || 500
 +
|}
  
 
{|
 
{|
Line 313: Line 395:
 
* |i|
 
* |i|
 
* |i|
 
* |i|
 +
 
   
 
   
 
* all but G-B
 
* all but G-B
Line 321: Line 404:
 
G |i|-|
 
G |i|-|
 
   
 
   
 +
 
   
 
   
 
</pre>
 
</pre>
Line 329: Line 413:
 
A |-|-|-|
 
A |-|-|-|
 
E |i|-|-|
 
E |i|-|-|
 +
 
</pre>
 
</pre>
 
|
 
|
Line 336: Line 421:
 
D/G |-|-|-|-|
 
D/G |-|-|-|-|
 
A/D |i|-|-|-|
 
A/D |i|-|-|-|
 +
 +
</pre>
 +
|
 +
<pre>
 +
B/E |i|-|-|
 +
G/B |-|-|-|
 +
D/G |-|-|-|
 +
A/D |-|-|-|
 +
E/A |-|-|r|
 
</pre>
 
</pre>
 
|}
 
|}
Line 350: Line 444:
 
=== Tritone ===
 
=== Tritone ===
 
{{Crumbs|blue|6
 
{{Crumbs|blue|6
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of itself
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 6 || itself || 45:32 = 1.40625        || 1.41421 || 590.22 || 600
 +
|}
  
 
{|
 
{|
Line 370: Line 469:
 
* |-|i|
 
* |-|i|
 
* |i|-|
 
* |i|-|
 +
 
   
 
   
 
* all but G-B
 
* all but G-B
Line 377: Line 477:
 
B |-|-|r|
 
B |-|-|r|
 
G |i|-|-|
 
G |i|-|-|
 +
 
   
 
   
 
   
 
   
Line 386: Line 487:
 
A |-|-|-|-|
 
A |-|-|-|-|
 
E |i|-|-|-|
 
E |i|-|-|-|
 +
 
</pre>
 
</pre>
 
|
 
|
Line 393: Line 495:
 
D/G |-|-|-|-|-|
 
D/G |-|-|-|-|-|
 
A/D |i|-|-|-|-|
 
A/D |i|-|-|-|-|
 +
 +
</pre>
 +
|
 +
<pre>
 +
B/E |i|-|
 +
G/B |-|-|
 +
D/G |-|-|
 +
A/D |-|-|
 +
E/A |-|m|
 
</pre>
 
</pre>
 
|}
 
|}
Line 407: Line 518:
 
=== Perfect 5th ===
 
=== Perfect 5th ===
 
{{Crumbs|blue|7
 
{{Crumbs|blue|7
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of [[#Perfect 4th|P4]]
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 7 || [[#Perfect 4th|{{P4|P4}}]] || 3:2 = 1.5        || 1.49831 || 701.96 || 700
 +
|}
  
 
{|
 
{|
Line 484: Line 600:
 
=== Minor 6th ===
 
=== Minor 6th ===
 
{{Crumbs|blue|8
 
{{Crumbs|blue|8
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of [[#Major 3rd|M3]]
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 8 || [[#Major 3rd|{{M3|M3}}]] || 8:5 = 1.6        || 1.58740 || 813.69 || 800
 +
|}
  
 
{|
 
{|
Line 553: Line 674:
 
=== Major 6th ===
 
=== Major 6th ===
 
{{Crumbs|blue|9
 
{{Crumbs|blue|9
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of [[#Minor 3rd|m3]]
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 9 || [[#Minor 3rd|{{Mn3|m3}}]] || 5:3 = 1.666...    || 1.68179 || 884.36 || 900
 +
|}
  
 
{|
 
{|
Line 574: Line 700:
 
* |i|-|-|-|-|
 
* |i|-|-|-|-|
 
   
 
   
 +
  
 
* all but G-B
 
* all but G-B
Line 582: Line 709:
 
A/D |-|-|
 
A/D |-|-|
 
E/A |-|m|
 
E/A |-|m|
 +
  
  
Line 590: Line 718:
 
G/B |)| |-|
 
G/B |)| |-|
 
D/G |i| |i|
 
D/G |i| |i|
 +
  
  
Line 600: Line 729:
 
A/D |-|-|-|
 
A/D |-|-|-|
 
E/A |i|-|-|
 
E/A |i|-|-|
 +
 +
</pre>
 +
|
 +
<pre>
 +
E |i|-|-|-|
 +
B |-|-|-|-|
 +
G |-|-|-|-|
 +
D |-|-|-|-|
 +
A |-|-|-|-|
 +
E |-|-|-|p|
 
</pre>
 
</pre>
 
|}
 
|}
Line 614: Line 753:
 
=== Minor 7th ===
 
=== Minor 7th ===
 
{{Crumbs|blue|10
 
{{Crumbs|blue|10
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of [[#Major 2nd|M2]]
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 10 || [[#Major 2nd|{{M2|M2}}]] || 9:5 = 1.80  || 1.78180 || 1017.60
 +
|| 1,000
 +
|}
  
 
{|
 
{|
Line 635: Line 780:
 
A/D |)|
 
A/D |)|
 
E/A |i|
 
E/A |i|
 +
  
  
Line 641: Line 787:
 
<pre>
 
<pre>
 
B/E |-|m|
 
B/E |-|m|
−
G/B |)|-|
+
G/B |-|-|
 
D/G |i|-|
 
D/G |i|-|
 +
  
  
Line 653: Line 800:
 
A/D |-|-|-|-|
 
A/D |-|-|-|-|
 
E/A |i|-|-|-|
 
E/A |i|-|-|-|
 +
 +
</pre>
 +
|
 +
<pre>
 +
E |i|-|-|
 +
B |-|-|-|
 +
G |-|-|-|
 +
D |-|-|-|
 +
A |-|-|-|
 +
E |-|-|r|
 
</pre>
 
</pre>
 
|}
 
|}
Line 667: Line 824:
 
=== Major 7th ===
 
=== Major 7th ===
 
{{Crumbs|blue|11
 
{{Crumbs|blue|11
−
| Ref: Intervals#Minor 2nd|m2
+
| Ref: Intervals#Minor 2nd|{{Mn2|m2}}
−
| Ref: Intervals#Major 2nd|M2
+
| Ref: Intervals#Major 2nd|{{M2|M2}}
−
| Ref: Intervals#Minor 3rd|m3
+
| Ref: Intervals#Minor 3rd|{{Mn3|m3}}
−
| Ref: Intervals#Major 3rd|M3
+
| Ref: Intervals#Major 3rd|{{M3|M3}}
−
| Ref: Intervals#Perfect 4th|P4
+
| Ref: Intervals#Perfect 4th|{{P4|P4}}
−
| Ref: Intervals#Tritone|tt
+
| Ref: Intervals#Tritone|{{tt|tt}}
−
| Ref: Intervals#Perfect 5th|P5
+
| Ref: Intervals#Perfect 5th|{{P5|P5}}
−
| Ref: Intervals#Minor 6th|m6
+
| Ref: Intervals#Minor 6th|{{Mn6|m6}}
−
| Ref: Intervals#Major 6th|M6
+
| Ref: Intervals#Major 6th|{{M6|M6}}
−
| Ref: Intervals#Minor 7th|m7
+
| Ref: Intervals#Minor 7th|{{Mn7|m7}}
−
| Ref: Intervals#Major 7th|M7
+
| Ref: Intervals#Major 7th|{{M7|M7}}
 
}}
 
}}
−
* Inversion of [[#Minor 2nd|m2]]
+
 
 +
{|class="wikitable"
 +
! Semitones !! Inversion !! Just Intonation<br>Ratio !! Equal<br>Temperament !! JI Cents !! ET Cents
 +
|-
 +
| 11 || [[#Minor 2nd|{{Mn2|m2}}]] || 15:8 = 1.875      || 1.88775
 +
|| 1088.27 || 1,100
 +
|}
 +
 
 +
{|
 +
|-
 +
|
 +
<pre>
 +
D/G |-|m|
 +
A/D |-|-|
 +
E/A |i|-|
 +
 
 +
 
 +
 
 +
</pre>
 +
|
 +
<pre>
 +
B/E |-|-|r|
 +
G/B |-|-|-|
 +
D/G |i|-|-|
 +
 
 +
 
 +
 
 +
</pre>
 +
|
 +
<pre>
 +
B/E |-|-|-|-|p|
 +
G/B |-|-|-|-|-|
 +
D/G |-|-|-|-|-|
 +
A/D |-|-|-|-|-|
 +
E/A |i|-|-|-|-|
 +
 
 +
</pre>
 +
|
 +
<pre>
 +
E |i|-|
 +
B |-|-|
 +
G |-|-|
 +
D |-|-|
 +
A |-|-|
 +
E |-|m|
 +
</pre>
 +
|}
  
 
<pre>
 
<pre>
Line 698: Line 901:
 
  |m6|M6|m7|M7|'''''U'''''-|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|
 
  |m6|M6|m7|M7|'''''U'''''-|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|
 
  |m3|M3|P4|tt|P5|m6|M6|m7|M7|'''''U'''''-|m2|M2|m3|M3|P4|tt|P5|m6|M6|
 
  |m3|M3|P4|tt|P5|m6|M6|m7|M7|'''''U'''''-|m2|M2|m3|M3|P4|tt|P5|m6|M6|
 +
 +
|{{Mn3|m3}}|{{M3|M3}}|{{P4|P4}}|{{tt|tt}}|{{P5|P5}}|{{Mn6|m6}}|{{M6|M6}}|{{Mn7|m7}}|{{M7|M7}}|'''''{{U|U}}'''''-|{{Mn2|m2}}|{{M2|M2}}|{{Mn3|m3}}|{{M3|M3}}|{{P4|P4}}|{{tt|tt}}|{{P5|P5}}|{{Mn6|m6}}|{{M6|M6}}|
 +
|{{Mn7|m7}}|{{M7|M7}}|{{U|U}}-|{{Mn2|m2}}|{{M2|M2}}|{{Mn3|m3}}|{{M3|M3}}|{{P4|P4}}|{{tt|tt}}|{{P5|P5}}|{{Mn6|m6}}|{{M6|M6}}|{{Mn7|m7}}|{{M7|M7}}|'''''{{U|U}}'''''-|{{Mn2|m2}}|{{M2|M2}}|{{Mn3|m3}}|{{M3|M3}}|
 +
|{{tt|tt}}|{{P5|P5}}|{{Mn6|m6}}|{{M6|M6}}|{{Mn7|m7}}|{{M7|M7}}|'''''{{U|U}}'''''-|{{Mn2|m2}}|{{M2|M2}}|{{Mn3|m3}}|{{M3|M3}}|{{P4|P4}}|{{tt|tt}}|{{P5|P5}}|{{Mn6|m6}}|{{M6|M6}}|{{Mn7|m7}}|{{M7|M7}}|{{U|U}}-|
 +
|{{Mn2|m2}}|{{M2|M2}}|{{Mn3|m3}}|{{M3|M3}}|{{P4|P4}}|{{tt|tt}}|{{P5|P5}}|{{Mn6|m6}}|{{M6|M6}}|{{Mn7|m7}}|{{M7|M7}}|'''''{{U|U}}'''''-|{{Mn2|m2}}|{{M2|M2}}|{{Mn3|m3}}|{{M3|M3}}|{{P4|P4}}|{{tt|tt}}|{{P5|P5}}|
 +
|{{Mn6|m6}}|{{M6|M6}}|{{Mn7|m7}}|{{M7|M7}}|'''''{{U|U}}'''''-|{{Mn2|m2}}|{{M2|M2}}|{{Mn3|m3}}|{{M3|M3}}|{{P4|P4}}|{{tt|tt}}|{{P5|P5}}|{{Mn6|m6}}|{{M6|M6}}|{{Mn7|m7}}|{{M7|M7}}|{{U|U}}-|{{Mn2|m2}}|{{M2|M2}}|
 +
|{{Mn3|m3}}|{{M3|M3}}|{{P4|P4}}|{{tt|tt}}|{{P5|P5}}|{{Mn6|m6}}|{{M6|M6}}|{{Mn7|m7}}|{{M7|M7}}|'''''{{U|U}}'''''-|{{Mn2|m2}}|{{M2|M2}}|{{Mn3|m3}}|{{M3|M3}}|{{P4|P4}}|{{tt|tt}}|{{P5|P5}}|{{Mn6|m6}}|{{M6|M6}}|
  
 
{{top}}
 
{{top}}

Latest revision as of 23:56, 22 February 2016

An interval is the "distance" in terms of pitch between two notes. This is usually defined in terms of the number of steps between them in the 12-tone chromatic scale. The actual pitch difference depends on the tuning system used. The two relative pitches also form a ratio that determine how harmonic they are.

Pitches and Ratios

Root
Note 		Other
Note 		Semitones
from
Root to
Other 		Interval
Short
Name 		Interval
Full
Name 		Just
Intonation
Ratio 		Root
Cycles
before
resync
C C 0 U Unison 1:1 1
C D♭ 1 m2 Minor 2nd 16:15 16
C D 2 M2 (Major) 2nd 9:8 9
C E♭ 3 m3 Minor 3rd 6:5 6
C E 4 M3 Major 3rd 5:4 5
C F 5 P4 (Perfect) 4th 4:3 4
C G♭ 6 A4, d5, tt Augmented 4th/
Diminished 5th/
Tritone 		45:32 45
C G 7 P5 (Perfect) 5th 3:2 3
C A♭ 8 m6 Minor 6th 8:5 8
C A 9 M6 Major 6th 5:3 5
C B♭ 10 m7 Minor 7th 9:5 9
C B 11 M7 Major 7th 15:8 15
C C 12 U Unison or Octave 2:1 2
Ordered by Most to Least Consonant (Harmonic)
Root
Note 		Other
Note 		Semitones
from
Root to
Other 		Interval
Short
Name 		Interval
Full
Name 		Just
Intonation
Ratio 		Root
Cycles
before
resync
C C 0 U Unison 1:1 1
C C 12 U Unison or Octave 2:1 2
C G 7 P5 (Perfect) 5th 3:2 3
C F 5 P4 (Perfect) 4th 4:3 4
C E 4 M3 Major 3rd 5:4 5
C A 9 M6 Major 6th 5:3 5
C E♭ 3 m3 Minor 3rd 6:5 6
C A♭ 8 m6 Minor 6th 8:5 8
C D 2 M2 (Major) 2nd 9:8 9
C B♭ 10 m7 Minor 7th 9:5 9
C B 11 M7 Major 7th 15:8 15
C D♭ 1 m2 Minor 2nd 16:15 16
C G♭ 6 A4, d5, tt Augmented 4th/
Diminished 5th/
Tritone 		45:32 45


Comparing Just Intonation to Equal Temperament
Interval
Name 		Just
Intonation
Ratio 		Equal
Temperament 		 %
Difference 		Just
Intonation
Cents 		12-TET
Cents 		Cent
Difference
m2 16:15 = 1.0666... 1.05946 -0.67563 111.73 100 -11.73
M2 9:8 = 1.125 1.12246 -0.22560 203.91 200 -3.91
m3 6:5 = 1.2 1.18921 -0.89941 315.64 300 -15.64
M3 5:4 = 1.25 1.25992 0.79368 386.31 400 +13.69
P4 4:3 = 1.333... 1.33484 0.11299 498.05 500 +1.96
tt 45:32 = 1.40625 1.41421 0.56630 590.22 600 +9.78
P5 3:2 = 1.5 1.49831 -0.11286 701.96 700 -1.96
m6 8:5 = 1.6 1.58740 -0.78743 813.69 800 -13.69
M6 5:3 = 1.666... 1.68179 0.90757 884.36 900 +15.64
m7 9:5 = 1.8 1.78180 -1.01125 1017.60 1,000 -17.60
M7 15:8 = 1.875 1.88775 0.67993 1088.27 1,100 +11.73

Minor 2nd

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
1 M7 16:15 = 1.0666... 1.05946 111.73 100 
* |i|-|-|-|-|
* |-|-|-|-|p|

* all but G-B

B |i|-|-|-|
G |-|-|-|p|

 

D/G |-|-|-|p|
A/D |-|-|-|-|
E/A |i|-|-|-|


B/E |-|-|-|-|p|
G/B |-|-|-|-|-|
D/G |i|-|-|-|-|


G |i|-|-|
D |-|-|-|
A |-|-|-|
E |-|-|r|

B/E |i|-|
G/B |-|-|
D/G |-|-|
A/D |-|m|

|-|-|-|-|-|-|-|-|-|x|o|-|-|-|-|-|-|-|-|
|-|-|-|o|-|-|-|-|-|-|-|-|-|-|x|o|-|-|-|
|-|-|-|-|-|-|x|o|-|-|-|-|-|-|-|-|-|-|-|
|o|-|-|-|-|-|-|-|-|-|-|x|o|-|-|-|-|-|-|
|-|-|-|-|x|o|-|-|-|-|-|-|-|-|-|-|-|o|-|
|-|-|-|-|-|-|-|-|-|x|o|-|-|-|-|-|-|-|-|

Major 2nd

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
2 m7 9:8 = 1.125 1.12246 203.91 200 
* |i|-|-|-|
* |-|-|-|p|
 
* all but G-B

B |i|-|-|
G |-|-|r|
 
 

D/G |-|-|-|-|p|
A/D |-|-|-|-|-|
E/A |i|-|-|-|-|
 

G/B |i|-|
D/G |-|-|
A/D |-|-|
E/A |-|m|

B/E |i| |m|
G/B |)| |-|
D/G |)| |-|
A/D |i| |i|

|-|-|-|-|-|-|-|-|-|x|-|o|-|-|-|-|-|-|-|
|-|-|-|-|o|-|-|-|-|-|-|-|-|-|x|-|o|-|-|
|-|-|-|-|-|-|x|-|o|-|-|-|-|-|-|-|-|-|-|
|-|o|-|-|-|-|-|-|-|-|-|x|-|o|-|-|-|-|-|
|-|-|-|-|x|-|o|-|-|-|-|-|-|-|-|-|-|-|o|
|-|-|-|-|-|-|-|-|-|x|-|o|-|-|-|-|-|-|-|

Minor 3rd

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
3 M6 6:5 = 1.2 1.18920 315.64 300 
* |i|-|-|
* |-|-|r|
 
* all but G-B

B |i|-|
G |-|m|
 
 

G |i| |m|
D |)| |-|
A |)| |-|
E |i| |i|

B/E |i| |m|
G/B |)| |-|
D/G |)| |-|
A/D |i| |i|

|o|-|-|-|-|-|-|-|-|x|-|-|o|-|-|-|-|-|-|
|-|-|-|-|-|o|-|-|-|-|-|-|-|-|x|-|-|o|-|
|-|-|-|-|-|-|x|-|-|o|-|-|-|-|-|-|-|-|-|
|-|-|o|-|-|-|-|-|-|-|-|x|-|-|o|-|-|-|-|
|-|-|-|-|x|-|-|o|-|-|-|-|-|-|-|-|-|-|-|
|o|-|-|-|-|-|-|-|-|x|-|-|o|-|-|-|-|-|-|

Major 3rd

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
4 m6 5:4 = 1.25 1.25991 386.31 400 
* |i|-|
* |-|m|
 
* all but G-B

B |i| |r|
G |i| |m|
 
 

G |-|m|
D |-|-|
A |-|-|
E |i|-|

B/E |-|-|r|
G/B |-|-|-|
D/G |-|-|-|
A/D |i|-|-|

|-|o|-|-|-|-|-|-|-|x|-|-|-|o|-|-|-|-|-|
|-|-|-|-|-|-|o|-|-|-|-|-|-|-|x|-|-|-|o|
|-|-|-|-|-|-|x|-|-|-|o|-|-|-|-|-|-|-|-|
|-|-|-|o|-|-|-|-|-|-|-|x|-|-|-|o|-|-|-|
|-|-|-|-|x|-|-|-|o|-|-|-|-|-|-|-|-|-|-|
|-|o|-|-|-|-|-|-|-|x|-|-|-|o|-|-|-|-|-|

Perfect 4th

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
5 P5 4:3 = 1.333... 1.33484 498.05 500 
* |i|
* |i|

 
* all but G-B

B |-|m|
G |i|-|
 

 

G |-|-|r|
D |-|-|-|
A |-|-|-|
E |i|-|-|


B/E |-|-|-|p|
G/B |-|-|-|-|
D/G |-|-|-|-|
A/D |i|-|-|-|


B/E |i|-|-|
G/B |-|-|-|
D/G |-|-|-|
A/D |-|-|-|
E/A |-|-|r|

|-|-|o|-|-|-|-|-|-|x|-|-|-|-|o|-|-|-|-|
|-|-|-|-|-|-|-|o|-|-|-|-|-|-|x|-|-|-|-|
|-|-|-|-|-|-|x|-|-|-|-|o|-|-|-|-|-|-|-|
|-|-|-|-|o|-|-|-|-|-|-|x|-|-|-|-|o|-|-|
|-|-|-|-|x|-|-|-|-|o|-|-|-|-|-|-|-|-|-|
|-|-|o|-|-|-|-|-|-|x|-|-|-|-|o|-|-|-|-|

Tritone

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
6 itself 45:32 = 1.40625 1.41421 590.22 600 
* |-|i|
* |i|-|

 
* all but G-B

B |-|-|r|
G |i|-|-|

 
 

G |-|-|-|p|
D |-|-|-|-|
A |-|-|-|-|
E |i|-|-|-|


B/E |-|-|-|-|p|
G/B |-|-|-|-|-|
D/G |-|-|-|-|-|
A/D |i|-|-|-|-|


B/E |i|-|
G/B |-|-|
D/G |-|-|
A/D |-|-|
E/A |-|m|

|-|-|-|o|-|-|-|-|-|x|-|-|-|-|-|o|-|-|-|
|-|-|-|-|-|-|-|-|o|-|-|-|-|-|x|-|-|-|-|
|o|-|-|-|-|-|x|-|-|-|-|-|o|-|-|-|-|-|-|
|-|-|-|-|-|o|-|-|-|-|-|x|-|-|-|-|-|o|-|
|-|-|-|-|x|-|-|-|-|-|o|-|-|-|-|-|-|-|-|
|-|-|-|o|-|-|-|-|-|x|-|-|-|-|-|o|-|-|-|

Perfect 5th

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
7 P4 3:2 = 1.5 1.49831 701.96 700 
* |-|-|r|
* |i|-|-|
 

* all but G-B

B |-|-|-|p|
G |i|-|-|-|

 
 

D/G |i|-|-|-|
A/D |-|-|-|-|
E/A |-|-|-|p|



B/E |i|-|-|
G/B |-|-|-|
D/G |-|-|r|



B |i|-|-|-|-|
G |)|-|-|-|p|
D |)|-|-|-|-|
A |)|-|-|-|-|
E |i|-|-|-|-|

E |i| |r|
B |)| |-|
G |)| |-|
D |)| |-|
A |i| |i|

|-|-|-|-|o|-|-|-|-|x|-|-|-|-|-|-|o|-|-|
|-|-|-|-|-|-|-|-|-|o|-|-|-|-|x|-|-|-|-|
|-|o|-|-|-|-|x|-|-|-|-|-|-|o|-|-|-|-|-|
|-|-|-|-|-|-|o|-|-|-|-|x|-|-|-|-|-|-|o|
|-|-|-|-|x|-|-|-|-|-|-|o|-|-|-|-|-|-|-|
|-|-|-|-|o|-|-|-|-|x|-|-|-|-|-|-|o|-|-|

Minor 6th

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
8 M3 8:5 = 1.6 1.58740 813.69 800 
* |-|-|-|p|
* |i|-|-|-|
 

* all but G-B

B |-|-|-|-|p|
G |i|-|-|-|-|

 
 

D/G |i|-|-|
A/D |-|-|-|
E/A |-|-|r|



B/E |i|-|
G/B |-|-|
D/G |-|m|



B/E |-|m|
G/B |-|-|
D/G |-|-|
A/D |-|-|
E/A |i|-|

|-|-|-|-|-|o|-|-|-|x|-|-|-|-|-|-|-|o|-|
|-|-|-|-|-|-|-|-|-|-|o|-|-|-|x|-|-|-|-|
|-|-|o|-|-|-|x|-|-|-|-|-|-|-|o|-|-|-|-|
|-|-|-|-|-|-|-|o|-|-|-|x|-|-|-|-|-|-|-|
|o|-|-|-|x|-|-|-|-|-|-|-|o|-|-|-|-|-|-|
|-|-|-|-|-|o|-|-|-|x|-|-|-|-|-|-|-|o|-|

Major 6th

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
9 m3 5:3 = 1.666... 1.68179 884.36 900 
* |-|-|-|-|p|
* |i|-|-|-|-|
 


* all but G-B

D/G |i|-|
A/D |-|-|
E/A |-|m|




B/E |i| |m|
G/B |)| |-|
D/G |i| |i|




B/E |-|-|r|
G/B |-|-|-|
D/G |-|-|-|
A/D |-|-|-|
E/A |i|-|-|


E |i|-|-|-|
B |-|-|-|-|
G |-|-|-|-|
D |-|-|-|-|
A |-|-|-|-|
E |-|-|-|p|

|-|-|-|-|-|-|o|-|-|x|-|-|-|-|-|-|-|-|o|
|-|-|-|-|-|-|-|-|-|-|-|o|-|-|x|-|-|-|-|
|-|-|-|o|-|-|x|-|-|-|-|-|-|-|-|o|-|-|-|
|-|-|-|-|-|-|-|-|o|-|-|x|-|-|-|-|-|-|-|
|-|o|-|-|x|-|-|-|-|-|-|-|-|o|-|-|-|-|-|
|-|-|-|-|-|-|o|-|-|x|-|-|-|-|-|-|-|-|o|

Minor 7th

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
10 M2 9:5 = 1.80 1.78180 1017.60 1,000 
D/G |i|
A/D |)|
E/A |i|




B/E |-|m|
G/B |-|-|
D/G |i|-|




B/E |-|-|-|p|
G/B |-|-|-|-|
D/G |-|-|-|-|
A/D |-|-|-|-|
E/A |i|-|-|-|


E |i|-|-|
B |-|-|-|
G |-|-|-|
D |-|-|-|
A |-|-|-|
E |-|-|r|

|-|-|-|-|-|-|-|o|-|x|-|-|-|-|-|-|-|-|-|
|o|-|-|-|-|-|-|-|-|-|-|-|o|-|x|-|-|-|-|
|-|-|-|-|o|-|x|-|-|-|-|-|-|-|-|-|o|-|-|
|-|-|-|-|-|-|-|-|-|o|-|x|-|-|-|-|-|-|-|
|-|-|o|-|x|-|-|-|-|-|-|-|-|-|o|-|-|-|-|
|-|-|-|-|-|-|-|o|-|x|-|-|-|-|-|-|-|-|-|

Major 7th

Semitones Inversion Just Intonation
Ratio 		Equal
Temperament 		JI Cents ET Cents
11 m2 15:8 = 1.875 1.88775 1088.27 1,100 
D/G |-|m|
A/D |-|-|
E/A |i|-|




B/E |-|-|r|
G/B |-|-|-|
D/G |i|-|-|




B/E |-|-|-|-|p|
G/B |-|-|-|-|-|
D/G |-|-|-|-|-|
A/D |-|-|-|-|-|
E/A |i|-|-|-|-|


E |i|-|
B |-|-|
G |-|-|
D |-|-|
A |-|-|
E |-|m|

|-|-|-|-|-|-|-|-|o|x|-|-|-|-|-|-|-|-|-|
|-|o|-|-|-|-|-|-|-|-|-|-|-|o|x|-|-|-|-|
|-|-|-|-|-|o|x|-|-|-|-|-|-|-|-|-|-|o|-|
|-|-|-|-|-|-|-|-|-|-|o|x|-|-|-|-|-|-|-|
|-|-|-|o|x|-|-|-|-|-|-|-|-|-|-|o|-|-|-|
|-|-|-|-|-|-|-|-|o|x|-|-|-|-|-|-|-|-|-|

All Intervals

|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|m6|M6|
|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|m3|M3|
|tt|P5|m6|M6|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|
|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|
|m6|M6|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|
|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|m6|M6|

|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|m6|M6|
|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|m3|M3|
|tt|P5|m6|M6|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|
|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|
|m6|M6|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|
|m3|M3|P4|tt|P5|m6|M6|m7|M7|U-|m2|M2|m3|M3|P4|tt|P5|m6|M6|
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