The Intervals page. Paragraphs in this page are: Interval physics Interval basics Interval table Inverted intervals Intervals in octaves Adding intervals Tips Conclusion Interval physics Let's remember "The Sound Paramaters" paragraph from The Sound page and focus on the last parameter: Clarity tone or noise If a sound has a distinguashable base frequency then it is called tone, with a specific pitch, therefore position, inside the acoustic range. The acoustic range is a virtual dimension inside our mind ordering sounds from bass to tremble. If the sound is a tone, then 2 different pitches have a distance between them. Hearing the tones sequentially gives gives a direction: up or down. Hearing the tones simultaneously produces an assonance. The assonance between 2 tones generates four tones: The initial two, the addition and the subtraction. When 2 tones have a slightly different pitch, then the subtraction is out of the hearing range and we experience a "low frequency oscillation" where sound seems to raise and lower volume periodically. This acoustic phenomenon helps fine-tuning musical instruments. Interval basics Definition. Bringing the interval issue more close to music, we define it in a single way: The interval is the diatonic distance between two tones. Measuring a diatonic distance means counting the whole tones and semitones between the two notes. The interval can also be the chromatic distance, meaning only semitones, and this is also a valid criteria, though secondary and useful only when needed. We will see why it is secondary soon enough. Calculating. In my teaching, I propose a 2 stage method: Quantity At this stage, we calculate the number of the small steps used to traverse the interval. If the step is one, it can be semitone, whole tone or trisemitone: s,T,3s. Because the step is one, the notes are adjustent, so the interval is a 2nd. Now let's see bigger intervals and diatonic cases: The intervals go from bigger to smaller. All are fourths nomatter what difference the steps have, because it is the amount of steps that counts at this stage. We also see that an interval changes if its edges change: Upper tone raises, interval gets bigger. Lower tone lowers, interval gets bigger. Upper tone lowers, interval gets smaller. Lower tone raises, interval gets smaller. Here we can see clearly why the chromatic distance is not the primary tool to calculate an interval. Though B-Eb and B-D# are the same considering pitch and chromatic distance (4s) the steps differ. These intervals have a different behaviour and roles. Note: Inside the Atonal Music System the chromatic distance may as well be the primary method. The B-Eb interval belongs to C Melodic minor, or C Harmonic minor among other Scales. C minors are relevant to Eb Major (3 flats key signature). B-D# belongs to E Major, B Major or B Mixolydian 6b among other Scales. These scales belong to sharp key signature tones. We can see the big harmonic distance between these 2 intervals that "appear" the same. Quality Aside from physics, intervals generate certain qualities in Music. From ancient ages, intervals were categorized using consonance and dissonance rate. This rate is relevant to the mathematical relation between tone frequencies: The simpler the frequency, the more consonant the interval. A 1/1 is the same tone, definetly consonant. A 1/2 is an octave, pure consonant also. 2/3 generate a perfect 5th, 3/4 a perfect 4th, clearly consonant relations. 5/6 and 6/7 are consonances but not perfect and all others are considered dissonances. 1 8 5 4 Perfects are the Perfect Consonances. 3 6 Major-minor are the Inperfect Consonances. 2 7 Major-minor, plus all Augmented and Diminished are Dissonances. The 2nd minor and its inversions is the most dissonant inverval of all mentioned above, as it has the most complex relation (15/16). But this also varies considering mode-scale tuning among Music Systems, so it is not the final criteria when analyzing music issues. The way of using the interval qualities varies between places, eras, and Music Systems in general. Note: Interval assonance (vertical) differs very much from interval sequence (horizontal). While The semitone is very dissonant vertically, horizontally it presents a very distinct "attraction" and introduces a "chromatic" way of thinking, that from a simple "attitude" inside Music Systems, has evolved to a Music System by itself (Panchromaticism). Interval table Knowing about interval quantities and interval qualities, we can now understand a complete table of intervals. Usual Interval Analysis Aug - Dim Interval Analysis 1P First Perfect 2m Second minor s 1Aug First Augmented s 2M Second Major T 3Dim Third Diminished Tss 3m Third minor Ts 2Aug Second Augmented 3s 3MThird Major TT 4Dim Fourth Diminished Tss 4P Fourth Perfect TTs 4Aug Fourth Augmented (Tritone) TTT 5Dim Fifth Diminished TTss 5P Fifth Perfect TTTs 6m Sixth minor TTTss 5Aug Fifth Augmented TTTT 6M Sixth Major TTTTs 7Dim Seventh Diminished TTTsss 7m Seventh minor TTTTss 6Aug Sixth Augmented TTTTT 7M Sevent Major TTTTTs 8Dim Eighth Diminished 8P Eighth Perfect TTTTTss An interval  in the right has the same chromatic distance (therefore pitch) with the one in the left, but their diatonic distance differs, so do their role considering modes, scales or simple note steps inside them. Inverted intervals I see that you have noticed by now that a 3rd minor (Ts) and a 6th Major (TTTTs) both sum an octave (TTTTTss) The same happens between a 2nd Major and 7th minor: T + TTTTss = TTTTTss! Inverted Intervals are those that when placed sequentially result to a perfect octave. C-D and D-C (ascending both) result to an octave. 2+7=9 C-E and E-C (ascending both) result to an octave. 3+6=9 ? C-F and F-C (ascending both) result to an octave. 4+5=9 !!! Yes, 9 is the magic number when inverting intervals. Furthermore: Perfect brings Perfect, Major brings minor, Augmented brings diminished. So, the inversion of a 4th Augmented is a 5th diminished. Aside of calculating faster, inverting intervals can save memorizing a lot, because upon asked about e.g. where a 6th minor belongs, we only have to calculate where the inverted 3rd major does. Intervals in octaves C-E is a third Major, but what if E was an octave higher? We simply add 7. so in this case it is a 10th. Qualities remain the same. So: 2 gets 9 3 gets 10 4 gets 11 5 gets 12 6 gets 13 7 gets 14 When creating big chords based on triads, this issue is very common. A 7-voice triad chord may have 1,3,5,7,9,11,13. Keep this in mind to avoid confusion. Adding intervals C-E is a 3rd. E-G is a 3rd. C-G is a 5th. 3rd + 3rd = 5th! This happens because "E" is counted twice. When adding intervals, the result is the addition minus 1. So, 3rd + 6th = 9-1 = 8th. This issue also explains theoretically the inverted interval calculation. The magic number there is 9 because of the "result -1" situation. Tips When calculating steps to figure out an interval, be sure to count steps first. Just parsing notes is half the success even with the hardest interval. Gb-E ascending: G A B C D E So it is a 6th. Half the problem is solved. Practice inverting the chords in order to memorize less and gain speed. E-Gb is a 3rd diminished, so the solution to the previous question is 6th Augmented. Conclusion As we will see in all other pages, intervals are not "just another way of measuring things". Intervals are basic tools for further music system structure and understanding them helps understand bigger issues in the long run.