A musical interval is a measure of the distance between two pitches. Mathematically, it can be seen as the ratio of the frequencies of the pitches. For example, a note that is higher than another note by the interval of one octave will have a frequency twice the frequency of the other tone.
Intervals can be any size, and are often measured in "cents", which are 1/1200th of an octave. That makes a standard western semitone, like the interval between F and F#, equal to 100 cents. This makes it easy to compare intervals to each other and to the prevailing tuning system at the same time.
Most of the time, when talking theory about them, intervals are resolved to be between a unison (1/1, 0 cents) and an octave (2/1, 1200 cents) so that they can easily be compared to each other. PolyHarp does not constrain you to this range. More about that in the Chord Bar section.
Intervals are used for describing chords in a Chord Bar, and how that chord may be transposed relative to the Base Key.
A chord is made out of a set of intervals, for example, a Major (M) chord in 12 EDO is I, III, V, and that chord may be used in a chord bar by transposing those intervals with another interval, like IM, VM, IV7, etc. Those chords themselves get "realized" into sets of pitches when the Base Key is specified. Thus , if the Base Key is A, IM becomes AM, V7 becomes E7, VIm becomes F#m, etc.
Another place Intervals are used is to transpose the Base Key. The Base Key can easily be transposed up or down by a preset interval, so that you can make a small, but complicated set of chords and transpose them to a related set of chords really quickly.
Another place is in specifying the repeat interval of a chord bar, which you can read about elsewhere.
PolyHarp uses a special notation to help you describe an interval. Intervals are used in PolyHarp when creating Chords Types, the Chord Bars, and retuning. Here are the ways you can describe an interval in PolyHarp:
Interval degree terms. Many of these terms are enharmonic: they describe the same interval, but in different keys. These resolve to a 12EDO intonation system. These term names can be in upper or lower case! To make typing easier, sharp can be expressed with '#' or 's', while flat can be expressed 'b' or 'f'.
| Term | cents | description |
|---|---|---|
| I | 0 cents | the root interval (no transposition!) |
| I♯ | 100 cents | n augmented root |
| II♭ | 100 cents | a diminished or minor 2nd |
| II | 200 cents | a major second |
| II♯ | 300 cents | an augmented second |
| III♭ | 300 cents | a minor third |
| III | 400 cents | a major third |
| III♯ | 500 cents | an augmented third |
| IV♭ | 400 cents | a diminished fourth |
| IV | 500 cents | a fourth |
| IV♯ | 600 cents | an augmented fourth |
| V♭ | 600 cents | a diminished fifth |
| V | 700 cents | a perfect fifth |
| V♯ | 800 cents | an augmented fifth |
| VI♭ | 800 cents | a diminished or minor sixth |
| VI | 900 cents | a sixth |
| VI♯ | 1000 cents | an augmented sixth |
| VII♭ | 1000 cents | a diminished or minor seventh |
| VII | 1000 cents | a seventh |
| VII♯ | 1100 cents | an augmented seventh |
| I♭ | 1100 cents | a diminished root |
| off,? | The interval is 0, or in other words, it turns it off. |
A just ratio is specified as a fraction like 4/3, 7/4 or 12/1. You can be a little lazy in PolyHarp, and specify a harmonic as N/, like 15/ , meaning 15/1.You can actually use non-integers here also, like 4.5/3.
550.0, 1330.234. Remember, these are intervals, not frequencies!
:), followed by the number of equal divisions. For example:
3:19, and would be equivalent to 189.47 cents.22:53, (498.11 cents). Again, you are not restricted to integers, so 4.5:12 is fine (and equivalent to 9:24, but you may want to use decimals instead for clarity). 3:4@7/6, which is the third degree of a 7/6 divided into 4 equal parts (and at 200.15 cents, darned close to II (at 200 cents)).0:19 instead of I, or 0 or 1/1, which are also names for a root.
Unlike normal numerals in base 10, which add up successive multiples of powers of 10 , this notation multiplies successive powers of primes. Each prime's power is separated by a ";", (ugly, but it's on the numeric keyboard), and which prime it is increases from right to left, the way powers of 10 increase right to left in decimal notation.
PolyHarp supports primes up to 127, which is rather a lot, considering consonance is happiest with small primes.
You don't have to type "0" if the number has no factors for that prime. Also, you can use negative powers and real numbers for some truly perverse intervals.
A side effect of this is that a simple ";" is how to specify "0" (;0 would be 1), and there is no way to create negative numbers.
So, for example, here are some popular numbers expressed this way:
Decimal 2 3 4 5 6 7 8 9 10 11 12 Power ;1 1;0 ;2 1;0;0 1;1 1;0;0;0 ;3 2;0 1;0;1 1;0;0;0;0 1;2 Or... ;1 1; ;2 1;; 1;1 1;;; ;3 2; 1;;1 1;;;; 1;2
and here are some more examples showing fractions and roots:
Decimal 3/2 cube root of 2 5/7 9/8 27/64 Power 1;-1 ;0.333333 -1;1;; 2;-3 3;-6
Remember, this is just another way to express a number, so you don't have to figure out factors between just ratios that have been transposed. Using a negative number in a numerator is like putting its absolute value as a denominator.
For example, to transpose 3/2 (1;/;1 or 1;-1/) by a 15/8 (1;1;/;3 or 1;1;-3/), you just add the "places" of the PoP numbers together: 1;2;/;4 or 1;2;-4/, which works out to 45/32.
Transposing by octaves just means fiddling with the 2's place (the last place).
These Pop numbers are difficult to compare or do any math with other than multiplying, but they provide some insight into comparing consonance of intervals that the decimal system doesn't!
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