Perfect Intervals

Intervals talk about the vibrational relationship between two notes. Any two notes, played in sequence in a melody or played together in harmony, have a relationship to each other that is distinctive to our ear. That relationship, called the interval, depends largely on the frequencies the two notes — in particular the ratio between the frequencies.

So, if one note has a frequency of 500 Hertz (the blue vibration below) and another note has a frequency of 1000 Hertz (the red vibration below), the ratio between the frequencies is 1000:500.

It is customary in an interval ratio to put the larger number first, so the notes are reversed in this case. It is also customary to reduce the ratio to the smallest numbers that can express the ratio, so in this case the interval ratio is 2:1.

When we hear two sounds that have a simple ratio such as 2:1, we perceive them as "harmonic" or "consonant" or "in tune". We really don't know why this happens, but if you add the two sounds from above together ...

... the valleys of the blue vibration are reinforced by the red vibration, and there is some sense of orderliness that our ear seems to like. Here is what two pitches of 1000 and 500 Hertz sound like:

Interval of 500 and 1,000 Hertz

Synthesized Flute from a Roland Sound Module.

I'm using a synthesizer that has a flute(-like) sounds for the audio samples in this section. I'll move to real flute sounds in a bit.

When we hear two sounds that have a more complex frequency ratio between them, we perceive them as more "dissonant", as having more "tension", or simply being "out of tune". If, for example, we changed the frequencies of our two pitches from 1000 and 500 Hertz to 943.876 and 500 Hertz, then we get a frequency ratio of 94.3876:50, which is far more complex than 2:1. It sounds like this:

Interval of 500Hz and 943.876Hz

Synthesized Flute from a Roland Sound Module.

A diagram showing intervals, from De Musica by Johannes Afflighemensis, 1100 CE

And finally, here is a sample of what's coming up on this and the next web pages: all the intervals within an octave. I'm playing Middle C as the low tone on the synthesizer, and descending through all the intervals within one octave:

Twelve Intervals in an Octave

Synthesized Flute from a Roland Sound Module.

Listen how the character of the intervals changes dramatically from interval to interval. Some sound beautifully consonant and other have tremendous tension.

Fundamental Note

If you close all the finger holes on a woodwind instrument and use a normally soft breath technique that avoids playing in the second register, you are playing the fundamental note of that instrument (or simply the “fundamental” of the instrument).

Playing the fundamental note on any particular Native American flute might be challenging: you might have trouble completely sealing the finger holes or the block might be adjusted so that it is difficult to play in the first register. However, those issues aside, you should be able to get the fundamental note of the flute fairly easily.

Root Note

The first note of a scale is the root note.

The root note might be the same note as the fundamental note of the flute, but there are many scales on the Native American flute that do not begin on the fundamental note.

The root note of a scale is sometimes called the “tonic” of the scale.

Octave

Two notes that have a frequency ratio of 2:1 are said to be separated by an interval of one octave. For example, an 600 Hz note is an octave above a 300 Hz note. You go up an octave by doubling the frequency and down an octave by cutting the frequency in half.

The first two notes of the song "Some-where Over the Rainbow" (the "Some" and the "-where" notes) are an octave apart. (All the songs on these interval pages that provide examples of intervals were suggested by James Oshinsky in [Oshinsky 2008]).

Most contemporary Native American flutes will get an octave interval with the fingerings for six hole flutes and for five-hole flutes.

On a Native American flute, an octave interval sounds like this (first two separate notes as in a melody, and then two notes together in harmony):

Octave Interval

Clint Goss. E minor flute of Spalted Maple by Barry Higgins.

The octave relationship is so consonant that music theory uses the same note names for two notes that are an octave apart. The note an octave above a C, F#, or Bb is still called a C, F#, or Bb.

There are various systems to denote the octave for a given note. On Flutopedia we use International Pitch Notation and add an octave number after the note. The two notes in the flute example above are E4 and E5. See Octave Notation for a lot more detail on how octaves are numbered in various systems.

The flute used on the audio example above and all the other audio examples on the pages in this section on intervals (unless otherwise mentioned) is an E minor Native American flute made of Spalted Birch by Barry Higgins of White Crow Flutes.

If you have a keyboard of any kind handy, it's easy to find octave intervals:

The black keys are clustered in groups of twos and threes. Pick any group of two black keys and press the right black key of the pair. Now find the same key in the next higher (or lower) pair of black keys. That's an octave!

Interval Names

The octave relationship is so consonant and harmonious that virtually all musical cultures recognize it as a core interval. But where does the name "octave" come from?

How a musical culture subdivides this perfect 2:1 frequency ratio into smaller intervals is central to how their music develops, sounds, and is understood by musicians:

• The early flutes from China dated between 5720 BCE and 5620 BCE played in a primary scale that divided the octave into five or seven intervals.
• The Sumerian flute at Ur dated about 2500 BCE played in a primary scale that divided the octave into seven intervals.
• A system that divides the octave into twelve steps was attributed to the Greek philosopher Pythagoras (about 570-495 BCE). However, it was later found to have been documented in Babylonian written texts as early as 1200 BCE ([West 1994]).
• The Arabic tone system from which present-day arabic scales are drawn divides the octave into 24 steps by inserting an additional tone halfway between each tone in the twelve-tone system ([Touma 1996]) (http://en.wikipedia.org/wiki/Arabian_tone_system). The system was developed originally as a 25-tone system by Abū Naṣr al-Fārābī (Arabic: أبو نصر محمد الفارابي) before 950 CE and published in The Great Book of Music (reprinted in [al-Farabi 1967]). It was refined to its present-day system of 24 steps in the late 18th or early 19th century and published by Mikhail Mishaqa (Arabic: ميخائيل مشاقة) ([Touma 1996]).

The primary systems in use in the Western classical music tradition involve dividing the octave into a seven-note scale called a diatonic scale. If you sing the first eight notes of the hymn "Joy to the world, the Lord Has Come", you are singing the Diatonic scale in descending order.

There are many ways to name the seven "scale degrees" of the diatonic scale:

• Sargam, the earliest known system, was developed as part of the East Indian classical music traditions of Hindustani music and Carnatic music. Portions of Hindu holy texts that date to 1300-1000 BCE were set to music using sargam. In its present-day form, sargam uses the names Sa, Ri or Re, Ga, Ma, Pa, Dha, and Ni for the seven notes of the diatonic scale. Each note, or "swara", is traditionally held to have originated from the sound of a different animal. Also, each swara is associated with one of the seven chakras of the body in ascending order.
• Durar Mufaṣṣalāt (Arabic: درر مفصّلات, literally "separated pearls" in English) is the Arabic system of naming notes. It uses the names dāl, rā', mīm, fā', ṣād, lām, and tā'.
• Solfège. The positions in the seven-note diatonic scale are named in a system called “solfège” and often called the "Do-Re-Mi scale". Solfège uses the names Do, Re, Mi, Fa, Sol, La, and Ti (or Si) for the seven notes of the diatonic scale. It may be heard in "Do-Re-Mi" from Rodgers and Hammerstein's score for The Sound of Music. There are many theories about the origins of solfège. The syllables may have arisen from The Hymn of St. John written by Paulus Diaconus in the 8th century. They may have come from the Arabic Durar Mufaṣṣalāt system during the middle ages, a theory proposed in [Meninski 1680].
• Byzantine music uses the syllables Pa, Vu', Ga, Di, Ke, Zo, and Ni derived from the Greek letters Alpha, Beta, Gamma, Delta, Epsilon, Zeta, and Eta (Greek: α β γ δ ε ζ η).
• Japanese music takes its syllables from the first line of an ancient poem, Iroha (Japanese: 伊呂波), written before 1079 CE. The first line of the poem in Man'yōgana, an ancient Japanese writing system, is 以呂波耳本へ止. It is written in present-day Hiragana characters as いろはにほへと and pronounced [ee roh hah nee hoh hey toh]. The syllables of this system are I, Ro, Ha, Ni, Ho, He, and To.
• Shakuhachi musical notation uses several systems for naming the degrees of the scale:
  • The "Fu Ho U" system first documented in 1608 ([Lee-RK 1988]). The syllables were chosen because their soft sound resembled the sound of a solo bamboo flute: Fu, Ho, U, E, Ya, and I ([Marett 1991], Volume 6, page 21).
  • The "Ro Tsu Re" system that is used by present-day shakuhachi players. The syllables are Ro, Tsu, Re, Chi, Ha, and Ri or Hi.
• Javanese musicians in Indonesia use a number-based system of syllables: Ji, Ro, Lu, Pi, Ma, and Nem. These are short for the words Siji, Loro, Telu, Papat, Lima, and Enem. Note that Pi (papat) is usually dropped becaues of the Javanese use of pentatonic scales.
• The Canntaireachd system is used to convey melodies in Scottish bagpipe music. Its a rather complex system of syllables based on Scottish Gaelic phonetics that uses vowels to represent pitch and consonants to indicate ornaments.
• Degree Names. The most straightforward way to name the steps in a diatonic scale is by their numerical position: First (Root), Second, Third, Fourth, Fifth, Sixth, and Seventh, and Eighth. The name "octave" comes from the root for the word "eight", since the octave note is really the "Eighth" note in this series. Scale degrees like this extend beyond the Eight into the next octave: Ninth, Tenth, Eleventh, etcetera.
• Arabic numerals are sometimes used with carats above them: , , , , etcetera. However, there are no convenient characters (not even in Unicode), so you have to use a special font or (as I have done here) import graphics. This gets messy.
• Roman numerals are often used in jazz notation: I, II, III, IV, etcetera. However, jazz music often uses these numerals to indicates chords rather than scale degrees, so the terminology is confusing.
• Function names. In this system, the notes in the seven-tone diatonic scale are named by their musical function: Tonic, Supertonic, Mediant, Subdominant, Dominant, Submediant, Subtonic or "Leading tone". These names come from a system where the tonic is the center of the scale, and the other scale degrees are built around the tonic:
  • The supertonic is one degreee above the tonic and the subtonic is one degreee below the tonic.
  • The mediant is two degreees above the tonic and the submediant is two degreees below the tonic.
  • The dominant is four degreees above the tonic and the subdominant is four degrees below the tonic (and, nicely, the subdominant is the degree below the dominant).

In practice, musicians are likely to encounter a mixture of the solfège, degree numbers, roman numerals, and function names in their musical travels. Flutopedia primarily uses the degree numbers, because it is easily extended to naming the notes of the twelve-tone chromatic that is the core of music theory in the Western classical music tradition. However, the solfège system is very well suited to singing melodies.

Unison

Before we look at more complex intervals, lets look at two sounds that have the same frequency. This interval is called a "unison".

It might sound trite to name an interval that has a 1:1 frequency ratio, but unisons are powerful in music. The first two notes of "My Country 'tis of Thee" are in unison.

Two instruments playing the same notes of a melody are called "doubling the melody", and more instruments are called a chorus. Here is a phrase on a Native American flute that is played solo, then doubled, then played with three and finally four flutes, and then backing down one by one to a single flute:

A Simple Melody Played in Unison

Clint Goss. E minor flute of Spalted Maple by Barry Higgins.

The root note is "Do" (or "Doh") in the solfège system and the "tonic" in the system of function names.

In the sargam system of East Indian musical tradition, this note is "Shadja" (shortened to "Sa"; Sanskrit: षड्जं). It originated from the peacock and is related to the chakra at the base of the spine (Sanskrit: मूलाधार, “mūlādhāra”).

Perfect Fifth

Two notes that have a frequency ratio of 3:2 are said to be separated by an interval of a "perfect fifth", or just a "fifth". For example, a 450 Hz note is a (perfect) fifth above a 300 Hz note.

The reason for the term "perfect" is clear when you hear the interval … the perfect fifth is usually perceived as "solid" and "pure", and is often used when you want to evoke the feeling of a "call over a long distance".

The first two "twinkles" in "Twinkle, Twinkle Little Star" are a perfect fifth apart. The first two notes of the song "Feelings" descend a perfect fifth.

Most contemporary six-hole Native American flutes will get a perfect fifth interval with the fingerings . Other pairs of perfect fifth intervals: and . The equivalent fingerings on five-hole flutes are and and .

Root Note	Perfect Fifth
A	E
Bb (A#)	F
B	F# (Gb)
C	G
C# (Db)	G# (Ab)
D	A
Eb (D#)	Bb (A#)
E	B
F	C
F# (Gb)	C# (Db)
G	D
G# (Ab)	Eb (D#)

The table on the right shows the pairs of notes that are a perfect fifth apart.

This is how these three interval leaps of a perfect fifth are written in Nakai Tablature:

On a Native American flute, here are some perfect fifth intervals in melody and harmony:

Perfect Fifth Intervals - Melody and Harmony

Clint Goss. E minor flute of Spalted Maple by Barry Higgins.

On a keyboard, a perfect fifth is easy to locate:

In the groups of two and three black keys, pick the right key of any pair. Now play the rightmost key in the next higher group of three black keys. That's a fifth!

The upper note of a pair that are a perfect fifth apart is the "Sol" in the solfège system and the "dominant" in the system of function names.

In the sargam system of East Indian musical tradition, this note is "Panchama" (shortened to "Pa"; Sanskrit: पंचमं). It originated from the cuckoo or nightingale and is related to the throat chakra (Sanskrit: विशुद्ध, “viśuddha”).

The present-day system for the shakuhachi flute uses the name "Chi" for this note - a word that literally means "air" or "breath", but which is usually used in a larger sense to mean English notion of "energy flow" or the Vedantic concept of "prāṇa" (Sanskrit: प्राण), a vital, life-sustaining force.

Perfect Fourth

The next interval we will look at has a frequency ratio of 4:3 and is called a "perfect fourth ", or just a "fourth". For example, a 400 Hz note is a (perfect) fourth above a 300 Hz note.

This interval also carries the term "perfect" because it has a similar feel (but, to most ears, somewhat less powerful) than a perfect fifth.

The first two notes of "Here Comes the Bride" are a perfect fourth apart.

Most contemporary six-hole Native American flutes will get a perfect fourth interval with the fingerings . Other pairs of perfect fourth intervals: and . The equivalent fingerings on five-hole flutes are and and .

Root Note	Perfect Fourth
A	D
Bb (A#)	Eb (D#)
B	E
C	F
C# (Db)	F# (Gb)
D	G
Eb (D#)	G# (Ab)
E	A
F	Bb (A#)
F# (Gb)	B
G	C
G# (Ab)	C# (Db)

The table on the right shows the pairs of notes that are a perfect fourth apart.

This is how these three interval leaps of a perfect fourth are written in Nakai Tablature:

On a Native American flute, here are some perfect fourth intervals in melody and harmony:

Perfect Fourth Intervals - Melody and Harmony

Clint Goss. E minor flute of Spalted Maple by Barry Higgins.

On a keyboard, a perfect fourth is easy to locate:

In the groups of two and three black keys, pick the right key of any pair. Now play the center key in the next higher group of three black keys. That's a perfect fourth.

The upper note of a pair that are a perfect fifth apart is the "Fa" in the solfège system and the "Subdominant" in the system of function names.

In the sargam system, this note is "Madhyama" (shortened to "Ma"; Sanskrit: मध्यमं). It originated from the dove or heron and is related to the heart chakra (Sanskrit: अनाहत, “anāhata”).