A harmony is independent of the melody if it is often doing something different from the melody. Even if it is not independent enough to be counterpoint, such harmony adds more depth and interest to the music than drones, parallel harmonies, or simple chordal accompaniments. So this type of harmony is extremely popular for hymns and other choral arrangements, and it is also very common in instrumental music and in instrumental accompaniments.
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If it often has different rhythms than the melody, it is independent.
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Even if it has the same rhythms as the melody, it is independent if it is often moving in a different direction from the melody; for example, the harmony part is going down when the melody is going up.
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If a harmony is truly independent, then even when it is moving in the same direction as the melody, it is usually moving by a different interval. For example, if the melody is going up by perfect fourth, it might go up by a single half step.
Independent harmonies are not quite counterpoint. In order to be considered true counterpoint or polyphony, the different parts must be not only independent, they must also sound like equally important melodies. Is there always a very clear line between independent harmony and counterpoint? No! Remember that all of the rules and definitions in music theory ("counterpoint", "harmony", "minor keys") were all made up to describe what good composers were already doing; they do not define what a composer is allowed to do. If the composer - or performer - likes, an independent line can easily drift back and forth between being a background, harmony part, and being so important that it becomes a countermelody.
But in much classical and popular music, there is one line that is clearly the melody. The harmonies or accompaniment parts are all clearly "background", but they still follow most of the important rules of counterpoint. The most important rule of counterpoint is that two lines should not move in parallel. In other words, when the melody goes down one step, the harmony should do something other than going down one step; it can go down by a different interval, or stay the same, but it is best if it goes up. When the melody goes up a perfect fourth, the harmony should do anything other than go up a perfect fourth. Independent harmonies also follow this rule.
For much homophonic music, following this basic rule about contrasting intervals is enough. In particular, there is a great deal of choral music (most traditional Western hymns, for example) in which all the parts have different intervals but use the same rhythms, so that everybody is singing the same word at the same time. This type of texture is sometimes called homorhythmic.
Other harmony or accompaniment parts are even more independent, and have a different rhythm from the melody also. Good examples of this are the bass line in most pop songs or the instrumental parts accompanying an opera aria. In these types of music, as well as in much jazz and symphonic music, there is one line that is clearly the melody, but the other parts aren't simply following along with the melody. They are "doing their own thing".
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You will need an audiotape or CD player.
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Choose some music with independent harmonies for your students to listen to. (There are some suggestions below.) If you have the class time for it, and you have not already covered monody, drones, chordal harmonies, and parallel harmonies, you may want to include some examples of these for contrast. Suggested recordings for these other textures can be found in those lessons.)
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Choose a song with independent harmonies for them to learn. Since this is fairly challenging, keep it simple unless you have older, musically trained students. If your students are up to the challenge, this type of harmony is not difficult to find; most SATB choral arrangements feature independent harmonies. If you would like a very simple example for young or musically inexperienced students, or if you are not experienced enough to tackle full-fledged harmonies, you may use "Train is a-Comin'", below. In this version, much of the song is in unison, with simple (but independent) harmonies in only a few places in the music. If you need to, listen to the melody, higher harmony, and lower harmony, and all the parts together.
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If you would like, arrange for an accompanist. An accompanist is not necessary for this style of singing (not even for a simple piece like "Train is a-Comin'"), but it may make things much easier or more comfortable for the singers.
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Have enough copies of the words and music for the students.
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Using the introduction above as a guide, talk with the class about independent harmonies. Introduce any definitions you want them to learn, and contrast this musical texture with any other textures you have studied or will study, including monody, drones, parallel harmonies, chordal harmonies, and counterpoint.
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Play some of your examples of music with independent harmonies or accompaniment. Ask the students to hum along with the melody the first time. Play the example again. Can they hum along with the bass line or another harmony or accompaniment part the second time? How different are the parts?
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If you have them, play some of your examples of monody, drones, parallel harmony, chordal accompaniment, and counterpoint, for contrast. If you have also studied these other textures, identify them as they are listening to them.
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If you have enough examples, play some more, asking the students to identify the pieces with independent harmonies. Can they identify the other textures as well?
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Divide the students into higher and lower voices and assign appropriate parts for the song they will sing.
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Teach each group its part (this may be done over the course of several sessions) and have them practice it alone before attempting to combine the groups.
The suggestions for recordings to look for are pretty vague because there is so much music in this category. It is very easy to find, so you should not spend a lot of time looking for specific recordings. Just make sure there is one clear and obvious melody in your selections, but that the accompaniment to it is interesting and independent of the melody. The choral selections will be more likely to be homorhythmic, so that the words can be easily understood, whereas instrumental accompaniments will tend to be even more independent.
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Almost any chorus from a Gilbert and Sullivan opera.
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Recordings of choirs singing traditional (nineteenth-century) hymns.
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If you have trouble hearing hymn harmonies, try listening to the chorales of Bach's Christmas Oratorio (Weinachts Oratorium). The chorales are not contrapuntal - the melody is clearly in the soprano part, and the different parts sing the same words at the same time - but it is unusually easy to hear that the parts are in fact quite different from each other.
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Pop music with a solo singer, a strong bass line, and interesting instrumental accompaniment.
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Most opera arias and many opera choruses.
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This is also one of the most common textures in orchestral music, particularly in classical-era and Romantic-era symphonies (Mozart, Haydn, Beethoven, Schubert, etc.) But be aware that in symphonic music, texture can change often and quickly.