The character for koshi means, "back; lower back; waist; hips; lumbar region." Gamaku is an Okinawan term, Uchinaguchi or hogan dialect, that refers to the musculature the surrounds the waist and connects the torso to the pelvis. Koshi is as can be determined by the meaning, Japanese term, the lower back as it connects to the waist, hips and the lumbar region. The koshi is a smaller part of the entire region so you can surmise that the koshi is the main controller of how power is generated in karate techniques.
Even referring to gamaku as the sides vs. koshi as the lower back is a more limited explanation of gamaku. Actually, gamaku is a method in utilizing the hara in martial arts. Every action taken is believed to originate at the hara and that generates a certain amount of power, i.e. it contributes one aspect of the principle that generates power in martial arts.
When you visualize gamaku or that midsection grouping or band of muscles connecting the entire body you will also envision how that is a part of the fundamental principles of martial system specifically those found in the principle category titled "physiokinetic."
To fully utilize the system of gamaku you need to master proper breathing, posture, spinal alignment, structure, relaxation, wave energy, centeredness, body-mind, centripetal and centrifugal force; sequential locking and relaxation, and rooting. This does not mean that the other principles in physiokinetics is not utilized but these are kind of primary. This also requires in a lesser degree all the other principles, i.e. theory, technique and philosophy that feed the mind-state, etc.
All this while under the stress of the attack, the physical stresses from the chemical dump and the fluidity of the current moment, etc. Like the term "chinkuchi" it is another historical model and theory that helps describe, from a time when such descriptions lacked modern science and medicine, how things work in martial systems. Like chinkuchi and other terms it actually describes for us how we can incorporate fundamental principles of martial systems properly and completely in our practice, training and application in self-defense, defense and combatives - as appropriate to the situation.
You might hypothesize that gamaku is an Okinawan term to mean "hara." Hara being the abdomen, belly, stomach along with one's mind, intentions or true motives. When you speak of the hara it should include the band of muscles that connect the belly, the hips and sides, the back and lower back, etc.
When you make it work it should result in a pelvic tile slightly forward resulting in the alignment of the spine (can best be observed in the Tai Chi Chuan forms). It forces a student to rotate hips with side muscles as when striking with either fist. Differences will occur depending on the technique and whether it is an arm and hand technique vs. a leg technique.
You might explain it as follows, i.e. "There is a very slight pelvic tile forward which helps minimize the lumbar and thoracic curves and also facilitates efficient transfer of energy through the gamaku area. This shifts the weight and center of gravity (tanden) slightly forward increasing the power of the blow."
"As far as koshi and gamaku in karate you move with your hara. Movement with hara requires development of gamaku through movement of the koshi. Once the gamaku is developed sufficiently movement is accomplished using gamaku and any movement of the koshi is a byproduct of using gamaku."
As with principles we can look at gamaku as one of three sub-principles that work the koshi, gamaku and hara into one wholehearted model used to connect how all the principles are aligned for maximizing application of martial arts.
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