チューバ
チューバが上手くならない. 上手くなるには練習せねばならない. 練習するにはカラオケへ行かねばならない. カラオケ代もバカにはならない. ということでチューバ1について色々と考えたことをまとめる.
チューバの基本概念
チューバとは, 大雑把に言えば 倍音の倍数と運指から音が決まる写像である. 当然奏法だったり色々音色を変える術はあるだろうが, 音だけで言えば非常に物理的な性質を持った楽器である2.
倍音
チューバの本質は, 両端が開いた管である. 管は 固有振動数 \(\omega_n\) を持ち, マウスピースでの唇の振動がそれと一致したときに, 管内で定常波となって共鳴する. 倍音の倍数 \(n\) に対し, 出る音の \(\sf{B_2}\) からの音高差 \(m\) [半音] は,
\[m = 12 \log_2 n\]と表される.
倍数 \(n = 1, \dots, 16\) について, \(m\) とそれに対応する音 \(T\) についてまとめた音が次の表3である. 一般的に用いられる倍音 \(n=2,3,4,5,6,8,\dots\) は音との音高差が比較的近いことがわかる.
| 倍音 \(n\) [倍] | \(\sf{B_2}\) との音高差 \(m\) [半音] | 音 \(T\) [半音] |
|---|---|---|
| \(1\) | \(0.00\) | \(\sf{B_2} \pm 0.00\) |
| \(2\) | \(12.00\) | \(\sf{B_1} \pm 0.00\) |
| \(3\) | \(19.02\) | \(\sf{F} + 0.02\) |
| \(4\) | \(24.00\) | \(\sf{B} \pm 0.00\) |
| \(5\) | \(27.86\) | \(\sf{d} - 0.14\) |
| \(6\) | \(31.02\) | \(\sf{f} + 0.02\) |
| \(7\) | \(33.69\) | \(\sf{a} - 0.31\) |
| \(8\) | \(36.00\) | \(\sf{b} \pm 0.00\) |
| \(9\) | \(38.04\) | \(\sf{c^1} + 0.04\) |
| \(10\) | \(39.86\) | \(\sf{d^1} - 0.14\) |
| \(11\) | \(41.51\) | \(\sf{e^1} - 0.49\) |
| \(12\) | \(43.02\) | \(\sf{f^1} + 0.02\) |
| \(13\) | \(44.41\) | \(\sf{g^1} - 0.59\) |
| \(14\) | \(45.69\) | \(\sf{as^1} - 0.31\) |
| \(15\) | \(46.88\) | \(\sf{a^1} - 0.12\) |
| \(16\) | \(48.00\) | \(\sf{b^1} \pm 0.00\) |
また, 次の図は \(n\) と \(m\) のグラフにチューバの倍音列を重ねたものである. 五線譜は線の間隔が一定ではないために無理がある図であるが, グラフと概ね一致していることがわかる.
運指
倍音に無い音を出すために, チューバには4個のロータリーバルブが付いている. 簡単のため, \(i\) 番ロータリーバルブの状態を \(F_i \in \{0,1\}\) で表し, \(F = F_2 \times 2^0 + F_1 \times 2^1 + F_3 \times 2^2 + F_4 \times 2^3\) とする. \(i\) 番レバーを押したとき, つまり \(F_i = 1\) のとき, 息の経路に \(i\) 番管が追加され, 管長が伸び, 固有振動数が大きくなる. \(F\) について, \(0\), \(1\), \(2\), \(3\) or \(4\), \(5\), \(6\) or \(8\), \(7\) or \(9\), \(10\), \(11\) or \(12\), \(13\), \(14\), \(15\) の順に音は半音ずつ下がっていく. それを利用することで, 倍音に無い音を出すことができる.
運指対応表(抜粋)
実用的な範囲4での \((n, F) \mapsto T\) の対応は以下.
| \(F\) | 運指 | \(n=\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) |
|---|---|---|---|---|---|---|---|---|---|---|
| \(0\) | ○○○○ | \(\sf{B_1}\) | \(\sf{F}\) | \(\sf{B}\) | \(\sf{d}\) | \(\sf{f}\) | \(\sf{b}\) | |||
| \(1\) | ○●○○ | \(\sf{A_1}\) | \(\sf{E}\) | \(\sf{A}\) | \(\sf{des}\) | \(\sf{e}\) | \(\sf{a}\) | |||
| \(2\) | ●○○○ | \(\sf{As_1}\) | \(\sf{Es}\) | \(\sf{As}\) | \(\sf{c}\) | \(\sf{es}\) | \(\sf{as}\) | |||
| \(3,4\) | ●●○○ or ○○●○ | \(\sf{G_1}\) | \(\sf{D}\) | \(\sf{G}\) | \(\sf{H}\) | \(\sf{d}\) | \(\sf{g}\) | \(\sf{a}\) | ||
| \(5\) | ○●●○ | \(\sf{Ges_1}\) | \(\sf{Des}\) | \(\sf{Ges}\) | \(\sf{des}\) | \(\sf{ges}\) | \(\sf{b}\) | |||
| \(6,8\) | ●○●○ or ○○○● | \(\sf{F_1}\) | \(\sf{C}\) | \(\sf{F}\) | \(\sf{A}\) | \(\sf{c}\) | \(\sf{f}\) | |||
| \(7,9\) | ●●●○ or ○●○● | \(\sf{E_1}\) | \(\sf{H_1}\) | \(\sf{H}\) |
逆引き運指表(抜粋)
特定の音 \(T\) を出したいときは以下の表.
| \(n\) | \(\sf{E_1}\) | \(\sf{F_1}\) | \(\sf{Ges_1}\) | \(\sf{G_1}\) | \(\sf{As_1}\) | \(\sf{A_1}\) | \(\sf{B_1}\) | \(\sf{H_1}\) | \(\sf{C}\) | \(\sf{Des}\) | \(\sf{D}\) | \(\sf{Es}\) | \(\sf{E}\) | \(\sf{F}\) | \(\sf{Ges}\) | \(\sf{G}\) | \(\sf{As}\) | \(\sf{A}\) | \(\sf{B}\) | \(\sf{H}\) | \(\sf{c}\) | \(\sf{des}\) | \(\sf{d}\) | \(\sf{es}\) | \(\sf{e}\) | \(\sf{f}\) | \(\sf{ges}\) | \(\sf{g}\) | \(\sf{as}\) | \(\sf{a}\) | \(\sf{b}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(10\) | \(5\) | ||||||||||||||||||||||||||||||
| \(9\) | \(3,4\) | ||||||||||||||||||||||||||||||
| \(8\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||
| \(6\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||
| \(5\) | \(6,8\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||||
| \(4\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | |||||||||||||||||||||||||
| \(3\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||
| \(2\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) |
全運指表
実用を無視して, 構造上考えられる全ての \((n, F) \mapsto T\) の対応をまとめてみる.
運指対応表
黄色字以外の音は理論的な音(ずれあり).
| \(F\) | 運指 | \(n=\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) | \(13\) | \(14\) | \(15\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(0\) | ○○○○ | \(\sf{B_2}\) | \(\textcolor{yellow}{\sf{B_1}}\) | \(\textcolor{yellow}{\sf{F}}\) | \(\textcolor{yellow}{\sf{B}}\) | \(\textcolor{yellow}{\sf{d}}\) | \(\textcolor{yellow}{\sf{f}}\) | \(\sf{a}\) | \(\textcolor{yellow}{\sf{b}}\) | \(\sf{c^1}\) | \(\sf{d^1}\) | \(\sf{e^1}\) | \(\sf{f^1}\) | \(\sf{g^1}\) | \(\sf{as^1}\) | \(\sf{a^1}\) | \(\sf{b^1}\) | |
| \(1\) | ○●○○ | \(\sf{A_2}\) | \(\textcolor{yellow}{\sf{A_1}}\) | \(\textcolor{yellow}{\sf{E}}\) | \(\textcolor{yellow}{\sf{A}}\) | \(\textcolor{yellow}{\sf{des}}\) | \(\textcolor{yellow}{\sf{e}}\) | \(\sf{as}\) | \(\textcolor{yellow}{\sf{a}}\) | \(\sf{h}\) | \(\sf{des^1}\) | \(\sf{es^1}\) | \(\sf{e^1}\) | \(\sf{ges^1}\) | \(\sf{g^1}\) | \(\sf{as^1}\) | \(\sf{a^1}\) | |
| \(2\) | ●○○○ | \(\sf{As_2}\) | \(\textcolor{yellow}{\sf{As_1}}\) | \(\textcolor{yellow}{\sf{Es}}\) | \(\textcolor{yellow}{\sf{As}}\) | \(\textcolor{yellow}{\sf{c}}\) | \(\textcolor{yellow}{\sf{es}}\) | \(\sf{g}\) | \(\textcolor{yellow}{\sf{as}}\) | \(\sf{b}\) | \(\sf{c^1}\) | \(\sf{d^1}\) | \(\sf{es^1}\) | \(\sf{f^1}\) | \(\sf{ges^1}\) | \(\sf{g^1}\) | \(\sf{as^1}\) | |
| \(3,4\) | ●●○○ or ○○●○ | \(\sf{G_2}\) | \(\textcolor{yellow}{\sf{G_1}}\) | \(\textcolor{yellow}{\sf{D}}\) | \(\textcolor{yellow}{\sf{G}}\) | \(\textcolor{yellow}{\sf{H}}\) | \(\textcolor{yellow}{\sf{d}}\) | \(\sf{ges}\) | \(\textcolor{yellow}{\sf{g}}\) | \(\textcolor{yellow}{\sf{a}}\) | \(\sf{h}\) | \(\sf{des^1}\) | \(\sf{d^1}\) | \(\sf{e^1}\) | \(\sf{f^1}\) | \(\sf{ges^1}\) | \(\sf{g^1}\) | |
| \(5\) | ○●●○ | \(\sf{Ges_2}\) | \(\textcolor{yellow}{\sf{Ges_1}}\) | \(\textcolor{yellow}{\sf{Des}}\) | \(\textcolor{yellow}{\sf{Ges}}\) | \(\sf{B}\) | \(\textcolor{yellow}{\sf{des}}\) | \(\sf{f}\) | \(\textcolor{yellow}{\sf{ges}}\) | \(\sf{as}\) | \(\textcolor{yellow}{\sf{b}}\) | \(\sf{c^1}\) | \(\sf{des^1}\) | \(\sf{es^1}\) | \(\sf{e^1}\) | \(\sf{f^1}\) | \(\sf{ges^1}\) | |
| \(6,8\) | ●○●○ or ○○○● | \(\sf{F_2}\) | \(\textcolor{yellow}{\sf{F_1}}\) | \(\textcolor{yellow}{\sf{C}}\) | \(\textcolor{yellow}{\sf{F}}\) | \(\textcolor{yellow}{\sf{A}}\) | \(\textcolor{yellow}{\sf{c}}\) | \(\sf{e}\) | \(\textcolor{yellow}{\sf{f}}\) | \(\sf{g}\) | \(\sf{a}\) | \(\sf{h}\) | \(\sf{c^1}\) | \(\sf{d^1}\) | \(\sf{es^1}\) | \(\sf{e^1}\) | \(\sf{f^1}\) | |
| \(7,9\) | ●●●○ or ○●○● | \(\sf{E_2}\) | \(\textcolor{yellow}{\sf{E_1}}\) | \(\textcolor{yellow}{\sf{H_1}}\) | \(\sf{E}\) | \(\sf{As}\) | \(\textcolor{yellow}{\sf{H}}\) | \(\sf{es}\) | \(\sf{e}\) | \(\sf{ges}\) | \(\sf{as}\) | \(\sf{b}\) | \(\sf{h}\) | \(\sf{des^1}\) | \(\sf{d^1}\) | \(\sf{es^1}\) | \(\sf{e^1}\) | |
| \(10\) | ●○○● | \(\sf{Es_2}\) | \(\sf{Es_1}\) | \(\sf{B_1}\) | \(\sf{Es}\) | \(\sf{G}\) | \(\sf{B}\) | \(\sf{d}\) | \(\sf{es}\) | \(\sf{f}\) | \(\sf{g}\) | \(\sf{a}\) | \(\sf{b}\) | \(\sf{c^1}\) | \(\sf{des^1}\) | \(\sf{d^1}\) | \(\sf{es^1}\) | |
| \(11,12\) | ●●○● or ○○●● | \(\sf{D_2}\) | \(\sf{D_1}\) | \(\sf{A_1}\) | \(\sf{D}\) | \(\sf{Ges}\) | \(\sf{A}\) | \(\sf{des}\) | \(\sf{d}\) | \(\sf{e}\) | \(\sf{ges}\) | \(\sf{as}\) | \(\sf{a}\) | \(\sf{h}\) | \(\sf{c^1}\) | \(\sf{des^1}\) | \(\sf{d^1}\) | |
| \(13\) | ○●●● | \(\sf{Des_2}\) | \(\sf{Des_1}\) | \(\sf{As_1}\) | \(\sf{Des}\) | \(\sf{F}\) | \(\sf{As}\) | \(\sf{c}\) | \(\sf{des}\) | \(\sf{es}\) | \(\sf{f}\) | \(\sf{g}\) | \(\sf{as}\) | \(\sf{b}\) | \(\sf{h}\) | \(\sf{c^1}\) | \(\sf{des^1}\) | |
| \(14\) | ●○●● | \(\sf{C_2}\) | \(\sf{C_1}\) | \(\sf{G_1}\) | \(\sf{C}\) | \(\sf{E}\) | \(\sf{G}\) | \(\sf{H}\) | \(\sf{c}\) | \(\sf{d}\) | \(\sf{e}\) | \(\sf{ges}\) | \(\sf{g}\) | \(\sf{a}\) | \(\sf{b}\) | \(\sf{h}\) | \(\sf{c^1}\) | |
| \(15\) | ●●●● | \(\sf{H_3}\) | \(\sf{H_2}\) | \(\sf{Ges_1}\) | \(\sf{H_1}\) | \(\sf{Es}\) | \(\sf{Ges}\) | \(\sf{B}\) | \(\sf{H}\) | \(\sf{des}\) | \(\sf{es}\) | \(\sf{f}\) | \(\sf{ges}\) | \(\sf{as}\) | \(\sf{a}\) | \(\sf{b}\) | \(\sf{h}\) |
逆引き運指表
黄色字以外の音は理論的な音(ずれあり).
| \(n\) | \(\sf{H_3}\) | \(\sf{C_2}\) | \(\sf{Des_2}\) | \(\sf{D_2}\) | \(\sf{Es_2}\) | \(\sf{E_2}\) | \(\sf{F_2}\) | \(\sf{Ges_2}\) | \(\sf{G_2}\) | \(\sf{As_2}\) | \(\sf{A_2}\) | \(\sf{B_2}\) | \(\sf{H_2}\) | \(\sf{C_1}\) | \(\sf{Des_1}\) | \(\sf{D_1}\) | \(\sf{Es_1}\) | \(\sf{E_1}\) | \(\sf{F_1}\) | \(\sf{Ges_1}\) | \(\sf{G_1}\) | \(\sf{As_1}\) | \(\sf{A_1}\) | \(\sf{B_1}\) | \(\sf{H_1}\) | \(\sf{C}\) | \(\sf{Des}\) | \(\sf{D}\) | \(\sf{Es}\) | \(\sf{E}\) | \(\sf{F}\) | \(\sf{Ges}\) | \(\sf{G}\) | \(\sf{As}\) | \(\sf{A}\) | \(\sf{B}\) | \(\sf{H}\) | \(\sf{c}\) | \(\sf{des}\) | \(\sf{d}\) | \(\sf{es}\) | \(\sf{e}\) | \(\sf{f}\) | \(\sf{ges}\) | \(\sf{g}\) | \(\sf{as}\) | \(\sf{a}\) | \(\sf{b}\) | \(\sf{h}\) | \(\sf{c^1}\) | \(\sf{des^1}\) | \(\sf{d^1}\) | \(\sf{es^1}\) | \(\sf{e^1}\) | \(\sf{f^1}\) | \(\sf{ges^1}\) | \(\sf{g^1}\) | \(\sf{as^1}\) | \(\sf{a^1}\) | \(\sf{b^1}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(16\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(15\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(14\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(13\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(12\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(11\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(10\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(6,8\) | \(\textcolor{yellow}{5}\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(9\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(6,8\) | \(5\) | \(\textcolor{yellow}{3,4}\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(8\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(\textcolor{yellow}{7,9}\) | \(\textcolor{yellow}{6,8}\) | \(\textcolor{yellow}{5}\) | \(\textcolor{yellow}{3,4}\) | \(\textcolor{yellow}{2}\) | \(\textcolor{yellow}{1}\) | \(\textcolor{yellow}{0}\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(7\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(6\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(\textcolor{yellow}{7,9}\) | \(\textcolor{yellow}{6,8}\) | \(\textcolor{yellow}{5}\) | \(\textcolor{yellow}{3,4}\) | \(\textcolor{yellow}{2}\) | \(\textcolor{yellow}{1}\) | \(\textcolor{yellow}{0}\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(5\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(\textcolor{yellow}{6,8}\) | \(5\) | \(\textcolor{yellow}{3,4}\) | \(\textcolor{yellow}{2}\) | \(\textcolor{yellow}{1}\) | \(\textcolor{yellow}{0}\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(4\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(\textcolor{yellow}{6,8}\) | \(\textcolor{yellow}{5}\) | \(\textcolor{yellow}{3,4}\) | \(\textcolor{yellow}{2}\) | \(\textcolor{yellow}{1}\) | \(\textcolor{yellow}{0}\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(3\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(\textcolor{yellow}{7,9}\) | \(\textcolor{yellow}{6,8}\) | \(\textcolor{yellow}{5}\) | \(\textcolor{yellow}{3,4}\) | \(\textcolor{yellow}{2}\) | \(\textcolor{yellow}{1}\) | \(\textcolor{yellow}{0}\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(2\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(\textcolor{yellow}{7,9}\) | \(\textcolor{yellow}{6,8}\) | \(\textcolor{yellow}{5}\) | \(\textcolor{yellow}{3,4}\) | \(\textcolor{yellow}{2}\) | \(\textcolor{yellow}{1}\) | \(\textcolor{yellow}{0}\) | ||||||||||||||||||||||||||||||||||||||||||||||||
| \(1\) | \(15\) | \(14\) | \(13\) | \(11,12\) | \(10\) | \(7,9\) | \(6,8\) | \(5\) | \(3,4\) | \(2\) | \(1\) | \(0\) |
本当に出るかは試してないし, そんな実力も無いので未検証. あとこんなこと考えても上達はしないので, 大人しく練習するべき. します.