{"id":16907,"date":"2026-08-12T15:09:00","date_gmt":"2026-08-12T05:09:00","guid":{"rendered":"https:\/\/geoffreychen.com\/?p=16907"},"modified":"2026-10-10T19:54:58","modified_gmt":"2026-10-10T08:54:58","slug":"why-does-twelve-tone-equal-temperament-need-the-twelfth-root-of-two","status":"publish","type":"post","link":"https:\/\/geoffreychen.com\/zh\/why-does-twelve-tone-equal-temperament-need-the-twelfth-root-of-two\/","title":{"rendered":"\u5341\u4e8c\u5e73\u5747\u5f8b\u4e3a\u4ec0\u4e48\u9700\u8981\u5341\u4e8c\u6b21\u65b9\u6839\uff1f"},"content":{"rendered":"<\/p>\n<p><em>\u300a\u542c\u89c1\u6570\u5b66\u300b\u00b7\u201c\u58f0\u97f3\u600e\u6837\u8fdb\u5165\u8eab\u4f53\u4e0e\u5fc3\u7075\u201d\u00b7 \u7b2c\u516d\u7bc7<\/em><\/p>\n<p>\u94a2\u7434\u4e0a\u4ece\u4efb\u610f\u4e00\u4e2a\u952e\u5411\u53f3\u79fb\u52a8\u5341\u4e8c\u4e2a\u534a\u97f3\uff0c\u4f1a\u5230\u8fbe\u540c\u540d\u7684\u9ad8\u516b\u5ea6\u97f3\u3002\u82e5\u4ece A4 \u7684 440 \u8d6b\u5179\u5f00\u59cb\uff0c\u5341\u4e8c\u6b65\u540e\u662f A5 \u7684 880 \u8d6b\u5179\u3002\u952e\u76d8\u89c6\u89c9\u4e0a\u50cf\u628a\u4e00\u6bb5\u8ddd\u79bb\u5207\u6210\u5341\u4e8c\u4efd\uff0c\u4e8e\u662f\u5f88\u81ea\u7136\u5730\u4ee5\u4e3a\u6bcf\u4e00\u6b65\u53ea\u9700\u589e\u52a0 <code>(880 - 440) \/ 12<\/code> \u8d6b\u5179\u3002<\/p>\n<p>\u8fd9\u6837\u8ba1\u7b97\uff0c\u7b2c\u4e00\u6b65\u589e\u52a0 36.67 \u8d6b\u5179\uff1b\u4f46\u4ece\u66f4\u4f4e\u7684 A3 220 \u8d6b\u5179\u5f00\u59cb\u82e5\u4ecd\u589e\u52a0\u540c\u6837\u6570\u503c\uff0c\u5341\u4e8c\u6b65\u4f1a\u5230 660 \u800c\u4e0d\u662f 440\u3002\u66f4\u6839\u672c\u7684\u95ee\u9898\u662f\uff0c\u76f8\u540c\u97f3\u4e50\u97f3\u7a0b\u4e0d\u5bf9\u5e94\u76f8\u540c\u9891\u7387\u5dee\uff0c\u800c\u5bf9\u5e94\u76f8\u540c\u9891\u7387\u6bd4\u3002\u5341\u4e8c\u5e73\u5747\u5f8b\u4e2d\u7684\u201c\u5e73\u5747\u201d\uff0c\u4e0d\u662f\u5728\u8d6b\u5179\u76f4\u7ebf\u4e0a\u7b49\u8ddd\uff0c\u800c\u662f\u5728\u5bf9\u6570\u97f3\u9ad8\u7a7a\u95f4\u4e2d\u7b49\u8ddd\u3002<\/p>\n<h2>\u6bcf\u4e00\u6b65\u90fd\u4e58\u540c\u4e00\u4e2a\u6570<\/h2>\n<p>\u8bbe\u76f8\u90bb\u534a\u97f3\u4e4b\u95f4\u7684\u9891\u7387\u6bd4\u4e3a <code>r<\/code>\u3002\u8d70\u4e00\u6b65\u4e58\u4ee5 <code>r<\/code>\uff0c\u8d70\u5341\u4e8c\u6b65\u4e58\u4ee5 <code>r^12<\/code>\u3002\u4e00\u4e2a\u516b\u5ea6\u9700\u8981\u9891\u7387\u7ffb\u500d\uff0c\u6240\u4ee5\uff1a<\/p>\n<p><code>r^12 = 2<\/code><\/p>\n<p>\u56e0\u6b64\uff1a<\/p>\n<p><code>r = 2^(1\/12) \u2248 1.059463<\/code><\/p>\n<p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u6bcf\u4e0a\u5347\u4e00\u4e2a\u5e73\u5747\u5f8b\u534a\u97f3\uff0c\u9891\u7387\u589e\u52a0\u7ea6 5.9463%\uff0c\u800c\u4e0d\u662f\u589e\u52a0\u56fa\u5b9a\u8d6b\u5179\u6570\u3002\u4ece A4 440 \u8d6b\u5179\u51fa\u53d1\uff0c\u4e0b\u4e00\u4e2a A\u266f\uff0fB\u266d\u7ea6\u4e3a <code>440 \u00d7 2^(1\/12) = 466.16<\/code> \u8d6b\u5179\uff1b\u518d\u4e0b\u4e00\u6b65\u7ea6\u4e3a 493.88\uff1b\u5341\u4e8c\u6b21\u76f8\u4e58\u6b63\u597d\u5230 880\u3002<\/p>\n<p>\u4e00\u822c\u516c\u5f0f\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n<p><code>f(n) = f\u2080 \u00d7 2^(n\/12)<\/code><\/p>\n<p>\u5176\u4e2d <code>n<\/code> \u662f\u4e0e\u53c2\u8003\u97f3\u76f8\u5dee\u7684\u534a\u97f3\u6570\uff0c\u53ef\u4ee5\u4e3a\u6b63\u4e5f\u53ef\u4ee5\u4e3a\u8d1f\u3002\u7535\u5b50\u8c03\u97f3\u5668\u3001\u5408\u6210\u5668\u548c\u97f3\u4e50\u8f6f\u4ef6\u7ecf\u5e38\u4f7f\u7528\u8fd9\u79cd\u5173\u7cfb\u751f\u6210\u7406\u8bba\u76ee\u6807\u503c\u3002\u516c\u5f0f\u6781\u5176\u7b80\u6d01\uff0c\u4f46\u5b83\u63cf\u8ff0\u7684\u662f\u4e00\u79cd\u88ab\u9009\u62e9\u7684\u8c03\u5f8b\u7ed3\u6784\uff0c\u4e0d\u662f\u81ea\u7136\u754c\u552f\u4e00\u5141\u8bb8\u7684\u97f3\u9636\u3002<\/p>\n<h2>\u97f3\u5206\u628a\u6bd4\u4f8b\u53d8\u6210\u53ef\u4ee5\u76f8\u52a0\u7684\u8ddd\u79bb<\/h2>\n<p>\u97f3\u4e50\u5bb6\u8c08\u4e00\u4e2a\u97f3\u504f\u9ad8 5 \u97f3\u5206\uff0c\u5e76\u4e0d\u662f\u8bf4\u5b83\u9ad8\u4e86\u56fa\u5b9a\u8d6b\u5179\u6570\u3002\u97f3\u5206\u91c7\u7528\u5bf9\u6570\u5b9a\u4e49\uff1a<\/p>\n<p><code>cents = 1200 \u00d7 log\u2082(f\u2082\/f\u2081)<\/code><\/p>\n<p>\u56e0\u6b64\u516b\u5ea6\u662f 1200 \u97f3\u5206\uff0c\u5e73\u5747\u5f8b\u534a\u97f3\u662f 100 \u97f3\u5206\u3002\u5bf9\u6570\u6709\u4e00\u4e2a\u5173\u952e\u6027\u8d28\uff1a\u9891\u7387\u6bd4\u76f8\u4e58\uff0c\u97f3\u5206\u8ddd\u79bb\u76f8\u52a0\u3002\u4e24\u4e2a\u534a\u97f3\u662f 200 \u97f3\u5206\uff0c\u4e03\u4e2a\u534a\u97f3\u662f 700 \u97f3\u5206\uff0c\u65e0\u8bba\u4ece\u54ea\u4e2a\u9891\u7387\u8d77\u6b65\u3002<\/p>\n<p>\u8fd9\u79cd\u8868\u793a\u4e0e\u79fb\u8c03\u7684\u9700\u8981\u543b\u5408\u3002\u5982\u679c\u4e00\u6bb5\u65cb\u5f8b\u6574\u4f53\u4e0a\u79fb\u4e09\u4e2a\u534a\u97f3\uff0c\u6bcf\u4e2a\u97f3\u90fd\u4e58\u4ee5\u540c\u4e00\u4e2a <code>2^(3\/12)<\/code>\uff1b\u97f3\u4e0e\u97f3\u4e4b\u95f4\u7684\u534a\u97f3\u8ddd\u79bb\u4e0d\u53d8\uff0c\u6307\u6cd5\u5f62\u72b6\u6216\u6570\u5b57\u8868\u793a\u4e5f\u53ef\u6574\u4f53\u642c\u79fb\u3002\u5341\u4e8c\u5e73\u5747\u5f8b\u628a\u8c03\u6027\u7a7a\u95f4\u505a\u6210\u4e00\u4e2a\u9ad8\u5ea6\u89c4\u5219\u7684\u683c\u5b50\u3002<\/p>\n<p>\u4f46\u89c4\u5219\u6027\u6709\u4ee3\u4ef7\u3002\u4e03\u4e2a\u5e73\u5747\u5f8b\u534a\u97f3\u5f62\u6210\u7684\u4e94\u5ea6\u4e3a <code>2^(7\/12)<\/code>\uff0c\u63a5\u8fd1\u800c\u4e0d\u7b49\u4e8e 3:2\uff1b\u56db\u4e2a\u534a\u97f3\u5f62\u6210\u7684\u5927\u4e09\u5ea6\u4e3a <code>2^(4\/12)<\/code>\uff0c\u7ea6 1.2599\uff0c\u6bd4 5:4 \u7684 1.25 \u66f4\u660e\u663e\u5730\u5bbd\u3002\u9664\u4e86\u516b\u5ea6\uff0c\u5341\u4e8c\u5e73\u5747\u5f8b\u5927\u591a\u6570\u97f3\u7a0b\u4e0d\u662f\u7b80\u5355\u6574\u6570\u6bd4\u3002\u5b83\u7684\u6210\u5c31\u4e0d\u662f\u8ba9\u6bcf\u4e2a\u5c40\u90e8\u6700\u7eaf\uff0c\u800c\u662f\u8ba9\u540c\u4e00\u79cd\u97f3\u7a0b\u5728\u6bcf\u4e2a\u8d77\u70b9\u627f\u62c5\u76f8\u540c\u504f\u5dee\u3002<\/p>\n<h2>\u4e3a\u4ec0\u4e48\u662f\u5341\u4e8c\uff0c\u4e0d\u662f\u6570\u5b66\u5f3a\u8feb\u7684\u552f\u4e00\u7b54\u6848<\/h2>\n<p>\u5982\u679c\u53ea\u8981\u6c42\u628a\u516b\u5ea6\u5e73\u5747\u5206\uff0c\u4efb\u4f55\u6b63\u6574\u6570 <code>N<\/code> \u90fd\u53ef\u4ea7\u751f <code>2^(1\/N)<\/code> \u7684 N \u7b49\u5206\u5f8b\u3002\u5341\u4e5d\u5e73\u5747\u5f8b\u3001\u4e09\u5341\u4e00\u5e73\u5747\u5f8b\u548c\u5176\u4ed6\u7cfb\u7edf\u90fd\u5728\u7406\u8bba\u4e0e\u5b9e\u8df5\u4e2d\u5b58\u5728\u3002\u5341\u4e8c\u4e4b\u6240\u4ee5\u5360\u4e3b\u5bfc\uff0c\u6d89\u53ca\u5386\u53f2\u4e50\u5668\u3001\u8bb0\u8c31\u3001\u4e94\u5ea6\u8fd1\u4f3c\u3001\u548c\u58f0\u5b9e\u8df5\u548c\u5236\u9020\u5236\u5ea6\u957f\u671f\u5171\u540c\u5f62\u6210\u7684\u8def\u5f84\u3002<\/p>\n<p>\u5341\u4e8c\u6b65\u7684\u4e00\u4e2a\u4f18\u52bf\u662f <code>2^(7\/12)<\/code> \u5f88\u63a5\u8fd1 3:2\uff1a\u8bef\u5dee\u53ea\u6709\u7ea6 2 \u97f3\u5206\u3002\u7531\u4e94\u5ea6\u63a8\u8fdb\u5f97\u5230\u7684\u5341\u4e8c\u97f3\u7ea7\u4e5f\u80fd\u5728\u4e0d\u589e\u52a0\u952e\u6570\u7684\u60c5\u51b5\u4e0b\u95ed\u5408\u3002\u5927\u4e09\u5ea6\u8fd1\u4f3c\u4e0d\u5982\u4e94\u5ea6\u7cbe\u786e\uff0c\u5374\u5728\u8bb8\u591a\u590d\u5408\u97f3\u54cd\u548c\u8bed\u5883\u4e2d\u53ef\u63a5\u53d7\u3002\u4e8e\u662f\u5341\u4e8c\u5e73\u5747\u5f8b\u63d0\u4f9b\u4e86\u4e00\u79cd\u5f88\u6709\u7ade\u4e89\u529b\u7684\u7ec4\u5408\uff1a\u952e\u6570\u6709\u9650\u3001\u4e94\u5ea6\u826f\u597d\u3001\u6240\u6709\u8c03\u5bf9\u79f0\u3001\u8f6c\u8c03\u65b9\u4fbf\u3002<\/p>\n<p>\u4f46\u201c\u5e7f\u6cdb\u91c7\u7528\u201d\u4e0d\u7b49\u4e8e\u201c\u81ea\u7136\u5fc5\u7136\u201d\u3002\u5370\u5ea6\u53e4\u5178\u97f3\u4e50\u3001\u963f\u62c9\u4f2f\u571f\u8033\u5176\u8c03\u5f0f\u3001\u5370\u5ea6\u5c3c\u897f\u4e9a\u7518\u7f8e\u5170\u4ee5\u53ca\u5404\u79cd\u5f53\u4ee3\u5fae\u5206\u97f3\u5b9e\u8df5\uff0c\u90fd\u4ee5\u4e0d\u540c\u65b9\u5f0f\u7ec4\u7ec7\u97f3\u9ad8\u3002\u5373\u4f7f\u5728\u897f\u65b9\u53e4\u5178\u97f3\u4e50\u4e2d\uff0c\u6b4c\u5531\u4e0e\u5f26\u4e50\u5408\u594f\u4e5f\u4f1a\u504f\u79bb\u5e73\u5747\u5f8b\u56fa\u5b9a\u70b9\u3002\u5341\u4e8c\u6b21\u65b9\u6839\u8bf4\u660e\u5341\u4e8c\u5e73\u5747\u5f8b\u5185\u90e8\u600e\u6837\u4e00\u81f4\uff0c\u4e0d\u6388\u4e88\u5b83\u5bf9\u6240\u6709\u97f3\u4e50\u7684\u4e3b\u6743\u3002<\/p>\n<h2>\u8096\u90a6\u300a\u4e8c\u5341\u56db\u9996\u524d\u594f\u66f2\u300b\u8ba9\u62bd\u8c61\u5bf9\u79f0\u8fdb\u5165\u4f5c\u54c1\u987a\u5e8f<\/h2>\n<p>\u8096\u90a6 Op. 28 \u5305\u542b\u4e8c\u5341\u56db\u9996\u524d\u594f\u66f2\uff0c\u4f7f\u7528\u5168\u90e8\u5927\u8c03\u548c\u5c0f\u8c03\u3002\u5b83\u4eec\u4e0d\u662f\u6309\u534a\u97f3\u4ece C \u9010\u6b65\u8d70\u5230 B\uff0c\u800c\u662f\u628a\u6bcf\u4e2a\u5927\u8c03\u4e0e\u5176\u5173\u7cfb\u5c0f\u8c03\u76f8\u90bb\uff0c\u518d\u6cbf\u4e94\u5ea6\u5173\u7cfb\u63a8\u8fdb\uff1aC \u5927\u8c03\u3001A \u5c0f\u8c03\uff1bG \u5927\u8c03\u3001E \u5c0f\u8c03\uff1bD \u5927\u8c03\u3001B \u5c0f\u8c03\u2026\u2026\u76f4\u5230 F \u5927\u8c03\u3001D \u5c0f\u8c03\u3002<a href=\"https:\/\/imslp.org\/wiki\/Preludes%2C_Op.28_%28Chopin%2C_Fr%C3%A9d%C3%A9ric%29\" target=\"_blank\" rel=\"noopener\">\u4f5c\u54c1\u76ee\u5f55\u4e0e\u4e50\u8c31<\/a>\u6e05\u695a\u5217\u51fa\u8fd9\u5957\u4e8c\u5341\u56db\u8c03\u5e8f\u5217\u3002<\/p>\n<p>\u5728\u73b0\u4ee3\u94a2\u7434\u4e0a\uff0c\u8fd9\u4e2a\u5faa\u73af\u80fd\u591f\u7528\u4e00\u5957\u56fa\u5b9a\u952e\u4f4d\u5b8c\u6210\u3002\u5e73\u5747\u5f8b\u4f7f\u4efb\u4f55\u4e00\u4e2a\u8c03\u7684\u97f3\u7a0b\u6a21\u5f0f\u53ef\u4ee5\u79fb\u5230\u53e6\u4e00\u4e2a\u8d77\u70b9\uff0c\u4e0d\u4f1a\u9047\u5230\u4e00\u4e2a\u5fc5\u987b\u907f\u5f00\u7684\u72fc\u97f3\u533a\u57df\u3002\u4f46\u8fd9\u4e0d\u8868\u793a\u4e8c\u5341\u56db\u9996\u542c\u8d77\u6765\u53ea\u6709\u540c\u4e00\u51e0\u4f55\u5f62\u72b6\u7684\u5e73\u79fb\u3002\u952e\u76d8\u7684\u9ed1\u767d\u952e\u5e03\u5c40\u6539\u53d8\u624b\u611f\uff0c\u97f3\u533a\u4e0e\u7ec7\u4f53\u6539\u53d8\u5171\u9e23\uff0c\u6bcf\u9996\u7684\u901f\u5ea6\u3001\u8282\u594f\u3001\u65cb\u5f8b\u548c\u529b\u5ea6\u5efa\u7acb\u5b8c\u5168\u4e0d\u540c\u7684\u4e16\u754c\u3002<\/p>\n<p>\u4f8b\u5982 C \u5927\u8c03\u7b2c\u4e00\u9996\u6d41\u52a8\u800c\u4e0d\u5b89\u5b9a\u7684\u548c\u5f26\u578b\uff0c\u4e0e\u7d27\u968f\u5176\u540e\u7684 A \u5c0f\u8c03\u7b2c\u4e8c\u9996\u9634\u5f71\u822c\u7684\u4f4e\u97f3\u52a8\u4f5c\uff0c\u5df2\u7ecf\u8bf4\u660e\u201c\u5173\u7cfb\u8c03\u201d\u4e0d\u662f\u60c5\u7eea\u590d\u5236\u3002E \u5c0f\u8c03\u7b2c\u56db\u9996\u7528\u4e0b\u884c\u548c\u58f0\u4e0e\u65cb\u5f8b\u5ef6\u8fdf\u5f62\u6210\u91cd\u91cf\uff1b\u964d D \u5927\u8c03\u7b2c\u5341\u4e94\u9996\u7684\u91cd\u590d\u97f3\u5728\u957f\u7bc7\u7ed3\u6784\u4e2d\u6539\u53d8\u610f\u4e49\uff1bD \u5c0f\u8c03\u7b2c\u4e8c\u5341\u56db\u9996\u628a\u6574\u4e2a\u5faa\u73af\u63a8\u5411\u5f3a\u70c8\u7ed3\u5c3e\u3002\u5341\u4e8c\u5e73\u5747\u5f8b\u63d0\u4f9b\u53ef\u8fdb\u5165\u7684\u8c03\u6027\u5750\u6807\uff0c\u8096\u90a6\u51b3\u5b9a\u5750\u6807\u4e2d\u53d1\u751f\u4ec0\u4e48\u3002<\/p>\n<h2>\u76f8\u540c\u7ed3\u6784\u53ef\u79fb\u8c03\uff0c\u4e0d\u7b49\u4e8e\u7ed3\u679c\u5b8c\u5168\u76f8\u540c<\/h2>\n<p>\u4ece\u6570\u5b66\u4e0a\u8bf4\uff0c\u5341\u4e8c\u5e73\u5747\u5f8b\u5177\u6709\u5e73\u79fb\u5bf9\u79f0\uff1a\u628a\u6240\u6709\u97f3\u7ea7\u7f16\u53f7\u540c\u65f6\u52a0\u4e0a\u540c\u4e00\u4e2a\u6574\u6570\uff08\u6a21 12\uff09\uff0c\u97f3\u7a0b\u7c7b\u522b\u4fdd\u6301\u4e0d\u53d8\u3002C-E-G \u7684\u7f16\u53f7\u5173\u7cfb\u79fb\u5230 D-F\u266f-A\uff0c\u4ecd\u662f\u5927\u4e09\u548c\u5f26\u7ed3\u6784\u3002\u8fd9\u4f7f\u4f5c\u66f2\u5206\u6790\u3001\u6570\u5b57\u63a5\u53e3\u548c\u7b97\u6cd5\u5904\u7406\u5f88\u65b9\u4fbf\u3002<\/p>\n<p>\u53ef\u662f\u4eba\u8033\u548c\u4e50\u5668\u4e0d\u662f\u62bd\u8c61\u7684\u6a21 12 \u96c6\u5408\u3002\u79fb\u8c03\u6539\u53d8\u7edd\u5bf9\u97f3\u533a\uff0c\u53ef\u80fd\u6539\u53d8\u6b4c\u624b\u80fd\u5426\u5531\u51fa\u3001\u94a2\u7434\u662f\u5426\u66f4\u660e\u4eae\u3001\u7ba1\u4e50\u5668\u6307\u6cd5\u662f\u5426\u987a\u624b\u3001\u5f26\u4e50\u662f\u5426\u80fd\u7528\u7a7a\u5f26\u3002\u7b49\u97f3\u7684 C\u266f \u4e0e D\u266d\u5728\u5e73\u5747\u5f8b\u952e\u76d8\u4e0a\u5171\u7528\u9891\u7387\uff0c\u5374\u53ef\u80fd\u56e0\u548c\u58f0\u62fc\u5199\u548c\u8fdb\u884c\u65b9\u5411\u800c\u627f\u62c5\u4e0d\u540c\u610f\u4e49\u3002\u6570\u5b66\u5bf9\u79f0\u4fdd\u5b58\u67d0\u4e9b\u5173\u7cfb\uff0c\u4e5f\u6709\u610f\u5ffd\u7565\u5176\u4ed6\u5173\u7cfb\u3002<\/p>\n<p>\u8fd9\u6b63\u662f\u6a21\u578b\u7684\u529b\u91cf\u4e0e\u8fb9\u754c\u3002\u53ea\u8981\u95ee\u9898\u662f\u201c\u97f3\u7a0b\u5f62\u72b6\u80fd\u5426\u4ece\u4e00\u4e2a\u8c03\u642c\u5230\u53e6\u4e00\u4e2a\u8c03\u201d\uff0c\u6a21 12 \u548c\u6307\u6570\u516c\u5f0f\u975e\u5e38\u6709\u6548\uff1b\u82e5\u95ee\u9898\u662f\u201c\u79fb\u8c03\u540e\u6f14\u594f\u4e3a\u4f55\u53d8\u96be\u3001\u97f3\u8272\u4e3a\u4f55\u53d8\u5316\u3001\u8c03\u6027\u4e3a\u4f55\u4ecd\u6709\u7ecf\u9a8c\u5dee\u5f02\u201d\uff0c\u5c31\u5fc5\u987b\u628a\u8eab\u4f53\u3001\u4e50\u5668\u4e0e\u8bb0\u8c31\u91cd\u65b0\u653e\u56de\u89e3\u91ca\u3002<\/p>\n<h2>\u516c\u5f0f\u4e0d\u662f\u97f3\u4e50\u7684\u539f\u56e0\uff0c\u800c\u662f\u5236\u5ea6\u9009\u62e9\u7684\u7cbe\u786e\u8868\u8fbe<\/h2>\n<p>\u5341\u4e8c\u6b21\u65b9\u6839\u5e38\u7ed9\u4eba\u4e00\u79cd\u53d1\u73b0\u5b87\u5b99\u79d8\u5bc6\u7684\u611f\u89c9\u3002\u5b9e\u9645\u4e0a\uff0c\u5b83\u6765\u81ea\u4e09\u4e2a\u660e\u786e\u524d\u63d0\uff1a\u628a\u516b\u5ea6\u89c4\u5b9a\u4e3a 2:1\uff0c\u628a\u5b83\u5206\u4e3a\u5341\u4e8c\u4e2a\u6b65\u9aa4\uff0c\u5e76\u8981\u6c42\u6bcf\u4e00\u6b65\u9891\u7387\u6bd4\u76f8\u7b49\u3002\u4e00\u65e6\u524d\u63d0\u63a5\u53d7\uff0c\u7b54\u6848\u53ea\u80fd\u662f <code>2^(1\/12)<\/code>\u3002\u6570\u5b66\u6ca1\u6709\u66ff\u6211\u4eec\u9009\u62e9\u5341\u4e8c\uff0c\u4e5f\u6ca1\u6709\u51b3\u5b9a\u6240\u6709\u6b65\u9aa4\u5fc5\u987b\u76f8\u7b49\uff1b\u5b83\u4fdd\u8bc1\u9009\u5b9a\u5236\u5ea6\u4e0d\u4f1a\u81ea\u76f8\u77db\u76fe\u3002<\/p>\n<p>\u8fd9\u79cd\u533a\u522b\u975e\u5e38\u91cd\u8981\u3002\u97f3\u4e50\u6280\u672f\u7ecf\u5e38\u628a\u5386\u53f2\u5f62\u6210\u7684\u9009\u62e9\u5c01\u88c5\u6210\u770b\u4f3c\u81ea\u7136\u7684\u9ed8\u8ba4\u503c\uff1aMIDI \u97f3\u7b26\u7f16\u53f7\u3001\u6570\u5b57\u94a2\u7434\u3001\u81ea\u52a8\u8c03\u97f3\u548c\u4f5c\u66f2\u8f6f\u4ef6\u90fd\u8ba9\u5341\u4e8c\u5e73\u5747\u5f8b\u53d8\u5f97\u65e0\u5f62\u3002\u4f7f\u7528\u8005\u6309\u4e0b\u952e\u5373\u53ef\u83b7\u5f97\u7a33\u5b9a\u7ed3\u679c\uff0c\u5f88\u5c11\u518d\u770b\u89c1\u80cc\u540e\u7684\u53d6\u820d\u3002<\/p>\n<p>\u7406\u89e3\u5341\u4e8c\u6b21\u65b9\u6839\uff0c\u4e0d\u662f\u4e3a\u4e86\u5426\u5b9a\u4fbf\u5229\uff0c\u800c\u662f\u91cd\u65b0\u770b\u89c1\u4fbf\u5229\u7684\u6761\u4ef6\u3002\u5b83\u628a\u65e0\u6cd5\u540c\u65f6\u7eaf\u6b63\u7684\u5173\u7cfb\u8f6c\u6210\u5747\u5300\u53ef\u79fb\u8c03\u7684\u7cfb\u7edf\uff0c\u8ba9\u8096\u90a6\u53ef\u4ee5\u8ba9\u4e8c\u5341\u56db\u4e2a\u8c03\u8fdb\u5165\u540c\u4e00\u5957\u94a2\u7434\u3002\u8fd9\u4e2a\u7cfb\u7edf\u975e\u5e38\u6210\u529f\uff0c\u5374\u4ecd\u662f\u4e00\u9879\u7ed3\u6784\u8bbe\u8ba1\u3002<\/p>\n<p>\u542c Op. 28 \u65f6\uff0c\u53ef\u4ee5\u4e00\u9762\u611f\u53d7\u6bcf\u9996\u524d\u594f\u66f2\u7684\u72ec\u7279\u6027\uff0c\u4e00\u9762\u6ce8\u610f\u5b83\u4eec\u5982\u4f55\u6784\u6210\u7a7f\u8d8a\u5168\u90e8\u8c03\u6027\u7684\u6574\u4f53\u3002\u5341\u4e8c\u5e73\u5747\u5f8b\u4f7f\u8def\u7ebf\u5728\u6280\u672f\u4e0a\u8fde\u7eed\uff1b\u4f5c\u66f2\u5bb6\u7684\u8282\u594f\u3001\u7ec7\u4f53\u548c\u65f6\u95f4\u5224\u65ad\u4f7f\u6bcf\u4e00\u7ad9\u4e0d\u53ef\u66ff\u6362\u3002\u65b9\u6839\u7ed9\u51fa\u5730\u56fe\u7684\u5c3a\u5ea6\uff0c\u97f3\u4e50\u624d\u4f7f\u5730\u56fe\u6210\u4e3a\u65c5\u7a0b\u3002<\/p>\n<h3>\u4e3b\u8981\u8d44\u6599\u4e0e\u8fdb\u4e00\u6b65\u8046\u542c<\/h3>\n<ul>\n<li><a href=\"https:\/\/link.springer.com\/article\/10.1007\/s00283-022-10260-4\" target=\"_blank\" rel=\"noopener\">\u6bd5\u8fbe\u54e5\u62c9\u65af\u97f3\u5dee\u4e0e\u8c03\u5f8b\u6bd4\u4f8b\u7684\u6570\u5b66\u5206\u6790<\/a><\/li>\n<li><a href=\"https:\/\/pubmed.ncbi.nlm.nih.gov\/15532672\/\" target=\"_blank\" rel=\"noopener\">\u5341\u4e8c\u5e73\u5747\u5f8b\u4e0e\u5176\u4ed6\u8c03\u5f8b\u7cfb\u7edf\u7684\u6570\u5b66\u8ba8\u8bba<\/a><\/li>\n<li><a href=\"https:\/\/imslp.org\/wiki\/Preludes%2C_Op.28_%28Chopin%2C_Fr%C3%A9d%C3%A9ric%29\" target=\"_blank\" rel=\"noopener\">\u8096\u90a6\u300a\u4e8c\u5341\u56db\u9996\u524d\u594f\u66f2\u300b\u4e50\u8c31\u4e0e\u4f5c\u54c1\u8d44\u6599<\/a><\/li>\n<\/ul>\n<p><strong>\u7ee7\u7eed\u9605\u8bfb\uff1a<\/strong><a href=\"https:\/\/geoffreychen.com\/zh\/category\/piano\/hearing-mathematics\/\">\u6d4f\u89c8\u300a\u542c\u89c1\u6570\u5b66\u300b\u7cfb\u5217\u6587\u7ae0<\/a><\/p>\n<div class=\"scoreflow-promotion\">\n<hr>\n<p><a href=\"https:\/\/apps.apple.com\/us\/app\/scoreflow\/id6796205656\" target=\"_blank\" rel=\"noopener\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/geoffreychen.com\/wp-content\/uploads\/2026\/10\/scoreflow-app-icon.jpg?resize=72%2C72&#038;ssl=1\" alt=\"ScoreFlow\" width=\"72\" height=\"72\" style=\"display:block;width:72px;height:72px;border-radius:16px;\" loading=\"lazy\"><\/a><\/p>\n<p><strong>ScoreFlow<\/strong><\/p>\n<p>\u5982\u679c\u4f60\u4e5f\u60f3\u628a\u8fd9\u4e9b\u5173\u4e8e\u8046\u542c\u3001\u7406\u89e3\u4e0e\u7ec3\u4e60\u7684\u60f3\u6cd5\u5e26\u5230\u7434\u952e\u4e0a\uff0c\u53ef\u4ee5\u8bd5\u8bd5\u6211\u5f00\u53d1\u7684 <a href=\"https:\/\/apps.apple.com\/us\/app\/scoreflow\/id6796205656\" rel=\"noopener\" target=\"_blank\">ScoreFlow<\/a>\uff0c\u8ba9\u4e50\u8c31\u9605\u8bfb\u4e0e\u65e5\u5e38\u7ec3\u4e60\u8854\u63a5\u5f97\u66f4\u81ea\u7136\u3002<\/p>\n<p><a href=\"https:\/\/apps.apple.com\/us\/app\/scoreflow\/id6796205656\" aria-label=\"\u5728 App Store \u4e0b\u8f7d: ScoreFlow\" style=\"display:inline-block;padding:12px 0;\" target=\"_blank\" rel=\"noopener\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/geoffreychen.com\/wp-content\/uploads\/2026\/10\/roadly-app-store-badge-zh-cn-hires.png?resize=150%2C50&#038;ssl=1\" alt=\"\u5728 App Store \u4e0b\u8f7d\" width=\"150\" height=\"50\" style=\"display:block;width:150px;height:50px;max-width:100%;border:0;\" loading=\"lazy\"><\/a><\/p>\n<\/div>\n<p>","protected":false},"excerpt":{"rendered":"","protected":false},"author":17897162,"featured_media":0,"comment_status":"closed","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_geoff_meta_description_en":"Explains why twelve-tone equal temperament uses 2^(1\/12): equal intervals mean equal frequency ratios on a log pitch grid, not equal Hz differences.","_geoff_meta_description_zh":"\u9610\u660e\u5341\u4e8c\u5e73\u5747\u5f8b\u4e3a\u4f55\u53d62^(1\/12)\uff1a\u7b49\u97f3\u7a0b\u6307\u5728\u5bf9\u6570\u97f3\u9ad8\u7a7a\u95f4\u4e2d\u7b49\u9891\u7387\u6bd4\u7684\u5747\u5206\uff0c\u800c\u975e\u7ebf\u6027\u8d6b\u5179\u5dee\u7684\u7b49\u8ddd\u3002","_wpcom_ai_launchpad_first_post":false,"_jetpack_hide_featured_image":false,"_jetpack_featured_image_duplicate_dismissed":0,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[787296608],"tags":[787296626,787296646,787296636,787296628,787296629,787296620,787296614,787296644],"class_list":["post-16907","post","type-post","status-publish","format-standard","hentry","category-hearing-mathematics","tag-hearing-mathematics","tag-how-sound-enters-the-body-and-mind","tag-logarithmic-frequency","tag-twelve-tone-equal-temperament","tag-twelve-tone-equal-temperament-zh","tag-hearing-mathematics-zh","tag-how-sound-enters-the-body-and-mind-zh","tag-logarithmic-frequency-zh","","has-post-thumbnail","fallback-thumbnail"],"translatepress_titles":{"en_AU":"Why Does Twelve-Tone Equal Temperament Need the Twelfth Root of 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