{"id":16914,"date":"2026-08-12T15:14:00","date_gmt":"2026-08-12T05:14:00","guid":{"rendered":"https:\/\/geoffreychen.com\/?p=16914"},"modified":"2026-08-12T23:18:14","modified_gmt":"2026-08-12T13:18:14","slug":"why-cant-a-piano-be-tuned-exactly-to-its-theoretical-frequencies","status":"publish","type":"post","link":"https:\/\/geoffreychen.com\/zh\/why-cant-a-piano-be-tuned-exactly-to-its-theoretical-frequencies\/","title":{"rendered":"\u4e3a\u4ec0\u4e48\u4e00\u67b6\u94a2\u7434\u4e0d\u80fd\u5b8c\u5168\u6309\u7167\u7406\u8bba\u9891\u7387\u8c03\u97f3\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\u4e03\u7bc7<\/em><\/p>\n<p>\u6709\u4e86 A4 \u7b49\u4e8e 440 \u8d6b\u5179\u548c <code>f(n)=440\u00d72^(n\/12)<\/code>\uff0c\u4f3c\u4e4e\u53ea\u8981\u628a\u516b\u5341\u516b\u4e2a\u7434\u952e\u9010\u4e00\u8c03\u5230\u8868\u4e2d\u9891\u7387\uff0c\u5c31\u80fd\u5f97\u5230\u4e00\u67b6\u5b8c\u7f8e\u94a2\u7434\u3002\u7535\u5b50\u6d4b\u91cf\u751a\u81f3\u53ef\u4ee5\u663e\u793a\u5c0f\u6570\u70b9\u540e\u7684\u504f\u5dee\u3002\u53ef\u662f\u4e13\u4e1a\u8c03\u5f8b\u5e08\u4e0d\u4f1a\u628a\u6bcf\u6839\u5f26\u5b64\u7acb\u5730\u6821\u51c6\u5230\u7406\u8bba\u57fa\u9891\u540e\u5c31\u7ed3\u675f\u3002\u5b9e\u9645\u8c03\u597d\u7684\u94a2\u7434\u901a\u5e38\u5728\u9ad8\u97f3\u533a\u6bd4\u7406\u60f3\u5e73\u5747\u5f8b\u66f4\u9ad8\uff0c\u5728\u4f4e\u97f3\u533a\u66f4\u4f4e\uff0c\u516b\u5ea6\u88ab\u6709\u610f\u201c\u62c9\u5bbd\u201d\u3002<\/p>\n<p>\u8fd9\u4e0d\u662f\u542c\u89c9\u4e60\u60ef\u59a8\u788d\u4e86\u79d1\u5b66\uff0c\u4e5f\u4e0d\u662f\u8c03\u5f8b\u5e08\u7528\u4e3b\u89c2\u611f\u89c9\u7834\u574f\u7cbe\u786e\u516c\u5f0f\u3002\u516c\u5f0f\u4ee5\u7406\u60f3\u5468\u671f\u6e90\u4e3a\u524d\u63d0\uff1b\u94a2\u7434\u5f26\u5374\u6709\u521a\u5ea6\uff0c\u7434\u69cc\u4f1a\u6539\u53d8\u6fc0\u53d1\u65b9\u5f0f\uff0c\u7434\u6865\u548c\u97f3\u677f\u4f1a\u8026\u5408\u80fd\u91cf\uff0c\u540c\u97f3\u5f26\u4e4b\u95f4\u9700\u8981\u534f\u8c03\uff0c\u6574\u4ef6\u4e50\u5668\u8fd8\u5fc5\u987b\u5728\u591a\u4e2a\u97f3\u533a\u542c\u8d77\u6765\u8fde\u7eed\u3002\u94a2\u7434\u8c03\u5f8b\u771f\u6b63\u4f18\u5316\u7684\u662f\u590d\u5408\u58f0\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u800c\u4e0d\u662f\u4e00\u5217\u62bd\u79bb\u6750\u6599\u7684\u57fa\u9891\u3002<\/p>\n<h2>\u7406\u60f3\u67d4\u5f26\u4e0e\u771f\u5b9e\u94a2\u4e1d\u7684\u5dee\u522b<\/h2>\n<p>\u7406\u60f3\u67d4\u5f26\u7684\u632f\u52a8\u6a21\u5f0f\u4e3a <code>f\u30012f\u30013f\u30014f\u2026\u2026<\/code>\u3002\u5982\u679c\u4f4e\u97f3\u57fa\u9891\u4e3a 110 \u8d6b\u5179\uff0c\u5b83\u7684\u7b2c\u4e8c\u5206\u97f3\u6b63\u597d 220\uff0c\u7b2c\u56db\u5206\u97f3\u6b63\u597d 440\uff1b\u9ad8\u516b\u5ea6 A \u7684\u57fa\u9891\u4e5f\u53ef\u6b63\u597d\u5bf9\u9f50 220\u3002\u8c03\u516b\u5ea6\u65f6\uff0c\u8ba9\u5bf9\u5e94\u5206\u97f3\u65e0\u62cd\u5730\u91cd\u5408\u4f3c\u4e4e\u5f88\u76f4\u63a5\u3002<\/p>\n<p>\u94a2\u7434\u5f26\u4e0d\u662f\u6ca1\u6709\u5f2f\u66f2\u963b\u529b\u7684\u51e0\u4f55\u7ebf\u3002\u94a2\u4e1d\u7684\u521a\u5ea6\u7ed9\u9ad8\u9636\u6a21\u5f0f\u589e\u52a0\u989d\u5916\u6062\u590d\u529b\uff0c\u4f7f\u5206\u97f3\u6bd4\u7406\u60f3\u6574\u6570\u500d\u7565\u9ad8\uff0c\u800c\u4e14\u9636\u6570\u8d8a\u9ad8\u504f\u5dee\u901a\u5e38\u8d8a\u660e\u663e\u3002\u8fd9\u79f0\u4e3a<strong>\u975e\u8c10\u6027<\/strong>\uff08inharmonicity\uff09\u3002\u4e00\u6839\u5f26\u7684\u201c\u7b2c\u4e8c\u5206\u97f3\u201d\u53ef\u80fd\u7565\u9ad8\u4e8e\u4e24\u500d\u57fa\u9891\uff0c\u7b2c\u56db\u5206\u97f3\u4e5f\u7565\u9ad8\u4e8e\u56db\u500d\u3002<\/p>\n<p>\u82e5\u628a\u9ad8\u516b\u5ea6\u57fa\u9891\u4e25\u683c\u8c03\u6210\u4f4e\u97f3\u57fa\u9891\u4e24\u500d\uff0c\u9ad8\u97f3\u7684\u67d0\u4e9b\u5206\u97f3\u5c31\u4f1a\u4e0e\u4f4e\u97f3\u504f\u9ad8\u7684\u5206\u97f3\u5bf9\u4e0d\u9f50\uff0c\u4ea7\u751f\u53ef\u95fb\u62cd\u52a8\u3002\u5c06\u9ad8\u97f3\u7565\u5411\u4e0a\u8c03\uff0c\u53ef\u4ee5\u8ba9\u8c03\u5f8b\u5e08\u5b9e\u9645\u6bd4\u8f83\u7684\u5206\u97f3\u5173\u7cfb\u66f4\u5e73\u9759\u3002\u5f80\u4f4e\u97f3\u533a\u63a8\u8fdb\u65f6\uff0c\u7c7b\u4f3c\u534f\u8c03\u4f1a\u4f7f\u57fa\u9891\u5411\u4e0b\u4f38\u5c55\u3002\u7531\u6b64\u5f62\u6210\u76f8\u5bf9\u7406\u8bba\u5e73\u5747\u5f8b\u5411\u4e24\u7aef\u5f20\u5f00\u7684\u8c03\u5f8b\u66f2\u7ebf\uff0c\u5e38\u88ab\u79f0\u4e3a Railsback stretch \u6216 Railsback curve\u3002<\/p>\n<p><a href=\"https:\/\/doi.org\/10.1121\/1.4931439\" target=\"_blank\" rel=\"noopener\">\u300a\u7f8e\u56fd\u58f0\u5b66\u5b66\u4f1a\u6742\u5fd7\u300b\u7684\u5f00\u653e\u7814\u7a76<\/a>\u628a\u5b9e\u6d4b\u94a2\u7434\u9891\u8c31\u3001\u975e\u8c10\u6027\u4e0e\u611f\u89c9\u4e0d\u534f\u548c\u6a21\u578b\u7ed3\u5408\uff0c\u5f97\u5230\u63a5\u8fd1\u7ecf\u9a8c Railsback \u66f2\u7ebf\u7684\u7ed3\u679c\u3002\u53e6\u4e00\u9879\u7531\u94a2\u7434\u8c03\u5f8b\u5e08\u548c\u7ba1\u5f26\u4e50\u624b\u53c2\u4e0e\u7684<a href=\"https:\/\/pubmed.ncbi.nlm.nih.gov\/36050167\/\" target=\"_blank\" rel=\"noopener\">\u4e3b\u89c2\u8c03\u5f8b\u5b9e\u9a8c<\/a>\u53d1\u73b0\uff0c\u542b\u771f\u5b9e\u975e\u8c10\u6027\u7684\u5408\u6210\u94a2\u7434\u97f3\u4f1a\u4ea7\u751f\u7c7b\u4f3c\u4f38\u5c55\u66f2\u7ebf\uff0c\u800c\u8c10\u6ce2\u7248\u672c\u7684\u7ed3\u679c\u663e\u8457\u4e0d\u540c\u3002<\/p>\n<h2>Railsback \u66f2\u7ebf\u4e0d\u662f\u6bcf\u67b6\u94a2\u7434\u5fc5\u987b\u590d\u5236\u7684\u6a21\u677f<\/h2>\n<p>\u628a\u201c\u94a2\u7434\u9700\u8981\u4f38\u5c55\u8c03\u5f8b\u201d\u7406\u89e3\u6210\u201c\u6240\u6709\u94a2\u7434\u5957\u7528\u540c\u4e00\u6761\u66f2\u7ebf\u201d\u53c8\u4f1a\u4ea7\u751f\u65b0\u8bef\u89e3\u3002\u975e\u8c10\u6027\u53d6\u51b3\u4e8e\u5f26\u957f\u3001\u76f4\u5f84\u3001\u5f20\u529b\u3001\u6750\u6599\u548c\u97f3\u9636\u8bbe\u8ba1\u3002\u5c0f\u578b\u7acb\u5f0f\u94a2\u7434\u4e3a\u83b7\u5f97\u4f4e\u97f3\uff0c\u53ef\u80fd\u4f7f\u7528\u66f4\u77ed\u3001\u66f4\u7c97\u6216\u7f20\u7ed5\u7684\u5f26\uff1b\u5927\u578b\u4e09\u89d2\u94a2\u7434\u6709\u4e0d\u540c\u5c3a\u5ea6\u3002\u4e2a\u522b\u7434\u952e\u8fd8\u4f1a\u6709\u5c40\u90e8\u5dee\u5f02\u3002<\/p>\n<p>Railsback \u66f2\u7ebf\u662f\u5bf9\u82e5\u5e72\u8c03\u5f8b\u7ed3\u679c\u7684\u4e00\u79cd\u7ecf\u5178\u6982\u62ec\uff0c\u4e0d\u662f\u72ec\u7acb\u4e8e\u4e50\u5668\u7684\u81ea\u7136\u6cd5\u3002\u8c03\u5f8b\u5e08\u5fc5\u987b\u542c\u5177\u4f53\u7434\u7684\u62cd\u901f\u3001\u97f3\u533a\u8fc7\u6e21\u3001\u540c\u97f3\u5f26\u548c\u5171\u9e23\uff0c\u7535\u5b50\u8f6f\u4ef6\u4e5f\u9700\u8981\u4f30\u8ba1\u8be5\u7434\u7684\u975e\u8c10\u6027\u53c2\u6570\u3002\u8c03\u5f8b\u7ed3\u679c\u751a\u81f3\u4f1a\u56e0\u7528\u9014\u3001\u97f3\u8272\u76ee\u6807\u548c\u6280\u672f\u4f20\u7edf\u7565\u6709\u4e0d\u540c\u3002<\/p>\n<p>\u56e0\u6b64\uff0c\u4e0d\u80fd\u4ece\u4e00\u5f20\u5e73\u5747\u66f2\u7ebf\u53cd\u63a8\u51fa\u201c\u9ad8\u97f3\u5fc5\u987b\u4e00\u5f8b\u9ad8\u591a\u5c11\u97f3\u5206\u201d\u3002\u771f\u6b63\u7a33\u5b9a\u7684\u662f\u95ee\u9898\u7ed3\u6784\uff1a\u7406\u8bba\u5e73\u5747\u5f8b\u7ed9\u51fa\u53c2\u8003\u5173\u7cfb\uff0c\u6750\u6599\u4f7f\u5206\u97f3\u504f\u79bb\uff0c\u8c03\u5f8b\u5728\u6574\u67b6\u7434\u4e0a\u5206\u914d\u504f\u5dee\uff0c\u4f7f\u5173\u952e\u6bd4\u8f83\u542c\u8d77\u6765\u8fde\u8d2f\u3002<\/p>\n<h2>\u4e00\u4e2a\u7434\u952e\u5e38\u4e0d\u53ea\u4e00\u6839\u5f26<\/h2>\n<p>\u94a2\u7434\u4e2d\u9ad8\u97f3\u533a\u7684\u4e00\u4e2a\u7434\u952e\u901a\u5e38\u9a71\u52a8\u4e24\u6839\u6216\u4e09\u6839\u540c\u97f3\u5f26\u3002\u5b83\u4eec\u82e5\u5b8c\u5168\u540c\u9891\uff0c\u53ef\u4ee5\u589e\u5f3a\u6301\u7eed\u58f0\uff1b\u7565\u6709\u5dee\u5f02\u5219\u4f1a\u4ea7\u751f\u62cd\u52a8\uff0c\u5f71\u54cd\u58f0\u97f3\u7684\u5c55\u5f00\u4e0e\u8870\u51cf\u3002\u8c03\u5f8b\u5e08\u5148\u5efa\u7acb\u4e3b\u8981\u97f3\u9ad8\uff0c\u518d\u628a\u540c\u7ec4\u5f26\u8c03\u5230\u9002\u5f53\u7edf\u4e00\u3002\u6240\u8c13\u201c\u540c\u97f3\u201d\u5728\u8fd9\u91cc\u4e5f\u662f\u52a8\u6001\u5173\u7cfb\uff1a\u542c\u8005\u9700\u8981\u611f\u5230\u4e00\u4e2a\u97f3\uff0c\u800c\u4e0d\u662f\u51e0\u4e2a\u4e92\u76f8\u6446\u52a8\u7684\u97f3\u6e90\uff0c\u540c\u65f6\u53c8\u4fdd\u7559\u94a2\u7434\u58f0\u7684\u6d3b\u529b\u3002<\/p>\n<p>\u5ef6\u97f3\u8e0f\u677f\u6309\u4e0b\u540e\uff0c\u672a\u88ab\u7434\u69cc\u76f4\u63a5\u6572\u51fb\u7684\u5f26\u4e5f\u53ef\u56e0\u5171\u4eab\u5206\u97f3\u800c\u5171\u632f\u3002\u7434\u6865\u628a\u80fd\u91cf\u4f20\u7ed9\u97f3\u677f\uff0c\u97f3\u677f\u53c8\u5c06\u66f4\u5927\u9762\u79ef\u7a7a\u6c14\u5e26\u52a8\u8d77\u6765\u3002\u623f\u95f4\u53cd\u5c04\u7ee7\u7eed\u6539\u53d8\u542c\u8005\u6536\u5230\u7684\u9891\u8c31\u3002\u8c03\u97f3\u5668\u5939\u5728\u67d0\u5904\u6d4b\u5f97\u7684\u57fa\u9891\u5e76\u4e0d\u5305\u542b\u6f14\u594f\u8005\u4e0e\u542c\u4f17\u7ecf\u5386\u7684\u5b8c\u6574\u5bf9\u8c61\u3002<\/p>\n<p>\u8fd9\u4e0d\u8868\u793a\u6d4b\u91cf\u65e0\u7528\u3002\u73b0\u4ee3\u8c03\u97f3\u5de5\u5177\u53ef\u4ee5\u6781\u51c6\u786e\u5730\u4f30\u8ba1\u57fa\u9891\u3001\u5206\u97f3\u548c\u975e\u8c10\u6027\uff0c\u5e2e\u52a9\u4fdd\u5b58\u76ee\u6807\u4e0e\u53d1\u73b0\u6f02\u79fb\u3002\u95ee\u9898\u5728\u4e8e\u628a\u201c\u6d4b\u5f97\u51c6\u201d\u8bef\u5199\u6210\u201c\u51b3\u5b9a\u8c03\u5230\u54ea\u91cc\u201d\u3002\u6570\u636e\u63cf\u8ff0\u5177\u4f53\u5173\u7cfb\uff1b\u6700\u7ec8\u6807\u51c6\u8fd8\u8981\u9009\u62e9\u54ea\u4e9b\u5206\u97f3\u9700\u8981\u534f\u8c03\u3001\u5404\u97f3\u533a\u600e\u6837\u8fde\u63a5\uff0c\u4ee5\u53ca\u8fd9\u67b6\u7434\u8981\u5728\u4ec0\u4e48\u7a7a\u95f4\u4e0e\u97f3\u4e50\u4e2d\u5de5\u4f5c\u3002<\/p>\n<h2>\u8c03\u5f8b\u5e08\u542c\u7684\u62cd\uff0c\u4e0d\u662f\u6a21\u7cca\u76f4\u89c9<\/h2>\n<p>\u8033\u8c03\u5e76\u975e\u51ed\u201c\u611f\u89c9\u5dee\u4e0d\u591a\u201d\u3002\u8c03\u5f8b\u5e08\u9009\u62e9\u7279\u5b9a\u97f3\u7a0b\u4e0e\u5206\u97f3\u914d\u5bf9\uff0c\u901a\u8fc7\u62cd\u901f\u5efa\u7acb\u6e29\u5ea6\u533a\uff0c\u518d\u5411\u6574\u4e2a\u952e\u76d8\u6269\u5c55\u3002\u5e73\u5747\u5f8b\u4e94\u5ea6\u3001\u56db\u5ea6\u3001\u4e09\u5ea6\u548c\u516d\u5ea6\u5404\u6709\u9884\u671f\u7684\u62cd\u52a8\u8d8b\u52bf\uff1b\u516b\u5ea6\u8fd8\u53ef\u4ee5\u6309 2:1\u30014:2\u30016:3 \u7b49\u4e0d\u540c\u5206\u97f3\u914d\u5bf9\u68c0\u9a8c\u3002\u4f4e\u97f3\u4e0e\u9ad8\u97f3\u7684\u51b3\u5b9a\u5fc5\u987b\u4ea4\u53c9\u9a8c\u8bc1\u3002<\/p>\n<p>\u8fd9\u662f\u4e00\u79cd\u628a\u6570\u5b66\u5173\u7cfb\u8f6c\u6210\u65f6\u95f4\u7ecf\u9a8c\u7684\u4e13\u4e1a\u6280\u80fd\u3002\u9891\u7387\u5dee\u5728\u7eb8\u4e0a\u662f\u6570\u5b57\uff0c\u5728\u8033\u4e2d\u6210\u4e3a\u6bcf\u79d2\u51e0\u6b21\u7684\u8d77\u4f0f\uff1b\u62cd\u901f\u6cbf\u534a\u97f3\u548c\u97f3\u533a\u5e94\u600e\u6837\u53d8\u5316\uff0c\u4e3a\u8c03\u5f8b\u63d0\u4f9b\u5197\u4f59\u68c0\u67e5\u3002\u4e00\u4e2a\u9519\u8bef\u5173\u7cfb\u4f1a\u5728\u591a\u4e2a\u68c0\u9a8c\u97f3\u7a0b\u4e2d\u66b4\u9732\uff0c\u800c\u4e0d\u662f\u4f9d\u8d56\u67d0\u6b21\u795e\u79d8\u5224\u65ad\u3002<\/p>\n<p>\u7535\u5b50\u8c03\u5f8b\u4e0e\u8033\u8c03\u4e5f\u4e0d\u662f\u4e92\u65a5\u9635\u8425\u3002\u9ad8\u8d28\u91cf\u5de5\u5177\u53ef\u4ee5\u6839\u636e\u5b9e\u6d4b\u975e\u8c10\u6027\u8ba1\u7b97\u4e2a\u522b\u4e50\u5668\u7684\u4f38\u5c55\u76ee\u6807\uff0c\u8c03\u5f8b\u5e08\u518d\u7528\u97f3\u7a0b\u3001\u97f3\u8272\u4e0e\u97f3\u533a\u8fde\u7eed\u6027\u9a8c\u8bc1\u3002\u673a\u5668\u6269\u5927\u53ef\u89c1\u4fe1\u606f\uff0c\u4eba\u8d1f\u8d23\u786e\u5b9a\u76ee\u6807\u548c\u6574\u4f53\u9a8c\u6536\u3002<\/p>\n<h2>\u5fb7\u5f6a\u897f\u300a\u6c89\u6ca1\u7684\u6559\u5802\u300b\u4e3a\u4f55\u9700\u8981\u6574\u67b6\u7434\u7684\u5171\u9e23<\/h2>\n<p>\u5fb7\u5f6a\u897f\u300a\u524d\u594f\u66f2\u300b\u7b2c\u4e00\u518c\u7b2c\u5341\u9996\u300a\u6c89\u6ca1\u7684\u6559\u5802\u300b\u8ba9\u94a2\u7434\u4ece\u8fdc\u5904\u7684\u7a7a\u4e94\u5ea6\u3001\u548c\u5f26\u5e73\u9762\u4e0e\u4f4e\u97f3\u5171\u9e23\u4e2d\u6784\u6210\u4e00\u5ea7\u4f3c\u4e4e\u5347\u51fa\u6c34\u9762\u7684\u5efa\u7b51\u3002<a href=\"https:\/\/imslp.org\/wiki\/Pr%C3%A9ludes%2C_Livre_1%2C_CD_125_%28Debussy%2C_Claude%29\" target=\"_blank\" rel=\"noopener\">\u539f\u7248\u94a2\u7434\u8c31<\/a>\u5199\u660e\u4e0d\u540c\u529b\u5ea6\u3001\u97f3\u533a\u548c\u8e0f\u677f\u5173\u7cfb\uff1b\u4f5c\u54c1\u7684\u6548\u679c\u663e\u7136\u4e0d\u53ea\u662f\u6309\u987a\u5e8f\u53d1\u51fa\u7406\u8bba\u9891\u7387\u3002<\/p>\n<p>\u5f00\u5934\u758f\u8fdc\u7684\u97f3\u7a0b\u9700\u8981\u9759\u9ed8\u4e0e\u8870\u51cf\u4e3a\u8fb9\u754c\u3002\u4e2d\u6bb5\u539a\u91cd\u548c\u5f26\u8de8\u8d8a\u5bbd\u5e7f\u97f3\u57df\uff0c\u4f4e\u97f3\u57fa\u9891\u3001\u4e0a\u65b9\u5206\u97f3\u3001\u97f3\u677f\u548c\u8e0f\u677f\u91ca\u653e\u7684\u5171\u9e23\u5171\u540c\u6784\u6210\u201c\u949f\u58f0\u201d\u822c\u7684\u60f3\u8c61\u3002\u82e5\u6bcf\u4e2a\u97f3\u88ab\u5f53\u6210\u4e92\u4e0d\u76f8\u5173\u7684\u7eaf\u6b63\u5f26\u6ce2\uff0c\u8bb8\u591a\u91cd\u91cf\u4e0e\u6df1\u5ea6\u90fd\u4f1a\u6d88\u5931\u3002<\/p>\n<p>\u4f38\u5c55\u8c03\u5f8b\u5728\u8fd9\u91cc\u4e0d\u662f\u4e3a\u4e86\u8ba9\u4f5c\u54c1\u66f4\u6d6a\u6f2b\uff0c\u800c\u662f\u8ba9\u771f\u5b9e\u94a2\u7434\u8de8\u97f3\u533a\u7684\u590d\u5408\u97f3\u66f4\u597d\u5730\u76f8\u63a5\u3002\u4f4e\u97f3\u7684\u9ad8\u9636\u5206\u97f3\u4e0e\u4e2d\u97f3\u3001\u9ad8\u97f3\u7684\u57fa\u9891\u4e0d\u65ad\u76f8\u9047\uff1b\u8c03\u5f8b\u5fc5\u987b\u8ba9\u6574\u4ef6\u4e50\u5668\u6210\u4e3a\u4e00\u4e2a\u53ef\u5171\u9e23\u7a7a\u95f4\u3002\u6f14\u594f\u8005\u968f\u540e\u7528\u89e6\u952e\u4e0e\u8e0f\u677f\u51b3\u5b9a\u54ea\u4e9b\u6f5c\u5728\u5173\u7cfb\u88ab\u5524\u9192\u3002<\/p>\n<h2>\u7406\u8bba\u9891\u7387\u4ecd\u7136\u91cd\u8981\uff0c\u4f46\u5b83\u53ea\u662f\u7b2c\u4e00\u5c42\u6807\u51c6<\/h2>\n<p>\u6ca1\u6709\u5341\u4e8c\u5e73\u5747\u5f8b\u516c\u5f0f\uff0c\u8c03\u5f8b\u7f3a\u5c11\u53ef\u79fb\u8c03\u7ed3\u6784\uff1b\u6ca1\u6709\u53c2\u8003\u97f3\uff0c\u4e50\u5668\u4e4b\u95f4\u96be\u4ee5\u534f\u8c03\uff1b\u6ca1\u6709\u9891\u8c31\u6d4b\u91cf\uff0c\u6211\u4eec\u4e5f\u5f88\u96be\u89e3\u91ca\u975e\u8c10\u6027\u3002\u7406\u8bba\u4e0d\u662f\u88ab\u5b9e\u8df5\u63a8\u7ffb\uff0c\u800c\u662f\u88ab\u5b9e\u8df5\u5177\u4f53\u5316\u3002<\/p>\n<p>\u66f4\u51c6\u786e\u7684\u5c42\u6b21\u662f\uff1a\u5e73\u5747\u5f8b\u89c4\u5b9a\u7406\u60f3\u97f3\u7ea7\u4e4b\u95f4\u7684\u5bf9\u6570\u5173\u7cfb\uff1b\u94a2\u7434\u5c3a\u5ea6\u8bbe\u8ba1\u51b3\u5b9a\u6bcf\u6839\u5f26\u7684\u6750\u6599\u6761\u4ef6\uff1b\u975e\u8c10\u6027\u6539\u53d8\u5206\u97f3\u4f4d\u7f6e\uff1b\u8c03\u5f8b\u5e08\u9009\u62e9\u6574\u67b6\u7434\u7684\u4f38\u5c55\u4e0e\u62cd\u901f\uff1b\u6f14\u594f\u548c\u623f\u95f4\u4f7f\u8fd9\u4e9b\u5173\u7cfb\u6210\u4e3a\u5b9e\u9645\u58f0\u97f3\u3002\u6bcf\u4e00\u5c42\u90fd\u589e\u52a0\u7ea6\u675f\uff0c\u4e5f\u589e\u52a0\u5224\u65ad\u3002<\/p>\n<p>\u6240\u4ee5\uff0c\u4e00\u67b6\u94a2\u7434\u4e0d\u80fd\u201c\u5b8c\u5168\u6309\u7167\u7406\u8bba\u9891\u7387\u201d\u8c03\u97f3\uff0c\u5e76\u975e\u56e0\u4e3a\u7406\u8bba\u4e0d\u7cbe\u786e\uff0c\u800c\u662f\u56e0\u4e3a\u7406\u8bba\u9891\u7387\u53ea\u5b9a\u4e49\u4e86\u57fa\u9891\u4e16\u754c\u4e2d\u7684\u4e00\u4e2a\u6a21\u578b\u3002\u771f\u5b9e\u94a2\u7434\u8981\u6c42\u66f4\u4e30\u5bcc\u7684\u7cbe\u786e\uff1a\u4e0d\u662f\u6bcf\u4e2a\u6570\u5b57\u5b64\u7acb\u6b63\u786e\uff0c\u800c\u662f\u8bb8\u591a\u4e0d\u5b8c\u5168\u7406\u60f3\u7684\u5206\u97f3\u5728\u6574\u67b6\u4e50\u5668\u4e0a\u5f62\u6210\u53ef\u4ee5\u88ab\u542c\u89c1\u3001\u6bd4\u8f83\u548c\u4f7f\u7528\u7684\u79e9\u5e8f\u3002<\/p>\n<p>\u4e0b\u6b21\u542c\u300a\u6c89\u6ca1\u7684\u6559\u5802\u300b\uff0c\u53ef\u4ee5\u6ce8\u610f\u4f4e\u97f3\u53d1\u51fa\u4ee5\u540e\uff0c\u4e0a\u65b9\u54ea\u4e9b\u58f0\u97f3\u50cf\u4ece\u5b83\u5185\u90e8\u51fa\u73b0\u3002\u90a3\u4e0d\u662f\u5355\u4e2a\u7434\u952e\u62e5\u6709\u7684\u6548\u679c\uff0c\u800c\u662f\u5f26\u3001\u97f3\u677f\u3001\u8e0f\u677f\u3001\u623f\u95f4\u548c\u542c\u89c9\u5171\u540c\u7ef4\u6301\u7684\u5173\u7cfb\u3002\u8c03\u5f8b\u7684\u76ee\u6807\u4e0d\u662f\u5f81\u670d\u6750\u6599\uff0c\u800c\u662f\u8ba9\u6750\u6599\u80fd\u591f\u5f7c\u6b64\u56de\u7b54\u3002<\/p>\n<h3>\u4e3b\u8981\u8d44\u6599\u4e0e\u8fdb\u4e00\u6b65\u8046\u542c<\/h3>\n<ul>\n<li><a href=\"https:\/\/doi.org\/10.1121\/1.4931439\" target=\"_blank\" rel=\"noopener\">Giordano\uff1a\u4ee5\u94a2\u7434\u97f3\u975e\u8c10\u6027\u548c\u611f\u89c9\u4e0d\u534f\u548c\u89e3\u91ca Railsback stretch<\/a><\/li>\n<li><a href=\"https:\/\/pubmed.ncbi.nlm.nih.gov\/36050167\/\" target=\"_blank\" rel=\"noopener\">Rigaudon \u7b49\uff1a\u975e\u8c10\u6027\u5bf9\u94a2\u7434\u97f3\u9ad8\u77e5\u89c9\u4e0e\u4e3b\u89c2\u8c03\u5f8b\u7684\u5f71\u54cd<\/a><\/li>\n<li><a href=\"https:\/\/imslp.org\/wiki\/Pr%C3%A9ludes%2C_Livre_1%2C_CD_125_%28Debussy%2C_Claude%29\" target=\"_blank\" rel=\"noopener\">\u5fb7\u5f6a\u897f\u300a\u524d\u594f\u66f2\u300b\u7b2c\u4e00\u518c\u53ca\u300a\u6c89\u6ca1\u7684\u6559\u5802\u300b\u4e50\u8c31\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<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>","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":true,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_geoff_meta_description_en":"Explains why a piano can\u2019t be tuned exactly to theoretical equal-tempered frequencies: real-string inharmonicity and coupling require stretched tuning for coherent beats across registers.","_geoff_meta_description_zh":"\u6587\u7ae0\u6307\u51fa\u94a2\u7434\u65e0\u6cd5\u5b8c\u5168\u6309\u7406\u8bba\u7b49\u5e73\u5747\u5f8b\u9891\u7387\u8c03\u51c6\uff1a\u771f\u5b9e\u94a2\u4e1d\u7684\u975e\u8c10\u6027\u53ca\u5f26\u3001\u97f3\u677f\u4e0e\u8e0f\u677f\u8026\u5408\u4f1a\u4ea7\u751f\u62cd\u52a8\uff0c\u9700\u8981\u5728\u5404\u97f3\u533a\u505a\u4f38\u5c55\u8c03\u5f8b\u4ee5\u4fdd\u6301\u6574\u4f53\u8fde\u8d2f\u3002","_wpcom_ai_launchpad_first_post":false,"_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":[787296631,787296643,787296626,787296646,787296637,787296634,787296620,787296614],"class_list":["post-16914","post","type-post","status-publish","format-standard","hentry","category-hearing-mathematics","tag-piano-tuning-zh","tag-inharmonicity-zh","tag-hearing-mathematics","tag-how-sound-enters-the-body-and-mind","tag-inharmonicity","tag-piano-tuning","tag-hearing-mathematics-zh","tag-how-sound-enters-the-body-and-mind-zh",""],"translatepress_titles":{"en_AU":"Why Can\u2019t a Piano Be Tuned Exactly to Its Theoretical 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