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HASH 2021-06-28 16. Nerves (Part 1) Cool Math for Hot Music ྠಡձ

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໨ඪ 1. άϥϑΛ֦ுͨ͠Β຺ମͩͶɺతͳཧղ 2. ͜ΕΛཧղ

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େ·͔ͳྲྀΕ 1. άϥϑͱʮͭͳ͕Γʯ 2. ୯ମͱʮͭͳ͕Γʯ 3. ຺ମͱʮͭͳ͕Γʯ 4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ

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1. άϥϑͱʮͭͳ͕Γʯ

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1. άϥϑͱʮͭͳ͕Γʯ • લͷ 15 ষͰ΍ͬͨʮάϥϑʯ͸௖఺Ͳ͏͠ͷʮͭͳ͕ΓʯΛҙࣝͨ͠΋ͷ • ٯͷݟํͱͯ͠ɺ௖఺Λհͨ͠ลͲ͏͠ͷʮͭͳ͕Γʯͱ΋ݟΕΔ

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1. άϥϑͱʮͭͳ͕Γʯ • ઢ෼ʢ1 ࣍ݩʣΛ఺ʢ0 ࣍ݩʣͰʮͭͳ͛Δʯͱάϥϑ͕Ͱ͖Δ • ࢛໘ମʢ3 ࣍ݩʣΛࡾ֯ܗʢ2 ࣍ݩʣͰʮͭͳ͛Δʯͱ Fig. 16.4 Έ͍ͨͳਤܗ͕Ͱ͖Δ • খ͞ͳਤܗʢ୯ମʣͷʮͭͳ͕ΓʯͰେ͖ͳਤܗΛߟ࡯͢Δ → Ґ૬زԿֶ

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2. ୯ମͱʮͭͳ͕Γʯ

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2. ୯ମͱʮͭͳ͕Γʯ • ୯ମʢsimplex, [ෳ]simplices | simplexesʣpp.129 Def.43 • ఺ɺઢ෼ɺࡾ֯ܗɺ࢛໘ମͷॱʹ 0 ୯ମɺ1 ୯ମɺ2 ୯ମɺ3 ୯ମͱ͍͏ pp.130 • n ࣍ݩਤܗͷʮ୯ମʯ΋ଘࡏ Hjhornbeck - ౤ߘऀࣗ਎ʹΑΔ࡞඼, CC දࣔ-ܧঝ 4.0, https://commons.wikimedia.org/w/index.php?curid=57828604ʹΑΔ

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2. ୯ମͱʮͭͳ͕Γʯ • ༨ஊɿ༗ݶཁૉ๏ • ݐங෺ͷ଱਒ੑͱ͔ΛධՁͨ͠Γ͢Δ਺஋ܭࢉख๏ • ߏ଄෺Λࡾ֯ܗͱ͔ʹ෼ׂͯ͠ܭࢉ͢Δ͜ͱ͕ଟ͔ͬͨ͸ͣ By I, Zureks, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2358783

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2. ୯ମͱʮͭͳ͕Γʯ • ͳͥ෼ׂ͢Δʁ • ؆୯ͳਤܗʹམͱ͠ࠐΜͰɺݩͷෳࡶɾେ͖ͳਤܗΛۙࣅ͍ͨ͠ • ྫɿ୯ମΛͭͳ͛ͯϔϞάϩϏϯͷߏ଄Λۙࣅ https://www.slideshare.net/Eniod/tutorial-of-topologicaldataanalysispart1basic

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3. ຺ମͱʮͭͳ͕Γʯ

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3. ຺ମͱʮͭͳ͕Γʯ • গ͠ผͷΞϓϩʔνͰʮۙࣅʯ͢Δ͜ͱΛߟ͑·͢ • ͜͜ʹԁ͕͋Γ·͢ • ԁ͸఺ͷू߹ X Ͱ͢

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3. ຺ମͱʮͭͳ͕Γʯ • ԁΛ n ຊͷʮހʯʹɺͨͩ͠ॏͳΔΑ͏ʹ෼ׂ͠·͢ʢX ͷඃ෴ɿcovering of Xʣ • ހ͸఺ͷू߹ C1, C2, ..., Cn Ͱ͢ʢ͜͜Ͱ C = { C1, C2, ..., Cn } ͱஔ͖·͢ʣ • ໌Β͔ʹ ∪C = X Ͱ͢ʢDef.43 L.2ʣ

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3. ຺ମͱʮͭͳ͕Γʯ • ހΛ఺ʢ0 ୯ମʣʹஔ͖׵͑ɺࡶʹந৅Խ͠·͢ • ݩʑॏͳ͍ͬͯͨՕॴʢ∩s ≠ ∅ʣ͕͋Ε͹ɺ఺ͱ఺Λઢ෼ sʢ1 ୯ମʣͰͭͳ͗·͢ • ໌Β͔ʹ ∪C = X Ͱ͢ʢDef.43 2 ߦ໨ʣ

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3. ຺ମͱʮͭͳ͕Γʯ • ͜ΕΛඃ෴ C ͷ຺ମʢNerveʣͱ͍͍ɺN(C) ͱॻ͖·͢ • ຺ମ͸୯ମʹʮͭͳ͕ΓʯΛ࣋ͨͤͨू߹Ͱ΋͋Γ·͢ʢந৅త୯ମෳମʣ N(C)

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3. ຺ମͱʮͭͳ͕Γʯ • ൃలɿ຺ମఆཧ • ͏·͘ඃ෴Λͱ຺ͬͯମΛ࡞Δͱɺݩͷू߹ X ͱ຺ମ N(C) ͸ʮ࣮࣭ಉ͡ʯˠ ͍͍ۙࣅ • ࣮࣭ಉ͡ʹϗϞτϐʔಉ஋ N(C) X

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3. ຺ମͱʮͭͳ͕Γʯ • ຺ମ N(C) ʹؚ·ΕΔ n ୯ମΛ Nn(C) ͱॻ͖·͢ • ྫɿN0(C) ͸ 0 ୯ମʢ఺ʣͷू߹ɺN1(C) ͸ 1 ୯ମʢઢ෼ʣͷू߹ • ͢ͳΘͪɺؔ਺ N1(C) → N0(C) ͸άϥϑʹͳΔʢڭՊॻ pp.133ʣ N(C) X

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3. ຺ମͱʮͭͳ͕Γʯ • Example 39 • C Major εέʔϧԻ͔ΒͳΔू߹ X = { c, d, e, f, g, a, b } Λஔ͖·͢ a b c d e f g

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3. ຺ମͱʮͭͳ͕Γʯ • Example 39 • C Major εέʔϧԻ͔ΒͳΔू߹ X = { c, d, e, f, g, a, b } Λஔ͖·͢ • Ⅰ ͷ࿨ԻΛ I = { c, e, g } ͱஔ͖·͢ a b c d e f g I

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3. ຺ମͱʮͭͳ͕Γʯ • Example 39 • ͦΜͳ۩߹ʹμΠΞτχοΫίʔυΛ͢΂ͯຒΊ͍͖ͯ·͢ • ͜͜Ͱ C = { I, II, III, IV, V, VI, VII } ͸໌Β͔ʹεέʔϧ X ͷඃ෴ a b c d e f g I III V VII II IV VI

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3. ຺ମͱʮͭͳ͕Γʯ • Example 39 • μΠΞτχοΫίʔυΛ 0 ୯ମͰந৅Խʢthe seven degrees as 0-simplices ) • ॏͳ͍ͬͯͨ෦෼Λ 1 ୯ମͰ݁Ϳʢthe 14 lines for all pairs of degrees that intersectʣ I III V VII II IV VI a b c d e f g I III V VII II IV VI

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3. ຺ମͱʮͭͳ͕Γʯ • Example 39 • 2 ॏͷॏͳΓ͸ 1 ୯ମͰܨ͛Δ → 3 ॏͷॏͳΓ͸ 2 ୯ମʢࡾ֯ܗʣͰܨ͛Δ • ஫ɿ1 ୯ମ Λ 3 ͭܨ͖͛ͨ݀͋ࡾ֯ܗͱɺ2 ୯ମͷ໘ੵ͖ͭࡾ֯ܗ͸ผ I III V VII II IV VI a b c d e f g I III V VII II IV VI

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3. ຺ମͱʮͭͳ͕Γʯ • Example 39 • ผͷදݱܗࣜͰ੔ཧͯ͠ΈΔʢલϖʔδͱʮͭͳ͕Γʯ͸ಉ͡ʣ • ࠨ୺ͱӈ୺͸ʮͭͳ͕Δʯ͸͚ͣͩΕͲɺI ͱ III ্͕Լٯ޲͖ʁ I III V VII II IV VI I III

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3. ຺ମͱʮͭͳ͕Γʯ • Example 39 • Ͳ͔͜Ͱ 1 ճͶ͡Βͳ͍ͱ I ͱ III ͕ʮͭͳ͕Βͳ͍ʯ • Fig. 16.2 ͰʮϝϏ΢εͷྠʯͱࣔࠦ͞Ε͍ͯΔͷ͸͜Ε

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4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ

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4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ • Example 39 • ϦʔϚϯͷػೳ࿨੠࿦ "Harmony, Simplified"ʢ1893ʣ • ౰࣌ͷ࡞ۂՈɿυϏϡογʔʢ1862-1918ʣγΣʔϯϕϧΫʢ1874-1951ʣ • ౰࣌ͷཧ࿦Ոɿθώλʔʢ1788-1867ʣγΣϯΧʔʢ1868-1935ʣ I III V VII II IV VI I III

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4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ • Example 39 • ڭՊॻ pp.130ʮୈ 1 ৚݅ͱͯ͠ɺI, IV, V ʹରͯ͠ T, S, D ͱఆΊɺྠֲઢ্Ͱ͸ҟͳΔ TSD Λ༩ ͑ΔΑ͏ϦʔϚϯ͸ఆΊͨʯ • ͔ͨ͠ʹ IV→I→V ͸ S→T→D ͱҟͳΔ I:T III V:D VII II IV:S VI I:T III ্ஈ͔Β

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4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ • Example 39 • ڭՊॻ pp.131ʮୈ 2 ৚݅ͱͯ͠ɺฒߦͳ౓਺ πY ͸ಉ͡ TSD ΛऔΔΑ͏ʹͨ͠ʯ • ͓ͦΒ͘ӳޠ൛ݪจ pp.73-74 ͷ Tp, Sp, Dp ͷ͜ͱ • Tonic parallel, Subdominant parallel, Dominant parallel I:T III:D V:D VII:? II:S IV:S VI:T I:T III:D ্ஈ͔Β

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4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ • Example 39 • Mazzola ઌੜͷओுʮୈ 2 ৚݅Λ V→III→I ͱॱ൪ʹద༻͍ͯͬͨ͠ΒɺD = T ʹͳͬͯໃ६͢Δ ΑͶʁ͜Ε͸ϝϏ΢εͷྠ͕޲͖Λ࣋ͨͳͯ͘දཪ൓ରʹͳͬͪΌ͏͔ΒͩΑʯ • ׂͱి೾ͳ͜ͱݴͬͯΔؾ͕͢Δʢ͜ͳΈʣ I:T III:D V:D VII:? II:S IV:S VI:T I:T III:D ্ஈ͔Β

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4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ • ༨ஊɿͦ΋ͦ΋ϦʔϚϯ͸ 1 ͭͷ࿨Իʹ 1 ͭͷػೳΛׂΓ౰ͯΑ͏ͱͨ͠ͷ͔ʁ • ϦʔϚϯ͸ 1 ͭͷ࿨Ի͕औΓಘΔػೳ͸ߴʑ 3ʮछྨʯͱ͚ͨͩ͠Ͱɺػೳ͕ 3 ͭͱ͸ݴ͍ͬͯ ͳ͍ɺͦ΋ͦ΋ 1 ͭͷ࿨Ի͕ඞͣಉ͡छྨͷػೳΛऔΔͱ΋அݴ͍ͯ͠ͳ͍ɺͱ͍͏ࢦఠ͕͋Δ • ػೳ͕ 3 ͚ͭͩɺಉ͡ػೳ͸ަ׵ՄೳɺͳͲ͸ޙͷཧ࿦ՈʹΑΔ֦େղऍ͔ʢౡԬ࿨੠ͳͲʣ • ࢀߟɿSoundQuestʮػೳ࿨੠ͷมભ ❶ ϦʔϚϯͷ“ݪ࡞”ʯ I III V VII II IV VI I III

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4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ • ؓ࿩ٳ୊ • ௕͞ n ͷ chain graph |n| ͱ຺ମͱͷؔ࿈ • ඃ෴ C1(n) = { {0, 1}, {1, 2}, ... , {n - 1, n } } • Chain graph |n| ͷ edge ͱ຺ମͷ 0 ୯ମɺ1 ʙ n - 1 ͷ௖఺ͱ 1 ୯ମ͕ରԠ 0 1 2 i - 1 i i + 1 n n - 1 n - 2 ... ... ... ...

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4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ • Ұൠʹ Ck(n) = { {0, 1, ..., k}, ... , {n - k, ..., n - 1, n} } • ྫͱͯ͠ C3(16) = { {0, 1, 2, 3}, {1, 2, 3, 4}, ..., {13, 14, 15, 16} } ͷ຺ମΛߟ͑Δͱɺ ࢛໘ମΛࡾ֯ܗͰషΓ߹Θͤͯʮͭͳ͛ͨʯΑ͏ͳਤܗʹͳΔ

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4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ • Ұൠʹ Ck(n) = { {0, 1, ..., k}, ... , {n - k, ..., n - 1, n} } • ྫͱͯ͠ C3(16) = { {0, 1, 2, 3}, {1, 2, 3, 4}, ..., {13, 14, 15, 16} } ͷ຺ମΛߟ͑Δͱɺ ࢛໘ମΛࡾ֯ܗͰషΓ߹Θͤͯʮͭͳ͛ͨʯΑ͏ͳਤܗʹͳΔ

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4. ϦʔϚϯͷػೳ࿨੠࿦ͱʮͭͳ͕Γʯ • Example 40 • ͜͏͍͏͜ͱʁʢࣗ৴͕ͳ͍ʣ IM7 IIIm7 V7 VIIm7-5 IIm7 IVM7 VIm7 IM7 IIIm7 V7 VIIm7-5 IIm7 IVM7 VIm7