<?xml version="1.0" encoding="US-ASCII" ?>
    <!-- Field: Doc-Info; Name: Generator; Value: GoFiler Complete; Version: 5.23a -->
    <!-- Field: Doc-Info; Name: VendorURI; Value: https://www.novaworks.com -->
    <!-- Field: Doc-Info; Name: Status; Value: 0x00000000 -->
    <!-- Field: Doc-Info; Name: Misc; Value: +6spw7xRiXhun8uCbn1MSaqGybP9w5dzWelBtm96HUnqPywhpaKhcSmiCx2mIa8F -->
<schema xmlns="http://www.w3.org/2001/XMLSchema" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:link="http://www.xbrl.org/2003/linkbase" xmlns:xbrli="http://www.xbrl.org/2003/instance" xmlns:xbrldt="http://xbrl.org/2005/xbrldt" xmlns:xbrldi="http://xbrl.org/2006/xbrldi" xmlns:dei="http://xbrl.sec.gov/dei/2024" xmlns:cef="http://xbrl.sec.gov/cef/2024" xmlns:dtr-types="http://www.xbrl.org/dtr/type/2022-03-31" xmlns:gcv="http://gabelli.com/20240711" elementFormDefault="qualified" targetNamespace="http://gabelli.com/20240711">
    <annotation>
      <appinfo>
        <link:linkbaseRef xlink:type="simple" xlink:href="gcv-20240711_pre.xml" xlink:role="http://www.xbrl.org/2003/role/presentationLinkbaseRef" xlink:arcrole="http://www.w3.org/1999/xlink/properties/linkbase" xlink:title="Presentation Links" />
        <link:linkbaseRef xlink:type="simple" xlink:href="gcv-20240711_lab.xml" xlink:role="http://www.xbrl.org/2003/role/labelLinkbaseRef" xlink:arcrole="http://www.w3.org/1999/xlink/properties/linkbase" xlink:title="Label Links" />
        <link:linkbaseRef xlink:type="simple" xlink:href="gcv-20240711_def.xml" xlink:role="http://www.xbrl.org/2003/role/definitionLinkbaseRef" xlink:arcrole="http://www.w3.org/1999/xlink/properties/linkbase" xlink:title="Definition Links" />
      </appinfo>
    </annotation>
    <import namespace="http://www.xbrl.org/2003/instance" schemaLocation="http://www.xbrl.org/2003/xbrl-instance-2003-12-31.xsd" />
    <import namespace="http://www.xbrl.org/2003/linkbase" schemaLocation="http://www.xbrl.org/2003/xbrl-linkbase-2003-12-31.xsd" />
    <import namespace="http://xbrl.sec.gov/dei/2024" schemaLocation="https://xbrl.sec.gov/dei/2024/dei-2024.xsd" />
    <import namespace="http://fasb.org/us-types/2024" schemaLocation="https://xbrl.fasb.org/us-gaap/2024/elts/us-types-2024.xsd" />
    <import namespace="http://www.xbrl.org/dtr/type/2022-03-31" schemaLocation="https://www.xbrl.org/dtr/type/2022-03-31/types.xsd" />
    <import namespace="http://xbrl.sec.gov/cef/2024" schemaLocation="https://xbrl.sec.gov/cef/2024/cef-2024.xsd" />
    <import namespace="http://xbrl.sec.gov/cef-pre/2024" schemaLocation="https://xbrl.sec.gov/cef/2024/cef-2024_pre.xsd" />
    <import namespace="http://fasb.org/srt/2024" schemaLocation="https://xbrl.fasb.org/srt/2024/elts/srt-2024.xsd" />
    <import namespace="http://fasb.org/srt-types/2024" schemaLocation="https://xbrl.fasb.org/srt/2024/elts/srt-types-2024.xsd" />
    <element id="gcv_CommonStocksMember" name="CommonStocksMember" abstract="true" nillable="true" xbrli:periodType="duration" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" />
    <element id="gcv_SeriesGCumulativePreferredStockMember" name="SeriesGCumulativePreferredStockMember" abstract="true" nillable="true" xbrli:periodType="duration" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" />
    <element id="gcv_CumulativePreferredStocksMember" name="CumulativePreferredStocksMember" abstract="true" nillable="true" xbrli:periodType="duration" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" />
    <element id="gcv_SeriesECumulativePreferredStockMember" name="SeriesECumulativePreferredStockMember" abstract="true" nillable="true" xbrli:periodType="duration" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" />
    <element id="gcv_SeriesBCumulativePreferredStockMember" name="SeriesBCumulativePreferredStockMember" abstract="true" nillable="true" xbrli:periodType="duration" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" />
</schema>
